Non-simple systoles on random hyperbolic surfaces for large genus

Yuxin He, Yang Shen, Yunhui Wu, and Yuhao Xue Tsinghua University, Beijing, China hyx21@mails.tsinghua.edu.cn yunhui_wu@tsinghua.edu.cn Fudan University, Shanghai, China shenwang@fudan.edu.cn Institut des Hautes Études Scientifiques (IHES), Bures-sur-Yvette, France xueyh@ihes.fr

Abstract.

In this paper, we investigate the asymptotic behavior of the non-simple systole, which is the length of a shortest non-simple closed geodesic, on a random closed hyperbolic surface on the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g$ endowed with the Weil-Petersson measure. We show that as the genus $g$ goes to infinity, the non-simple systole of a generic hyperbolic surface in $\mathcal{M}_{g}$ behaves exactly like $\log g$ .

1. Introduction

The study of closed geodesics on hyperbolic surfaces has deep connection to their spectral theory, dynamics and hyperbolic geometry. Let $X=X_{g}$ be a closed hyperbolic surface of genus $g\geq 2$ . The systole of $X$ , the length of a shortest closed geodesic on $X$ , is always realized by a simple closed geodesic, i.e. a closed geodesic without self-intersections. The non-simple systole $\ell^{ns}_{sys}(X)$ of $X$ is defined as

\ell^{ns}_{sys}(X)=\min\{\ell_{\alpha}(X);\ \textit{$\alpha\subset X$ is a non-simple closed geodesic}\}

where $\ell_{\alpha}(X)$ is the length of $\alpha$ in $X$ . It is known that $\ell^{ns}_{sys}(X)$ is always realized as a figure-eight closed geodesic in $X$ (see e.g. [Bus10, Theorem 4.2.4]). In this work, we view the non-simple systole as a random variable on moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g$ endowed with the Weil-Petersson probability measure $\mathop{\rm Prob}\nolimits_{\rm WP}^{g}$ . This subject was initiated by Mirzakhani in [Mir10, Mir13], based on her celebrated thesis works [Mir07a, Mir07b]. Firstly it is known that for all $g\geq 2$ , $\inf\limits_{X\in\mathcal{M}_{g}}\ell^{ns}_{sys}(X)=2\mathop{\rm arccosh}(3)\sim 3.52...$ (see e.g. [Yam82, Bus10]) and $\sup\limits_{X\in\mathcal{M}_{g}}\ell^{ns}_{sys}(X)\asymp\log g$ (see e.g. [BS94, Tor23]). In this paper, we show that as $g$ goes to infinity, a generic hyperbolic surface in $\mathcal{M}_{g}$ has non-simple systole behaving like $\log g$ . More precisely, let $\omega:\{2,3,\cdots\}\to\mathbb{R}^{>0}$ be any function satisfying

(1)

\lim\limits_{g\to\infty}\omega(g)=+\infty\ \textit{and}\ \lim\limits_{g\to\infty}\frac{\omega(g)}{\log\log g}=0.

Theorem 1.

For any $\omega(g)$ satisfying (1), the following limit holds:

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ |\ell^{ns}_{sys}(X)-(\log g-\log\log g)|<\omega(g)\right)=1.

Remark.

It was shown in [NWX23, Theorem 4] that for any $\epsilon>0$ ,

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ (1-\epsilon)\log g<\ell^{ns}_{sys}(X)<2\log g\right)=1.

As a direct consequence of Theorem 1, as $g$ goes to infinity, the asymtotpic behavior of the expected value of $\ell^{ns}_{sys}(\cdot)$ over $\mathcal{M}_{g}$ can also be determined.

Theorem 2.

The following limit holds:

\lim\limits_{g\to\infty}\frac{\int_{\mathcal{M}_{g}}\ell^{ns}_{sys}(X)dX}{\mathop{\rm Vol_{\rm WP}}(\mathcal{M}_{g})\log g}=1.

Proof.

Take $\omega(g)=\log\log\log g$ and set $V_{g}=\mathop{\rm Vol_{\rm WP}}(\mathcal{M}_{g})$ . Define

A_{\omega}(g):=\{X\in\mathcal{M}_{g};\ |\ell^{ns}_{sys}(X)-(\log g-\log\log g)|<\omega(g)\}.

By Theorem 1 we know that

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ X\in A_{\omega}(g)\right)=1.

Then firstly it is clear that

\displaystyle\liminf\limits_{g\to\infty}\frac{\int_{\mathcal{M}_{g}}\ell^{ns}_{sys}(X)dX}{V_{g}\cdot\log g}\geq\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ X\in A_{\omega}(g)\right)=1.

For the other direction, since $\sup\limits_{X\in\mathcal{M}_{g}}\ell^{ns}_{sys}(X)\leq C\cdot\log g$ for some universal constant $C>0$ (see e.g. Lemma 9),

	$\displaystyle\limsup\limits_{g\to\infty}\frac{\int_{\mathcal{M}_{g}}\ell^{ns}_{sys}(X)dX}{V_{g}\cdot\log g}=\limsup\limits_{g\to\infty}\left(\frac{\int_{A_{\omega}(g)}\ell^{ns}_{sys}(X)}{V_{g}\cdot\log g}+\frac{\int_{A^{c}_{\omega}(g)}\ell^{ns}_{sys}(X)}{V_{g}\cdot\log g}\right)$
	$\displaystyle\leq 1+C\cdot\limsup\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ X\notin A_{\omega}(g)\right)=1.$

The proof is complete. ∎

Remark.

(1)

Mirzkhani-Petri in [MP19] showed that

$\lim\limits_{g\to\infty}\frac{\int_{\mathcal{M}_{g}}\ell_{\rm sys}(X)dX}{\mathop{\rm Vol_{\rm WP}}(\mathcal{M}_{g})}=1.61498...$

where $\ell_{\rm sys}(X)$ is the systole of $X$ .

(2)

Based on [NWX23], joint with Parlier, the third and forth named authors in [PWX22] showed that

\lim\limits_{g\to\infty}\frac{\int_{\mathcal{M}_{g}}\ell_{\mathop{\rm sys}}^{\rm sep}(X)dX}{\mathop{\rm Vol_{\rm WP}}(\mathcal{M}_{g})\log g}=2

where $\ell_{\mathop{\rm sys}}^{\rm sep}(X)$ is the length of a shortest separating simple closed geodesic in $X$ , an unbounded function over $\mathcal{M}_{g}$ .

The geometry and spectra of random hyperbolic surfaces under this Weil-Petersson measure have been widely studied in recent years. For examples, one may see [GPY11] for Bers’ constant, [Mir13, WX22b] for diameter, [MP19] for systole, [Mir13, NWX23, PWX22] for separating systole, [Mir13, WX22b, LW21, AM23] for first eigenvalue, [GMST21] for eigenfunction, [Mon22] for Weyl law, [Rud22, RW23] for GOE, [WX22a] for prime geodesic theorem, [Nau23] for determinant of Laplacian. One may also see [LS20, MT21, Hid22, SW22, HW22, HHH22, HT22, DS23, Gon23, MS23] and the references therein for more related topics.

Strategy on the proof of Theorem 1. The proof of Theorem 1 mainly consists of two parts.

A relative easier part is to prove the lower bound, that is to show that

(2)

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ \ell^{ns}_{sys}(X)>\log g-\log\log g-\omega(g)\right)=1.

We know that $\ell^{ns}_{sys}(X)$ is realized by a figure-eight closed geodesic that is always filling in a unique pair of pants. And the length of such a figure-eight closed geodesic can be determined by the lengths of the three boundary geodesics of the pair of pants (see e.g. formula (17)). For $L=L_{g}=\log g-\log\log g-\omega(g)$ and $X\in\mathcal{M}_{g}$ , denote by $N_{\rm f-8}(X,L)$ the number of figure-eight closed geodesics of length $\leq L$ in $X$ . We view it as a random variable on $\mathcal{M}_{g}$ . Then using Mirzakhani’s integration formula and change of variables, a direct computation shows that its expected value $\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}(X,L)]$ satisfies that as $g\to\infty$ ,

(3)

\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}(X,L)]\sim\frac{Le^{L}}{8\pi^{2}g}\to 0.

Here we say $f(g)\sim h(g)$ if $\lim\limits_{g\to\infty}\frac{f(g)}{h(g)}=1$ . Thus, we have

\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{\rm f-8}(X,L_{g})\geq 1\right)\leq\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}(X,L_{g})]\to 0

as $g\to\infty$ . This in particular implies (2).

The hard part of Theorem 1 is the upper bound, that is to show that

(4)

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ \ell^{ns}_{sys}(X)<\log g-\log\log g+\omega(g)\right)=1.

Set $L=L_{g}=\log g-\log\log g+\omega(g),$ and for any $X\in\mathcal{M}_{g}$ we define the following particular set and quantity:

\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)=\left\{(\gamma_{1},\gamma_{2},\gamma_{3});\ \begin{matrix}(\gamma_{1},\gamma_{2},\gamma_{3})\text{ is a pair of ordered simple closed}\\ \text{curves such that }X\setminus\cup_{i=1}^{3}\gamma_{i}\simeq S_{0,3}\bigcup S_{g-2,3},\\ \ell_{\gamma_{1}}(X)\leq L,\ \ell_{\gamma_{2}}(X)+\ell_{\gamma_{3}}(X)\leq L\\ \text{ and }\ell_{\gamma_{1}}(X),\ell_{\gamma_{2}}(X),\ell_{\gamma_{3}}(X)\geq 10\log L\end{matrix}\right\}

and

N_{(0,3),\star}^{(g-2,3)}(X,L)=\#\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L).

It is not hard to see that there exists some universal constant $c>0$ such that

(5)		$\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(N_{\rm f-8}(X,L+c)=0\right)$
	$\displaystyle\leq$	$\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{(0,3),\star}^{(g-2,3)}(X,L)=0\right)$
	$\displaystyle\leq$	$\displaystyle\frac{\mathbb{E}_{\rm WP}^{g}\left[(N_{(0,3),\star}^{(g-2,3)}(X,L))^{2}\right]-\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]^{2}}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]^{2}}.$

To prove (4), it suffices to show

(6)

\lim\limits_{g\to\infty}\mathrm{RHS}\textit{\rm{ of \eqref{i-link f-8 with Nstar}}}=0.

Using Mirzakhani’s integration formula and known bounds on Weil-Petersson volumes, direct computations show that (see Proposition 20),

(7)

\mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]=\frac{1}{2\pi^{2}g}Le^{L}\left(1+O\left(\frac{\log L}{L}\right)\right)\sim\frac{e^{\omega(g)}}{2\pi^{2}}.

Next we consider $\mathbb{E}_{\rm WP}^{g}\left[(N_{(0,3),\star}^{(g-2,3)}(X,L))^{2}\right]$ . For any $\Gamma\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ , denote by $P(\Gamma)$ the unique pair of pants bounded by $\Gamma$ and let $\overline{P(\Gamma)}$ be its completion in $X$ . Split $(N_{(0,3),\star}^{(g-2,3)}(X,L))^{2}$ as follows:

	$\displaystyle A(X,L)=\#\left\{(\Gamma_{1},\Gamma_{2});\ P(\Gamma_{1})=P(\Gamma_{2})\right\},$
	$\displaystyle B(X,L)=\#\left\{(\Gamma_{1},\Gamma_{2});\ \overline{P(\Gamma_{1})}\cap\overline{P(\Gamma_{2})}=\emptyset\right\},$
	$\displaystyle C(X,L)=\#\left\{(\Gamma_{1},\Gamma_{2});\ P(\Gamma_{1})\neq P(\Gamma_{2}),\ {P(\Gamma_{1})}\cap{P(\Gamma_{2})}\neq\emptyset\right\},$
	$\displaystyle D(X,L)=\#\left\{(\Gamma_{1},\Gamma_{2});\ P(\Gamma_{1})\cap P(\Gamma_{2})=\emptyset,\ \overline{P(\Gamma_{1})}\cap\overline{P(\Gamma_{2})}\neq\emptyset\right\}.$

Then it is clear that

(8)			$\displaystyle\mathbb{E}_{\textnormal{WP}}^{g}\left[(N_{(0,3),\star}^{(g-2,3)}(X,L))^{2}\right]=\mathbb{E}_{\textnormal{WP}}^{g}\left[A(X,L)\right]$
(8)			$\displaystyle+\mathbb{E}_{\textnormal{WP}}^{g}\left[B(X,L)\right]+\mathbb{E}_{\textnormal{WP}}^{g}\left[C(X,L)\right]+\mathbb{E}_{\rm WP}^{g}\left[D(X,L)\right].$

Since for a pair of pants $P$ in $X$ , there exist at most $6$ different $\Gamma^{\prime}$ s such that $P=P(\Gamma)$ , it follows from (7) that

(9)

\displaystyle\mathbb{E}_{\textnormal{WP}}^{g}\left[A(X,L)\right]\leq 6\cdot\mathbb{E}_{\textnormal{WP}}^{g}\left[\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)\right]\prec\frac{Le^{L}}{g}\asymp e^{\omega(g)}.

Using Mirzakhani’s integration formula and known bounds on Weil-Petersson volumes, direct computations can show that (see Proposition 21 and 34)

(10)

\displaystyle\mathbb{E}_{\textnormal{WP}}^{g}\left[B(X,L)\right]=\frac{1}{4\pi^{4}g^{2}}L^{2}e^{2L}\left(1+O\left(\frac{\log L}{L}\right)\right)\sim\mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]^{2}

and

(11)

\mathbb{E}_{\rm WP}^{g}\left[D(X,L)\right]\prec\frac{e^{2L}}{g^{2}L^{6}}=o(1).

Combining (7) and (10) gives the following important cancellation:

(12)

\left|\mathbb{E}_{\textnormal{WP}}^{g}\left[B(X,L)\right]-\mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]^{2}\right|\prec\frac{L^{2}e^{2L}}{g^{2}}\cdot\frac{\log L}{L}=e^{2\omega(g)}\cdot o(1).

The crucial part is to bound $\mathbb{E}_{\rm WP}^{g}\left[C(X,L)\right]$ . For this part, our method is inspired by the ones in [MP19, WX22b, NWX23]. Denote by $S(\Gamma_{1},\Gamma_{2})$ the compact subsurface in $X$ of geodesic boundary such that $\Gamma_{1}\cup\Gamma_{2}$ is filling in it. Then one can divide $C(X,L)$ as follows:

	$\displaystyle C_{0,4}(X,L)=\#\left\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L);\ S(\Gamma_{1},\Gamma_{2})\simeq S_{0,4}\right\},$
	$\displaystyle C_{1,2}(X,L)=\#\left\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L);\ S(\Gamma_{1},\Gamma_{2})\simeq S_{1,2}\right\},$
	$\displaystyle C_{\geq 3}(X,L)=\#\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L);\ \|\chi(S(\Gamma_{1},\Gamma_{2}))\|\geq 3\}.$

Through using the method in [WX22b] and the counting result on filling multi-geodesics in [WX22b, WX22a], we can show that (see Proposition 23)

(13)

\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{\geq 3}(X,L)\right]\prec\left(L^{67}e^{(2+\epsilon)\cdot L}\frac{1}{g^{3}}+\frac{L^{3}e^{8L}}{g^{11}}\right)=o(1).

For $C_{0,4}(X,L)$ and $C_{1,2}(X,L)$ , through classifying all the accurate relative positions of $(\Gamma_{1},\Gamma_{2})$ in both $S_{1,2}$ and $S_{0,4}$ , applying the McShane-Mirzakhani identity in [Mir07a] as for counting closed geodesics (we warn here that both the general counting result and the counting result in [WX22b, WX22a] on closed geodesics are inefficient to deal with these two cases), and then using Mirzakhani’s integration formula and known bounds on Weil-Petersson volumes, we can show that (see Proposition 24 and 28)

(14)

\mathbb{E}_{\rm WP}^{g}\left[C_{1,2}(X,L)\right]\prec\frac{e^{2L}}{g^{2}}=o(1)

and

(15)

\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}(X,L)\right]\prec\frac{Le^{2L}}{g^{2}}=o(1).

Then combining all these equations (7)—(15), one may finish the proof of (6), thus get (4) which is the upper bound in Theorem 1.

Notations. For any two nonnegative functions $f$ and $h$ (may be of multi-variables), we say $f\prec h$ if there exists a uniform constant $C>0$ such that $f\leq Ch$ . And we also say $f\asymp h$ if $f\prec h$ and $h\prec f$ .

Plan of the paper. Section 2 will provide a review of relevant and necessary background materials. In Section 3 we compute the expectation of the number of figure-eight closed geodesics of length $\leq L$ over $\mathcal{M}_{g}$ which will imply (2), i.e. the lower bound in Theorem 1. In Section 4 we prove (4), i.e. the upper bound in Theorem 1. In which we apply the counting result on closed geodesics in [WX22b], and also apply the McShane-Mirzakhani identity in [Mir07a] to count closed geodesics for $C_{1,2}(X,L)$ and $C_{0,4}(X,L)$ .

Acknowledgement We would like to thank all the participants in our seminar on Teichmüller theory for helpful discussions on this project. The third named author is partially supported by the NSFC grant No. $12171263$ .

2. Preliminaries

In this section, we set our notations and recall certain relevant necessary results used in this paper, including Weil-Petersson metric, Mirzakhani’s integration formula, bounds on Weil-Petersson volumes, figure-eight closed geodesics and three countings on closed geodesics.

2.1. Moduli space and Weil-Petersson metric

Denote by $S_{g,n}$ an oriented topological surface with genus $g$ of $n$ punctures or boundaries where $2g-2+n\geq 1$ . Let $\mathcal{T}_{g,n}$ be the Teichmüller space of surfaces with genus $g$ of $n$ punctures, and $\mathop{\rm Mod}_{g,n}$ be the mapping class group of $S_{g,n}$ fixing the order of punctures. The moduli space of Riemann surfaces is $\mathcal{M}_{g,n}=\mathcal{T}_{g,n}/\mathop{\rm Mod}_{g,n}$ . Denote $\mathcal{T}_{g}=\mathcal{T}_{g,0}$ and $\mathcal{M}_{g}=\mathcal{M}_{g,0}$ for simplicity. Given $L=(L_{1},\cdots,L_{n})\in\mathbb{R}^{n}_{\geq 0}$ , let $\mathcal{T}_{g,n}(L)$ be the Teichmüller space of bordered hyperbolic surfaces with $n$ geodesic boundaries of lengths $L_{1},\cdots,L_{n}$ , and $\mathcal{M}_{g,n}(L)=\mathcal{T}_{g,n}(L)/\mathop{\rm Mod}_{g,n}$ be the weighted moduli space. In particular, $\mathcal{T}_{g,n}(0,\cdots,0)=\mathcal{T}_{g,n}$ and $\mathcal{M}_{g,n}(0,\cdots,0)=\mathcal{M}_{g,n}$ .

Given a pants decomposition $\{\alpha_{i}\}_{i=1}^{3g-3+n}$ of $S_{g,n}$ , the Fenchel-Nielsen coordinates on the Teichmüller space $\mathcal{T}_{g,n}(L)$ is given by the map $X\mapsto(\ell_{\alpha_{i}}(X),\tau_{\alpha_{i}}(X))_{i=1}^{3g-3+n}$ . Here $\ell_{\alpha_{i}}(X)$ is the length of $\alpha_{i}$ on $X$ and $\tau_{\alpha_{i}}(X)$ is the twist along $\alpha_{i}$ (measured by length). The following magic formula is due to Wolpert [Wol82]:

Theorem 3 (Wolpert).

The Weil-Petersson symplectic form $\omega_{\mathrm{WP}}$ on $\mathcal{T}_{g,n}(L)$ is given by

\omega_{\mathrm{WP}}=\sum_{i=1}^{3g-3+n}d\ell_{\alpha_{i}}\wedge d\tau_{\alpha_{i}}.

The form $\omega_{\mathrm{WP}}$ is mapping class group invariant. So it induces the so-called Weil-Petersson volume form on $\mathcal{M}_{g,n}$ given by

d\mathop{\rm Vol}\nolimits_{\mathrm{WP}}=\tfrac{1}{(3g-3+n)!}\cdot\underbrace{\omega_{\mathrm{WP}}\wedge\cdots\wedge\omega_{\mathrm{WP}}}_{3g-3+n\ \text{copies}}.

Denote by $V_{g,n}(L)$ the total volume of $\mathcal{M}_{g,n}(L)$ under the Weil-Petersson metric which is finite. Following [Mir13], we view a function $f:\mathcal{M}_{g}\to\mathbb{R}$ as a random variable on $\mathcal{M}_{g}$ with respect to the probability measure $\mathop{\rm Prob}\nolimits_{\rm WP}^{g}$ on $\mathcal{M}_{g}$ , and we also denote by $\mathbb{E}_{\rm WP}^{g}[f]$ the expected value of $f$ over $\mathcal{M}_{g}$ . Namely,

\mathop{\rm Prob}\nolimits_{\rm WP}^{g}(\mathcal{A}):=\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\mathbf{1}_{\mathcal{A}}dX,\quad\mathbb{E}_{\rm WP}^{g}[f]:=\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}f(X)dX

where $\mathcal{A}\subset\mathcal{M}_{g}$ is a Borel subset, $\mathbf{1}_{\mathcal{A}}:\mathcal{M}_{g}\to\{0,1\}$ is its characteristic function, and $dX$ is short for $d\mathop{\rm Vol}_{\mathrm{WP}}(X)$ .

2.2. Mirzakhani’s integration formula

In this subsection, we recall Mirzakhani’s integration formula in [Mir07a].

Let $\gamma$ be a non-trivial and non-peripheral closed curve on topological surface $S_{g,n}$ and $X\in\mathcal{T}_{g,n}$ . Denote $\ell(\gamma)=\ell_{\gamma}(X)$ to be the hyperbolic length of the unique closed geodesic in the homotopy class of $\gamma$ on $X$ . Let $\Gamma=(\gamma_{1},\cdots,\gamma_{k})$ be an ordered k-tuple where the $\gamma_{i}$ ’s are distinct disjoint homotopy classes of nontrivial, non-peripheral, unoriented simple closed curves on $S_{g,n}$ . Let $\mathcal{O}_{\Gamma}$ be the orbit containing $\Gamma$ under the $\mathop{\rm Mod}_{g,n}$ -action:

\mathcal{O}_{\Gamma}=\{(h\cdot\gamma_{1},\cdots,h\cdot\gamma_{k});\ h\in\mathop{\rm Mod}\nolimits_{g,n}\}.

Given a function $F:\mathbb{R}_{\geq 0}^{k}\to\mathbb{R}$ , one may define a function on $\mathcal{M}_{g,n}$ :

	$\displaystyle F^{\Gamma}:\mathcal{M}_{g,n}$	$\displaystyle\to$	$\displaystyle\mathbb{R}$
	$\displaystyle X$	$\displaystyle\mapsto$	$\displaystyle\sum_{(\alpha_{1},\cdots,\alpha_{k})\in\mathcal{O}_{\Gamma}}F(\ell_{\alpha_{1}}(X),\cdots,\ell_{\alpha_{k}}(X)).$

Note that although $\ell_{\alpha_{i}}(X)$ can be only defined on $\mathcal{T}_{g,n}$ , after taking sum $\sum_{(\alpha_{1},\cdots,\alpha_{k})\in\mathcal{O}_{\Gamma}}$ , the function $F^{\Gamma}$ is well-defined on the moduli space $\mathcal{M}_{g,n}$ .

For any $x=(x_{1},\cdots,x_{k})\in\mathbb{R}_{\geq 0}^{k}$ , we set $\mathcal{M}(S_{g,n}(\Gamma);\ell_{\Gamma}=x)$ to be the moduli space of the hyperbolic surfaces (possibly disconnected) homeomorphic to $S_{g,n}\setminus\cup_{j=1}^{k}\gamma_{j}$ with $\ell(\gamma_{i}^{1})=\ell(\gamma_{i}^{2})=x_{i}$ for every $i=1,\cdots,k$ , where $\gamma_{i}^{1}$ and $\gamma_{i}^{2}$ are the two boundary components of $S_{g,n}\setminus\cup_{j=1}^{k}\gamma_{j}$ given by cutting along $\gamma_{i}$ . Assume $S_{g,n}\setminus\cup_{j=1}^{k}\gamma_{j}\cong\cup_{i=1}^{s}S_{g_{i},n_{i}}$ . Consider the Weil-Petersson volume

V_{g,n}(\Gamma,x)=\mathop{\rm Vol}\nolimits_{\mathrm{WP}}\left(\mathcal{M}(S_{g,n}(\Gamma);\ell_{\Gamma}=x)\right)=\prod_{i=1}^{s}V_{g_{i},n_{i}}(x^{(i)})

where $x^{(i)}$ is the list of those coordinates $x_{j}$ of $x$ such that $\gamma_{j}$ is a boundary component of $S_{g_{i},n_{i}}$ . The following integration formula is due to Mirzakhani. One may see [Mir07a, Theorem 7.1], [Mir13, Theorem 2.2], [MP19, Theorem 2.2], [Wri20, Theorem 4.1] for different versions.

Theorem 4 (Mirzakhani’s integration formula).

For any $\Gamma=(\gamma_{1},\cdots,\gamma_{k})$ , the integral of $F^{\Gamma}$ over $M_{g,n}$ with respect to the Weil-Petersson metric is given by

\int_{\mathcal{M}_{g,n}}F^{\Gamma}dX=C_{\Gamma}\int_{\mathbb{R}_{\geq 0}^{k}}F(x_{1},\cdots,x_{k})V_{g,n}(\Gamma,x)x_{1}\cdots x_{k}dx_{1}\cdots dx_{k}

where the constant $C_{\Gamma}\in(0,1]$ only depends on $\Gamma$ .

Remark.

One may see [Wri20, Theorem 4.1] for the detailed explanation and expression for the constant $C_{\Gamma}$ . We will give the exact value of $C_{\Gamma}$ only when required in this paper.

2.3. Weil-Petersson volumes

Denote $V_{g,n}(x_{1},\cdots,x_{n})$ to be the Weil-Petersson volume of $\mathcal{M}_{g,n}(x_{1},\cdots,x_{n})$ and $V_{g,n}=V_{g,n}(0,\cdots,0)$ . In this subsection we only list the bounds for $V_{g,n}(x_{1},\cdots,x_{n})$ that we will need in this paper.

Theorem 5 ([Mir07a, Theorem 1.1]).

The initial volume $V_{0,3}(x,y,z)=1$ . The volume $V_{g,n}(x_{1},\cdots,x_{n})$ is a polynomial in $x_{1}^{2},\cdots,x_{n}^{2}$ with degree $3g-3+n$ . Namely we have

V_{g,n}(x_{1},\cdots,x_{n})=\sum_{\alpha;\,|\alpha|\leq 3g-3+n}C_{\alpha}\cdot x^{2\alpha}

where $C_{\alpha}>0$ lies in $\pi^{6g-6+2n-|2\alpha|}\cdot\mathbb{Q}$ . Here $\alpha=(\alpha_{1},\cdots,\alpha_{n})$ is a multi-index and $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}$ , $x^{2\alpha}=x_{1}^{2\alpha_{1}}\cdots x_{n}^{2\alpha_{n}}$ .

Theorem 6.

(1)

([Mir13, Lemma 3.2]). For any $g,n\geq 0$

$V_{g-1,n+4}\leq V_{g,n+2}$

and

$b_{0}\leq\frac{V_{g,n+1}}{(2g-2+n)V_{g,n}}\leq b_{1}$

for some universal constants $b_{0},b_{1}>0$ independent of $g,n$ .
(2)

([Mir13, Theorem 3.5]).

$\frac{(2g-2+n)V_{g,n}}{V_{g,n+1}}=\frac{1}{4\pi^{2}}+O_{n}\left(\frac{1}{g}\right),$

$\frac{V_{g,n}}{V_{g-1,n+2}}=1+O_{n}\left(\frac{1}{g}\right).$

Where the implied constants for $O_{n}(\cdot)$ are related to $n$ and independent of $g$ .

Part (2) above can also be derived by [MZ15] of Mirzakhani-Zograf in which the precise asymptotic behavior of $V_{g,n}$ is provided for given $n$ .

Set $r=2g-2+n$ . We also use the following quantity $W_{r}$ to approximate $V_{g,n}$ :

(16)

W_{r}:=\begin{cases}V_{\frac{r}{2}+1,0}&\text{if $r$ is even},\\[5.0pt] V_{\frac{r+1}{2},1}&\text{if $r$ is odd}.\end{cases}

The estimation about sum of products of Weil-Petersson volumes can be found in e.g. [Mir13, MP19, GMST21, NWX23]. Here we use the following version:

Theorem 7 ([NWX23, Lemma 24]).

Assume $q\geq 1$ , $n_{1},\cdots,n_{q}\geq 0$ , $r\geq 2$ . Then there exists two universal constants $c,D>0$ such that

\sum_{\{g_{i}\}}V_{g_{1},n_{1}}\cdots V_{g_{q},n_{q}}\leq c\left(\frac{D}{r}\right)^{q-1}W_{r}

where the sum is taken over all $\{g_{i}\}_{i=1}^{q}\subset\mathbb{N}$ such that $2g_{i}-2+n_{i}\geq 1$ for all $i=1,\cdots,q$ , and $\sum_{i=1}^{q}(2g_{i}-2+n_{i})=r$ .

The following asymptotic behavior of $V_{g,n}(x_{1},\cdots,x_{n})$ was firstly studied in [MP19, Proposition 3.1]. We use the following version in [NWX23]. One may also see more sharp ones in [AM22].

Theorem 8 ([NWX23, Lemma 20]).

There exists a constant $c(n)>0$ independent of $g$ and $x_{i}$ ’s such that

\left(1-c(n)\frac{\sum_{i=1}^{n}x_{i}^{2}}{g}\right)\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}\leq\frac{V_{g,n}(x_{1},\cdots,x_{n})}{V_{g,n}}\leq\prod_{i=1}^{n}\frac{\sinh(x_{i}/2)}{x_{i}/2}.

2.4. Figure-eight closed geodesics

Let $X$ be a hyperbolic surface. We say a closed geodesic in $X$ is a figure-eight closed geodesic if it has exactly one self-intersection point. Given a figure-eight closed geodesic $\alpha$ , it is filling in a pair of pants $P(x,y,z)$ with three geodesic boundary of lengths $x,y,z$ as shown in Figure 1. The length $L(x,y,z)$ of the figure-eight closed geodesic $\alpha$ is given by (see e.g. [Bus10, Equation (4.2.3)]):

(17)

\cosh\left(\tfrac{L(x,y,z)}{2}\right)=\cosh(\tfrac{z}{2})+2\cosh(\tfrac{x}{2})\cosh(\tfrac{y}{2}).

It is clear that $L(x,y,z)\geq 2\mathop{\rm arccosh}3$ .

Refer to caption — Figure 1. A figure-eight closed geodesic $\alpha$ in the pair of pants $P(x,y,z)$ .

Remark.

In a pair of pants $P(x,y,z)$ , there are exactly three different figure-eight closed geodesics of lengths $L(x,y,z)$ , $L(z,x,y)$ and $L(y,z,x)$ . Here $L(x,y,z)$ is the length of the figure-eight closed geodesic winding around $x$ and $y$ as shown in Figure 1.

It is known (see e.g. [Tor23, Lemma 5.2] or [Bus10, Section 5.2]) that the length of the shortest figure-eight closed geodesic in any $X\in\mathcal{M}_{g}$ is bounded.

Lemma 9.

There exists a universal constant $c>0$ independent of $g$ such that for any $X\in\mathcal{M}_{g}$ , the shortest figure-eight closed geodesic in $X$ has length $\leq c\log g$ .

Outline of the proof of Lemma 9.

Let $\gamma\subset X$ be a systolic curve. It is known that $\ell_{\gamma}(X)\prec\log g$ . Next consider the maximal collar around $\gamma$ and then one may get a pair of pants such that its boundary contains $\gamma$ and each of the three boundary geodesics has length $\prec\log g$ . Then the conclusion follows by (17). One may see [Tor23, Bus10] for more details. ∎

Remark.

From [NWX23, Theorem 4] we know that the growth rate $\log g$ in the upper bound in Lemma 9 holds for generic hyperbolic surfaces in $\mathcal{M}_{g}$ as $g\to\infty$ . In this paper, we study its precise asymptotic behavior.

Recall that the non-simple systole $\ell^{ns}_{sys}(X)$ of $X\in\mathcal{M}_{g}$ is defined as

\ell^{ns}_{sys}(X)=\min\{\ell_{\alpha}(X);\ \textit{$\alpha\subset X$ is a non-simple closed geodesic}\}.

We focus on figure-eight closed geodesics because the non-simple systole of a hyperbolic surface is always achieved by a figure-eight closed geodesic (see e.g. [Bus10, Theorem 4.2.4]). That is,

\ell^{ns}_{sys}(X)=\textit{length of a shortest figure-eight closed geodesic in $X$.}

In particular, as introduced above we have

\sup\limits_{X\in\mathcal{M}_{g}}\ell^{ns}_{sys}(X)\asymp\log g.

2.5. Three countings on closed geodesics

In this subsection, we mainly introduce three results about counting closed geodesics in hyperbolic surfaces. In this paper, we only consider primitive closed geodesics without orientations.

2.5.1. On closed hyperbolic surfaces

First, by the Collar Lemma (see e.g. [Bus10, Theorem 4.1.6]), we have that a closed hyperbolic surface of genus $g$ has most $3g-3$ pairwisely disjoint simple closed geodesics of length $\leq 2\mathop{\rm arcsinh}1\approx 1.7627$ . It is also known from [Bus10, Theorem 6.6.4] that for all $L>0$ and $X\in\mathcal{M}_{g}$ , there are at most $(g-1)e^{L+6}$ closed geodesics in $X$ of length $\leq L$ which are not iterates of closed geodesics of length $\leq 2\mathop{\rm arcsinh}1$ . As a consequence, we have

Theorem 10.

For any $L>0$ and $X\in\mathcal{M}_{g}$ , there are at most $(g-1)e^{L+7}$ primitive closed geodesics in $X$ of length $\leq L$ .

2.5.2. On compact hyperbolic surfaces with geodesic boundaries

For compact hyperbolic surfaces with geodesic boundaries, the following result for filling closed multi-geodesics is useful.

Definition.

Let $Y\in\mathcal{M}_{g,n}(L_{1},\cdots,L_{n})$ be a hyperbolic surface with boundaries. Let $\Gamma=(\gamma_{1},\cdots,\gamma_{k})$ be an ordered $k$ -tuple where $\gamma_{i}$ ’s are non-peripheral closed geodesics in $Y$ . We say $\Gamma$ is filling in $Y$ if each component of the complement $Y\setminus\cup_{i=1}^{k}\gamma_{i}$ is homeomorphic to either a disk or a cylinder which is homotopic to a boundary component of $Y$ .

In particular, a filling $1$ -tuple is a filling closed geodesic in $Y$ .

Define the length of a $k$ -tuple $\Gamma=(\gamma_{1},\cdots,\gamma_{k})$ to be the total length of $\gamma_{i}$ ’s, that is,

\ell_{\Gamma}(Y):=\sum_{i=1}^{k}\ell_{\gamma_{i}}(Y).

Define the counting function $N_{k}^{\text{fill}}(Y,L)$ for $L\geq 0$ to be

N_{k}^{\text{fill}}(Y,L):=\#\left\{\Gamma=(\gamma_{1},\cdots,\gamma_{k});\ \begin{aligned} &\Gamma\ \text{is a filling}\ k\text{-tuple in}\ Y\\ &\text{and}\ \ell_{\Gamma}(Y)\leq L\end{aligned}\right\}.

Theorem 11 ([WX22b, Theorem 4] or [WX22a, Theorem 18]).

For any $k\in\mathbb{Z}_{\geq 1}$ , $0<\varepsilon<\frac{1}{2}$ and $m=2g-2+n\geq 1$ , there exists a constant $c(k,\varepsilon,m)>0$ only depending on $k,\varepsilon$ and $m$ such that for all $L>0$ and any compact hyperbolic surface $Y$ of genus $g$ with $n$ boundary simple closed geodesics, the following holds:

N_{k}^{\text{fill}}(Y,L)\leq c(k,\varepsilon,m)\cdot(1+L)^{k-1}e^{L-\frac{1-\varepsilon}{2}\ell(\partial Y)}.

Where $\ell(\partial Y)$ is the total length of the boundary closed geodesics of $Y$ .

2.5.3. On $S_{1,2}$ and $S_{0,4}$

For the cases of $S_{1,2}$ and $S_{0,4}$ , we will apply the McShane-Mirzakhani identity to give more refined counting results on certain specific types of simple closed geodesics. The McShane-Mirzakhani identity states as:

Theorem 12 ([Mir07a, Theorem 1.3]).

For $Y\in\mathcal{M}_{g,n}(L_{1},\cdots,L_{n})$ with $n$ geodesic boundaries $\beta_{1},\cdots,\beta_{n}$ of length $L_{1},\cdots,L_{n}$ respectively, we have

\sum_{\{\gamma_{1},\gamma_{2}\}}\mathcal{D}(L_{1},\ell(\gamma_{1}),\ell(\gamma_{2}))+\sum_{i=2}^{n}\sum_{\gamma}\mathcal{R}(L_{1},L_{i},\ell(\gamma))=L_{1}

where the first sum is over all unordered pairs of simple closed geodesics $\{\gamma_{1},\gamma_{2}\}$ bounding a pair of pants with $\beta_{1}$ , and the second sum is over all simple closed geodesics $\gamma$ bounding a pair of pants with $\beta_{1}$ and $\beta_{i}$ . Here $\mathcal{D}$ and $\mathcal{R}$ are given by

\mathcal{D}(x,y,z)=2\log\left(\frac{e^{\frac{x}{2}}+e^{\frac{y+z}{2}}}{e^{\frac{-x}{2}}+e^{\frac{y+z}{2}}}\right)

and

\mathcal{R}(x,y,z)=x-\log\left(\frac{\cosh(\frac{y}{2})+\cosh(\frac{x+z}{2})}{\cosh(\frac{y}{2})+\cosh(\frac{x-z}{2})}\right).

The functions $\mathcal{D}(x,y,z)$ and $\mathcal{R}(x,y,z)$ have the following elementary properties:

Lemma 13 ([NWX23, Lemma 27]).

Assume that $x,y,z>0$ , then the following properties hold:

(1)

$\mathcal{R}(x,y,z)\geq 0$ and $\mathcal{D}(x,y,z)\geq 0$ .
(2)

$\mathcal{R}(x,y,z)$ is decreasing with respect to $z$ and increasing with respect to $y$ . $\mathcal{D}(x,y,z)$ is decreasing with respect to $y$ and $z$ and increasing with respect to $x$ .
(3)

We have

$\frac{x}{\mathcal{R}(x,y,z)}\leq 100(1+x)(1+e^{\frac{z}{2}}e^{-\frac{x+y}{2}}),$

and

$\frac{x}{\mathcal{D}(x,y,z)}\leq 100(1+x)(1+e^{\frac{y+z}{2}}e^{-\frac{x}{2}}).$

As a direct consequence of Theorem 12 and the monotonicity in Part (2) of Lemma 13, we have

Theorem 14.

On a surface $Y\in\mathcal{M}_{0,4}(L_{1},L_{2},L_{3},L_{4})$ , the number of simple closed geodesics of length $\leq L$ which bound a pair of pants with the two boundaries of lengths $L_{1}$ and $L_{2}$ has the upper bound

(18)

\leq\min\left\{\frac{L_{1}}{\mathcal{R}(L_{1},L_{2},L)},\ \frac{L_{2}}{\mathcal{R}(L_{2},L_{1},L)},\ \frac{L_{3}}{\mathcal{R}(L_{3},L_{4},L)},\ \frac{L_{4}}{\mathcal{R}(L_{4},L_{3},L)}\right\}.

On a surface $Y\in\mathcal{M}_{1,2}(L_{1},L_{2})$ , the number of simple closed geodesics of length $\leq L$ which bound a pair of pants with the two boundaries of lengths $L_{1}$ and $L_{2}$ has the upper bound

(19) $\leq\min\left\{\frac{L_{1}}{\mathcal{R}(L_{1},L_{2},L)},\ \frac{L_{2}}{\mathcal{R}(L_{2},L_{1},L)}\right\}.$
On a surface $Y\in\mathcal{M}_{1,2}(L_{1},L_{2})$ , the number of unordered pairs of simple closed geodesics of total length $\leq L$ which bound a pair of pants with the boundary of length $L_{1}$ has the upper bound

(20) $\leq\min\left\{\frac{L_{1}}{\mathcal{D}(L_{1},L,0)},\ \frac{L_{2}}{\mathcal{D}(L_{2},L,0)}\right\}.$

Proof.

These three bounds follow from Theorem 12 and the monotonicity of $\mathcal{D}(x,y,z)$ and $\mathcal{R}(x,y,z)$ . For (20), we also apply the fact that $\mathcal{D}(x,y,z)$ only depends on $y+z$ but not on $y,z$ respectively. ∎

Then by applying the estimates in Lemma 13, one may get upper bounds for (18), (19) and (20).

3. Lower bound

In this section we compute the expectation of the number of figure-eight closed geodesics of length $\leq L$ over $\mathcal{M}_{g}$ , and hence gives the lower bound of the length of non-simple systole for random hyperbolic surfaces.

Let $X\in\mathcal{M}_{g}$ and denote

\mathcal{N}_{\rm f-8}(X,L):=\left\{\alpha\subset X;\ \begin{array}[]{l}\alpha\ \text{is a figure-eight closed geodesic in}\ X\\ \text{and}\ \ell(\alpha)\leq L\end{array}\right\}

and

N_{\rm f-8}(X,L):=\#\mathcal{N}_{\rm f-8}(X,L)

to be the number of figure-eight closed geodesics in $X$ with length $\leq L$ . Then the non-simple systole of $X$ has length $>L$ if and only if $N_{\rm f-8}(X,L)=0$ .

The main result of this section is as follows.

Proposition 15.

For any $L\geq 2\mathop{\rm arccosh}3$ and $g>2$ ,

\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}(X,L)]=\frac{1}{8\pi^{2}g}(L-3-4\log 2)e^{L}+O\left(\tfrac{1}{g}L^{2}e^{\frac{1}{2}L}+\tfrac{1}{g^{2}}L^{4}e^{L}\right)

where the implied constant is independent of $L$ and $g$ .

Let’s firstly assume Proposition 15 and prove the following direct consequence which is the lower bound in Theorem 1.

Theorem 16.

For any function $\omega(g)$ satisfying

\lim\limits_{g\to\infty}\omega(g)=+\infty\ \textit{and}\ \lim\limits_{g\to\infty}\frac{\omega(g)}{\log\log g}=0,

then we have

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ \ell_{\mathop{\rm sys}}^{\rm ns}(X)>\log g-\log\log g-\omega(g)\right)=1.

Proof.

Taking $L=L_{g}=\log g-\log\log g-\omega(g)$ in Proposition 15, it is clear that

\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}(X,L_{g})]\to 0,\textit{ as $g\to\infty$}.

Then it follows by Markov’s inequality that

\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{\rm f-8}(X,L_{g})\geq 1\right)\leq\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}(X,L_{g})]\to 0,\ \textit{as $g\to\infty$}.

This also means that

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{\rm f-8}(X,L_{g})=0\right)=1.

Then the conclusion follows because for any $X\in\mathcal{M}_{g}$ , $\ell^{ns}_{sys}(X)$ is equal to the length of a shortest figure-eight closed geodesic in $X$ . ∎

For a figure-eight closed geodesic $\alpha\subset X\in\mathcal{M}_{g}$ , it is always filling in a unique pair of pants $P(x,y,z)$ as shown in Figure 1. If two of the boundary geodesics in $P(x,y,z)$ are the same simple closed geodesics in $X$ , then the completion $\overline{P(x,y,z)}\subset X$ of $P(x,y,z)$ is a hyperbolic torus with one geodesic boundary; otherwise $\overline{P(x,y,z)}\subset X$ is still a pair of pants. So the complement $X\setminus\overline{P(x,y,z)}$ may have one or two or three components (see Figure 2 for an illustation when $g=3$ ). We classify all figure-eight closed geodesics by the topology of $X\setminus\overline{P(x,y,z)}$ . Denote

\displaystyle\mathcal{N}_{\rm f-8}^{(g-2,3)}(X,L)

\displaystyle:=

\displaystyle\left\{\alpha\in\mathcal{N}_{\rm f-8}(X,L);\ X\setminus\overline{P(x,y,z)}\cong S_{g-2,3}\right\},

\displaystyle\mathcal{N}_{\rm f-8}^{(g-1,1)}(X,L)

\displaystyle:=

\displaystyle\left\{\alpha\in\mathcal{N}_{\rm f-8}(X,L);\ X\setminus\overline{P(x,y,z)}\cong S_{g-1,1}\right\},

	$\displaystyle\mathcal{N}_{\rm f-8}^{(g_{1},1)(g_{2},2)}(X,L)$	$\displaystyle:=$	$\displaystyle\{\alpha\in\mathcal{N}_{\rm f-8}(X,L);\$
			$\displaystyle X\setminus\overline{P(x,y,z)}\cong S_{g_{1},1}\cup S_{g_{2},2}\},$

	$\displaystyle\mathcal{N}_{\rm f-8}^{(g_{1},1)(g_{2},1)(g_{3},1)}(X,L)$	$\displaystyle:=$	$\displaystyle\{\alpha\in\mathcal{N}_{\rm f-8}(X,L);\$
			$\displaystyle X\setminus\overline{P(x,y,z)}\cong S_{g_{1},1}\cup S_{g_{2},1}\cup S_{g_{3},1}\},$

and

	$\displaystyle N_{\rm f-8}^{(g-2,3)}(X,L):=\#\mathcal{N}_{\rm f-8}^{(g-2,3)}(X,L),$
	$\displaystyle N_{\rm f-8}^{(g-1,1)}(X,L):=\#\mathcal{N}_{\rm f-8}^{(g-1,1)}(X,L),$
	$\displaystyle N_{\rm f-8}^{(g_{1},1)(g_{2},2)}(X,L):=\#\mathcal{N}_{\rm f-8}^{(g_{1},1)(g_{2},2)}(X,L),$
	$\displaystyle N_{\rm f-8}^{(g_{1},1)(g_{2},1)(g_{3},1)}(X,L):=\#\mathcal{N}_{\rm f-8}^{(g_{1},1)(g_{2},1)(g_{3},1)}(X,L).$

Then

	$\displaystyle N_{\rm f-8}(X,L)$	$\displaystyle=$	$\displaystyle N_{\rm f-8}^{(g-2,3)}(X,L)+\sum_{(g_{1},g_{2})}N_{\rm f-8}^{(g_{1},1)(g_{2},2)}(X,L)$
			$\displaystyle+N_{\rm f-8}^{(g-1,1)}(X,L)+\sum_{(g_{1},g_{2},g_{3})}N_{\rm f-8}^{(g_{1},1)(g_{2},1)(g_{3},1)}(X,L)$

where the first sum is taken over all $(g_{1},g_{2})$ with $g_{1}+g_{2}=g-1$ and $g_{1},g_{2}\geq 1$ ; the second sum is taken over all $(g_{1},g_{2},g_{3})$ with $g_{1}+g_{2}+g_{3}=g$ and $g_{1}\geq g_{2}\geq g_{3}\geq 1$ .

We now compute $\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g-2,3)}]$ , $\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g-1,1)}]$ and sum of all possible $\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g_{1},1)(g_{2},2)}]$ and $\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g_{1},1)(g_{2},1)(g_{3},1)}]$ in the following Lemmas.

Lemma 17.

For any $L\geq 2\mathop{\rm arccosh}3$ and $g>2$ ,

\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g-2,3)}(X,L)]=\frac{1}{8\pi^{2}g}(L-3-4\log 2)e^{L}+O\left(\tfrac{1}{g}Le^{\frac{1}{2}L}+\tfrac{1}{g^{2}}L^{4}e^{L}\right)

where the implied constant is independent of $L$ and $g$ .

Proof.

Instead of a figure-eight closed geodesic, we consider the unique pair of pants $P(x,y,z)$ (with three boundary lengths equal to $x,y,z$ ) in which a figure-eight closed geodesic is filling. In each pair of pants, there are exactly three figure-eight closed geodesics. And in the pair of pants $P(x,y,z)$ , the number of figure-eight closed geodesics of length $\leq L$ is equal to $\mathbf{1}_{\{L(x,y,z)\leq L\}}+\mathbf{1}_{\{L(z,x,y)\leq L\}}+\mathbf{1}_{\{L(y,z,x)\leq L\}}$ where $L(x,y,z)$ is the length function given in (17). So the counting $N_{\rm f-8}^{(g-2,3)}(X,L)$ of figure-eight closed geodesics can be replaced by the counting of pairs of pants $P(x,y,z)$ ’s satisfying $X\setminus\overline{P(x,y,z)}\cong S_{g-2,3}$ . And by Mirzakhani’s integration formula Theorem 4 (here $C_{\Gamma}=1$ for the pair $\Gamma=(x,y,z)$ ), we have

(22)		$\displaystyle\quad\quad\quad\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g-2,3)}(X,L)]$
	$\displaystyle=\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\text{pairs of pants}\ P(x,y,z)}\mathbf{1}_{\{L(x,y,z)\leq L\}}+\mathbf{1}_{\{L(z,x,y)\leq L\}}+\mathbf{1}_{\{L(y,z,x)\leq L\}}dX$
	$\displaystyle=\frac{1}{6}\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}\text{ordered pair}\\ (x,y,z)\end{subarray}}\mathbf{1}_{\{L(x,y,z)\leq L\}}+\mathbf{1}_{\{L(z,x,y)\leq L\}}+\mathbf{1}_{\{L(y,z,x)\leq L\}}dX$
	$\displaystyle=\frac{1}{6}\int_{x,y,z\geq 0}\left(\mathbf{1}_{\{L(x,y,z)\leq L\}}+\mathbf{1}_{\{L(z,x,y)\leq L\}}+\mathbf{1}_{\{L(y,z,x)\leq L\}}\right)$
	$\displaystyle\quad\quad\frac{V_{g-2,3}(x,y,z)V_{0,3}(x,y,z)}{V_{g}}xyz\ dxdydz$
	$\displaystyle=\frac{1}{2}\int_{x,y,z\geq 0}\mathbf{1}_{\{L(x,y,z)\leq L\}}\frac{V_{g-2,3}(x,y,z)}{V_{g}}xyz\ dxdydz$

where in the last equation we apply that $V_{0,3}(x,y,z)=1$ and the product $V_{g-2,3}(x,y,z)xyz$ is symmetric with respect to $x,y,z$ . By Part (2) of Theorem 6 we have

(23)

\frac{V_{g-2,3}}{V_{g}}=\frac{1}{8\pi^{2}g}\cdot\left(1+O\left(\frac{1}{g}\right)\right).

This together with Theorem 8 imply that

(24)

\frac{V_{g-2,3}(x,y,z)}{V_{g}}xyz=\frac{1}{\pi^{2}g}\sinh(\tfrac{1}{2}x)\sinh(\tfrac{1}{2}y)\sinh(\tfrac{1}{2}z)\cdot\left(1+O(\tfrac{1+x^{2}+y^{2}+z^{2}}{g})\right).

Applying (24) into the last line of (22), by the fact that $L(x,y,z)>\tfrac{1}{2}(x+y+z)$ , the remainder term can be bounded as

			$\displaystyle\left\|\frac{1}{2}\int_{\{L(x,y,z)\leq L\}}\frac{1}{\pi^{2}g}\sinh(\tfrac{1}{2}x)\sinh(\tfrac{1}{2}y)\sinh(\tfrac{1}{2}z)O(\tfrac{1+x^{2}+y^{2}+z^{2}}{g})dxdydz\right\|$
			$\displaystyle\prec\frac{1}{g^{2}}\int_{\{x+y+z\leq 2L\}}(1+x^{2}+y^{2}+z^{2})e^{\tfrac{1}{2}(x+y+z)}dxdydz$
			$\displaystyle\prec\frac{1}{g^{2}}L^{4}e^{L}.$

And the main term is

(26)

\frac{1}{2}\int_{x,y,z\geq 0}\mathbf{1}_{\{L(x,y,z)\leq L\}}\frac{1}{\pi^{2}g}\sinh(\tfrac{1}{2}x)\sinh(\tfrac{1}{2}y)\sinh(\tfrac{1}{2}z)dxdydz.

We change the variables $(x,y,z)$ into $(x,y,t)$ with $t=L(x,y,z)$ . By (17),

\tfrac{1}{2}\sinh(\tfrac{1}{2}t)dt=\tfrac{1}{2}\sinh(\tfrac{1}{2}z)dz+*dx+*dy.

(27)		$\displaystyle\frac{1}{2}\int_{x,y,z\geq 0}\mathbf{1}_{\{L(x,y,z)\leq L\}}\frac{1}{\pi^{2}g}\sinh(\tfrac{1}{2}x)\sinh(\tfrac{1}{2}y)\sinh(\tfrac{1}{2}z)dxdydz$
	$\displaystyle=\int_{\textbf{Cond}}\frac{1}{2\pi^{2}g}\sinh(\tfrac{1}{2}x)\sinh(\tfrac{1}{2}y)\sinh(\tfrac{1}{2}t)dxdydt$

where the integration region Cond is

\begin{cases}x,y,z\geq 0\\ L(x,y,z)\leq L\end{cases}\Longleftrightarrow\begin{cases}x\geq 0,\ \cosh(\tfrac{1}{2}x)\leq\frac{\cosh(\tfrac{1}{2}t)-1}{2\cosh(\tfrac{1}{2}y)}\\ y\geq 0,\ 2\cosh(\tfrac{1}{2}y)\leq\cosh(\tfrac{1}{2}t)-1\\ 0\leq t\leq L,\ \cosh(\tfrac{1}{2}t)\geq 3\end{cases}.

We consider the integral for $x,y$ and $t$ in order. First taking an integral for $x$ , we get

\int_{x\geq 0}\mathbf{1}_{\left\{\cosh(\tfrac{1}{2}x)\leq\frac{\cosh(\tfrac{1}{2}t)-1}{2\cosh(\tfrac{1}{2}y)}\right\}}\cdot\frac{1}{2\pi^{2}g}\sinh(\tfrac{1}{2}x)dx=\frac{1}{\pi^{2}g}\left(\frac{\cosh(\tfrac{1}{2}t)-1}{2\cosh(\tfrac{1}{2}y)}-1\right).

Then taking an integral for $y$ , we get

	$\displaystyle\int_{y\geq 0}\mathbf{1}_{\left\{2\cosh(\tfrac{1}{2}y)\leq\cosh(\tfrac{1}{2}t)-1\right\}}\cdot\sinh(\tfrac{1}{2}y)\frac{1}{\pi^{2}g}\left(\frac{\cosh(\tfrac{1}{2}t)-1}{2\cosh(\tfrac{1}{2}y)}-1\right)dy$
	$\displaystyle=\frac{1}{2\pi^{2}g}\left(\cosh(\tfrac{1}{2}t)-1\right)\left(2\log(\tfrac{\cosh(\tfrac{1}{2}t)-1}{2})\right)-\frac{2}{\pi^{2}g}\left(\tfrac{\cosh(\tfrac{1}{2}t)-1}{2}-1\right)$
	$\displaystyle=\frac{4}{\pi^{2}g}\sinh^{2}(\tfrac{1}{4}t)\log(\sinh(\tfrac{1}{4}t))-\frac{2}{\pi^{2}g}\sinh^{2}(\tfrac{1}{4}t)+\frac{2}{\pi^{2}g}.$

Finally taking an integral for $t$ , it is clear that $\log(\sinh(\tfrac{1}{4}t))=\tfrac{1}{4}t-\log 2+O(e^{-\frac{1}{2}t})$ , so we get

(28)		$\displaystyle\int_{\textbf{Cond}}\frac{1}{2\pi^{2}g}\sinh(\tfrac{1}{2}x)\sinh(\tfrac{1}{2}y)\sinh(\tfrac{1}{2}t)dxdydt$
	$\displaystyle=\int_{2\mathop{\rm arccosh}3}^{L}\sinh(\tfrac{1}{2}t)\bigg{(}\frac{4}{\pi^{2}g}\sinh^{2}(\tfrac{1}{4}t)\log(\sinh(\tfrac{1}{4}t))$
	$\displaystyle\quad\quad-\frac{2}{\pi^{2}g}\sinh^{2}(\tfrac{1}{4}t)+\frac{2}{\pi^{2}g}\bigg{)}dt$
	$\displaystyle=\int_{2\mathop{\rm arccosh}3}^{L}\sinh(\tfrac{1}{2}t)\bigg{(}\frac{4}{\pi^{2}g}\sinh^{2}(\tfrac{1}{4}t)(\tfrac{1}{4}t-\log 2+O(e^{-\frac{1}{2}t}))$
	$\displaystyle\quad\quad-\frac{2}{\pi^{2}g}\sinh^{2}(\tfrac{1}{4}t)+\frac{2}{\pi^{2}g}\bigg{)}dt$
	$\displaystyle=\frac{1}{8\pi^{2}g}(L-3-4\log 2)e^{L}+O(\tfrac{1}{g}Le^{\frac{1}{2}L}).$

So combining (22), (3), (27) and (28), we obtain

\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g-2,3)}(X,L)]=\frac{1}{8\pi^{2}g}(L-3-4\log 2)e^{L}+O\left(\tfrac{1}{g}Le^{\frac{1}{2}L}+\tfrac{1}{g^{2}}L^{4}e^{L}\right)

as desired. ∎

Remark.

Similar computations were taken in [AM23].

Lemma 18.

For any $L\geq 2\mathop{\rm arccosh}3$ and $g>2$ we have

\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g-1,1)}(X,L)]\prec\frac{1}{g}L^{2}e^{\frac{1}{2}L},

\sum_{(g_{1},g_{2})}\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g_{1},1)(g_{2},2)}(X,L)]\prec\frac{1}{g^{2}}L^{2}e^{L},

\sum_{(g_{1},g_{2},g_{3})}\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g_{1},1)(g_{2},1)(g_{3},1)}(X,L)]\prec\frac{1}{g^{3}}L^{2}e^{L},

Proof.

In a pair of pants $P(x,y,z)$ , there are exactly three figure-eight closed geodesics. And from (17) we know that the condition that a figure-eight closed geodesic in $P(x,y,z)$ has length $\leq L$ can imply that $x+y+z\leq 2L$ and all $x,y,z\leq L$ . So

N_{\rm f-8}^{(g-1,1)}(X,L)\leq 3\cdot\#\left\{P(x,y,z);\ \begin{array}[]{l}X\setminus\overline{P(x,y,z)}\cong S_{g-1,1}\\ x,y\ \text{are the same loop in}\ X\\ x=y\leq L,z\leq L\end{array}\right\},

N_{\rm f-8}^{(g_{1},1)(g_{2},2)}(X,L)\leq 3\cdot\#\left\{P(x,y,z);\ \begin{array}[]{l}X\setminus\overline{P(x,y,z)}\cong S_{g_{1},1}\cup S_{g_{2},2}\\ x+y+z\leq 2L\end{array}\right\},

N_{\rm f-8}^{(g_{1},g_{2},g_{3})}(X,L)\leq 3\cdot\#\left\{P(x,y,z);\ \begin{array}[]{l}X\setminus\overline{P(x,y,z)}\cong S_{g_{1},1}\cup S_{g_{2},1}\cup S_{g_{3},1}\\ x+y+z\leq 2L\end{array}\right\}.

Then applying Mirzakhani’s integration formula Theorem 4 and Theorem 8,

$\displaystyle\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g-1,1)}(X,L)]$	$\displaystyle\leq$	$\displaystyle 3\int_{\begin{subarray}{c}x,z\geq 0\\ x,z\leq L\end{subarray}}\frac{V_{g-1,1}(z)V_{0,3}(x,x,z)}{V_{g}}xz\ dxdz$
	$\displaystyle\prec$	$\displaystyle\int_{\begin{subarray}{c}x,z\geq 0\\ x,z\leq L\end{subarray}}x\sinh(\tfrac{1}{2}z)\frac{V_{g-1,1}}{V_{g}}dxdz$
	$\displaystyle\prec$	$\displaystyle\frac{V_{g-1,1}}{V_{g}}L^{2}e^{\frac{1}{2}L},$

and

(30)		$\displaystyle\sum_{(g_{1},g_{2})}\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g_{1},1)(g_{2},2)}(X,L)]$
	$\displaystyle\leq 3\sum_{(g_{1},g_{2})}\int_{\begin{subarray}{c}x,y,z\geq 0\\ x+y+z\leq 2L\end{subarray}}\frac{V_{g_{1},1}(x)V_{g_{2},2}(y,z)V_{0,3}(x,y,z)}{V_{g}}xyz\ dxdydz$
	$\displaystyle\prec\sum_{(g_{1},g_{2})}\int_{\begin{subarray}{c}x,y,z\geq 0\\ x+y+z\leq 2L\end{subarray}}\sinh(\tfrac{1}{2}x)\sinh(\tfrac{1}{2}y)\sinh(\tfrac{1}{2}z)\frac{V_{g_{1},1}V_{g_{2},2}}{V_{g}}dxdydz$
	$\displaystyle\prec\sum_{(g_{1},g_{2})}\frac{V_{g_{1},1}V_{g_{2},2}}{V_{g}}L^{2}e^{L},$

and similarly

(31)

\sum_{(g_{1},g_{2},g_{3})}\mathbb{E}_{\rm WP}^{g}[N_{\rm f-8}^{(g_{1},1)(g_{2},1)(g_{3},1)}(X,L)]\prec\sum_{(g_{1},g_{2},g_{3})}\frac{V_{g_{1},1}V_{g_{2},1}V_{g_{3},1}}{V_{g}}L^{2}e^{L}.

Then applying Theorem 6 and Theorem 7 for $q=2,3$ we have

(32)

\frac{V_{g-1,1}}{V_{g}}\prec\frac{1}{g},

(33)

\sum_{(g_{1},g_{2})}\frac{V_{g_{1},1}V_{g_{2},2}}{V_{g}}\prec\frac{1}{g}\frac{W_{2g-3}}{V_{g}}\prec\frac{1}{g^{2}},

(34)

\sum_{(g_{1},g_{2},g_{3})}\frac{V_{g_{1},1}V_{g_{2},1}V_{g_{3},1}}{V_{g}}\prec\frac{1}{g^{2}}\frac{W_{2g-3}}{V_{g}}\prec\frac{1}{g^{3}}.

Then the conclusion follows from all these equations (3)-(34). ∎

Proof of Proposition 15.

The conclusion clearly follows from (3), Lemma 17 and Lemma 18. ∎

4. Upper bound

In this section, we will prove the upper bound of the length of non-simple systole for random hyperbolic surfaces. That is, we show

Theorem 19.

For any function $\omega(g)$ satisfying

\lim\limits_{g\to\infty}\omega(g)=+\infty\ \textit{and}\ \lim\limits_{g\to\infty}\frac{\omega(g)}{\log\log g}=0,

then we have

\lim\limits_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ \ell_{\mathop{\rm sys}}^{\rm ns}(X)<\log g-\log\log g+\omega(g)\right)=1.

In order to prove Theorem 19, it suffices to show that

(35)

\lim_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{\rm f-8}(X,L_{g})=0\right)=0

where

L_{g}=\log g-\log\log g+\omega(g).

Instead of working on $N_{\rm f-8}(X,L_{g})$ , we consider $N_{(0,3),\star}^{(g-2,3)}(X,L_{g})$ defined as follows.

Definition.

For any $L>1$ and $X\in\mathcal{M}_{g}$ , denote by

\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)=\left\{(\gamma_{1},\gamma_{2},\gamma_{3});\ \begin{matrix}(\gamma_{1},\gamma_{2},\gamma_{3})\text{ is a pair of ordered simple closed}\\ \text{curves such that }X\setminus\cup_{i=1}^{3}\gamma_{i}\simeq S_{0,3}\bigcup S_{g-2,3},\\ \ell_{\gamma_{1}}(X)\leq L,\ \ell_{\gamma_{2}}(X)+\ell_{\gamma_{3}}(X)\leq L\\ \text{ and }\ell_{\gamma_{1}}(X),\ell_{\gamma_{2}}(X),\ell_{\gamma_{3}}(X)\geq 10\log L\end{matrix}\right\}

and

N_{(0,3),\star}^{(g-2,3)}(X,L)=\#\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L).

It follows by Equation (17) that each $(\gamma_{1},\gamma_{2},\gamma_{3})\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ bounds a pair of pants $P$ that contains a figure-eight closed geodesic of length $\leq L+c$ for some uniform constant $c>0$ : acutally the desired figure-eight closed geodesic is the one winding around $\gamma_{2}$ and $\gamma_{3}$ . Then we have

(36)			$\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{\rm f-8}(X,L_{g})=0\right)$
(36)		$\displaystyle\leq$	$\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{(0,3),\star}^{(g-2,3)}(X,L_{g}-c)=0\right).$

For any $L>1$ , we view $N_{(0,3),\star}^{(g-2,3)}(X,L)$ as a nonnegative integer-valued random variable on $\mathcal{M}_{g}$ . Then by the standard Cauchy-Schwarz inequality we know that

\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{(0,3),\star}^{(g-2,3)}(X,L)>0\right)\geq\frac{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]^{2}}{\mathbb{E}_{\rm WP}^{g}\left[(N_{(0,3),\star}^{(g-2,3)}(X,L))^{2}\right]}

implying

(37)			$\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{(0,3),\star}^{(g-2,3)}(X,L)=0\right)$
(37)			$\displaystyle\leq\frac{\mathbb{E}_{\rm WP}^{g}\left[(N_{(0,3),\star}^{(g-2,3)}(X,L))^{2}\right]-\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]^{2}}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]^{2}}.$

For any $\Gamma=(\gamma_{1},\gamma_{2},\gamma_{3})\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ , denote by $P(\Gamma)$ the pair of pants bounded by the three closed geodesics in $\Gamma$ . Set

	$\displaystyle\mathcal{A}(X,L)=\left\{(\Gamma_{1},\Gamma_{2})\in\left(\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)\right)^{2};\ P(\Gamma_{1})=P(\Gamma_{2})\right\},$
	$\displaystyle\mathcal{B}(X,L)=\left\{(\Gamma_{1},\Gamma_{2})\in\left(\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)\right)^{2};\ \overline{P(\Gamma_{1})}\cap\overline{P(\Gamma_{2})}=\emptyset\right\},$
	$\displaystyle\mathcal{C}(X,L)=\left\{(\Gamma_{1},\Gamma_{2})\in\left(\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)\right)^{2};\ \begin{matrix}P(\Gamma_{1})\neq P(\Gamma_{2}),\\ {P(\Gamma_{1})}\cap{P(\Gamma_{2})}\neq\emptyset\end{matrix}\right\},$
	$\displaystyle\mathcal{D}(X,L)=\left\{(\Gamma_{1},\Gamma_{2})\in\left(\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)\right)^{2};\ \begin{matrix}P(\Gamma_{1})\cap P(\Gamma_{2})=\emptyset,\\ \overline{P(\Gamma_{1})}\cap\overline{P(\Gamma_{2})}\neq\emptyset\end{matrix}\right\}.$

Assume $\Gamma_{1}=(\gamma_{11},\gamma_{12},\gamma_{13})$ and $\Gamma_{2}=(\gamma_{21},\gamma_{22},\gamma_{23})$ , as shown in Figure 3:

(1)

In the first picture, $\gamma_{1,i}=\gamma_{2,4-i}(i=1,2,3)$ . Then we have $\Gamma_{1}\neq\Gamma_{2}$ and $P(\Gamma_{1})=P(\Gamma_{2})$ . Hence $(\Gamma_{1},\Gamma_{2})\in\mathcal{A}(X,L)$ ;
(2)

In the second picture, $\overline{P(\Gamma_{1})}\cap\overline{P(\Gamma_{2})}=\emptyset$ . Hence $(\Gamma_{1},\Gamma_{2})\in\mathcal{B}(X,L)$ and $X\setminus(\Gamma_{1}\cup\Gamma_{2})\simeq S_{0,3}\cup S_{0,3}\cup S_{g-4,n+6}$ ;
(3)

In the third picture, ${P(\Gamma_{1})}\cap{P(\Gamma_{2})}\neq\emptyset$ . Hence $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L)$ ;
(4)

In the forth picture, $P(\Gamma_{1})\cap P(\Gamma_{2})=\emptyset$ and $\gamma_{1i}=\gamma_{2i}\ (i=1,2)$ . Hence $(\Gamma_{1},\Gamma_{2})\in\mathcal{D}(X,L).$

Denote by

	$\displaystyle A(X,L)=\#\mathcal{A}(X,L),\ B(X,L)=\#\mathcal{B}(X,L),$
	$\displaystyle C(X,L)=\#\mathcal{C}(X,L),\ D(X,L)=\#\mathcal{D}(X,L).$

It is clear that

(38)			$\displaystyle\mathbb{E}_{\textnormal{WP}}^{g}\left[(N_{(0,3),\star}^{(g-2,3)}(X,L))^{2}\right]=\mathbb{E}_{\textnormal{WP}}^{g}\left[A(X,L)\right]$
(38)			$\displaystyle+\mathbb{E}_{\textnormal{WP}}^{g}\left[B(X,L)\right]+\mathbb{E}_{\textnormal{WP}}^{g}\left[C(X,L)\right]+\mathbb{E}_{\rm WP}^{g}\left[D(X,L)\right].$

Since for any pair of pants $P$ in $X$ , there exist at most $6$ different $\Gamma^{\prime}$ s such that $P=P(\Gamma)$ , it follows that

(39)

\displaystyle\mathbb{E}_{\textnormal{WP}}^{g}\left[A(X,L)\right]\leq 6\cdot\mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right].

Now we split the proof of Theorem 19 into the following several subsections. The estimate for $\mathbb{E}_{\textnormal{WP}}^{g}\left[C(X,L)\right]$ is the hard part. And the estimates for $\mathbb{E}_{\textnormal{WP}}^{g}\left[A(X,L)\right],\ \mathbb{E}_{\textnormal{WP}}^{g}\left[B(X,L)\right]$ and $\mathbb{E}_{\textnormal{WP}}^{g}\left[D(X,L)\right]$ are relative easier.

4.1. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]$

Recall that by Proposition 15 we have

(40)

\mathbb{E}_{\rm WP}^{g}\left[N_{\rm f-8}(X,L_{g})\right]\sim\frac{L_{g}e^{L_{g}}}{8\pi^{2}g}

where $L_{g}=\log g-\log\log g+\omega(g)$ and $\omega(g)=o(\log g)$ . We will see that $\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]$ is of the same growth rate.

Proposition 20.

Assume $L>1$ and $L=O\left(\log g\right)$ , then

\mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]=\frac{1}{2\pi^{2}g}Le^{L}\left(1+O\left(\frac{\log L}{L}\right)\right)

where the implied constant is independent of $L$ and $g$ .

Proof.

For any $L>1$ , let $D_{L}\subset\mathbb{R}_{\geq 0}^{3}$ be a domain defined by

D_{L}:=\left\{(x,y,z)\in\mathbb{R}^{3}_{\geq 0};\ x\leq L,\ y+z\leq L,\ x,y,z\geq 10\log L\right\}.

Assume $\phi_{L}:\mathbb{R}^{3}_{+}\to\mathbb{R}_{\geq 0}$ is the characteristic function of $D_{L}$ , i.e.

\phi_{L}(u)=\begin{cases}0&u\notin D_{L},\\ 1&u\in D_{L}.\end{cases}

Assume $\Gamma=(\gamma_{1},\gamma_{2},\gamma_{3})$ is a pair of ordered simple closed curves in $X$ such that

X\setminus\left(\mathop{\cup}\limits_{i=1}^{3}\gamma_{i}\right)\simeq S_{0,3}\cup S_{g-2,3}.

Apply Mirzakhani’s integration formula Theorem 4 to the ordered simple closed multi-curve $\sum\limits_{i=1}^{3}\gamma_{i}$ and function $\phi_{L}$ , we have

(41)		$\displaystyle\ \ \ \ \mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]$
	$\displaystyle=\frac{1}{V_{g}}\int\limits_{\mathcal{M}_{g}}\phi_{L}^{\Gamma}(X)dX$
	$\displaystyle=\frac{1}{V_{g}}\int\limits_{D_{L}}V_{0,3}(x,y,z)V_{g-2,3}(x,y,z)xyzdxdydz.$

Recall that Equation (23) says that

(42)

\frac{V_{g-2,3}}{V_{g}}=\frac{1}{8\pi^{2}g}\left(1+O\left(\frac{1}{g}\right)\right).

By Theorem 8, for any $0<x,y,z\leq L$ , as $g\to\infty$ , we have

(43)

\displaystyle V_{g-2,3}(x,y,z)=V_{g-2,3}\frac{8\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}}{xyz}\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)

where the implied constant is independent of $g$ and $L$ . So we have

(44)			$\displaystyle\ \ \ \ \mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]$
		$\displaystyle=\frac{1}{\pi^{2}g}\int\limits_{D_{L}}\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}dxdydz\cdot\left(1+O\left(\frac{1}{g}\right)\right)\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)$

where the implied constant is independent of $L$ and $g$ . From direct calculations we have

(45)

\displaystyle\int_{10\log L}^{L}\sinh\frac{x}{2}dx=2\left(\cosh\frac{L}{2}-\cosh(5\log L)\right)=e^{\frac{L}{2}}+O(L^{5})

and

(48)		$\displaystyle\ \ \ \ \int\limits_{\mbox{\tiny$\begin{array}[]{c}y+z\leq L\\ y,z\geq 10\log L\end{array}$}}\sinh\frac{y}{2}\sinh\frac{z}{2}dydz$
	$\displaystyle=\int_{10\log L}^{L-10\log L}\sinh\frac{z}{2}\int_{10\log L}^{L-z}\sinh\frac{y}{2}dydz$
	$\displaystyle=2\int_{10\log L}^{L-10\log L}\sinh\frac{z}{2}\left(\cosh\frac{L-z}{2}-\cosh(5\log L)\right)dz$
	$\displaystyle=2\int_{10\log L}^{L-10\log L}\sinh\frac{z}{2}\cosh\frac{L-z}{2}dz+O\left(e^{\frac{L}{2}}\right)$
	$\displaystyle=\frac{1}{2}Le^{\frac{L}{2}}+O\left(e^{\frac{L}{2}}\log L\right),$

where the implied constants are independent of $L$ . From (45) and (48) we have

(49)		$\displaystyle\ \ \ \ \int_{D_{L}}\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}dxdydz$
(52)		$\displaystyle=\int_{10\log L}^{L}\sinh\frac{x}{2}dx\times\int\limits_{\mbox{\tiny$\begin{array}[]{c}y+z\leq L\\ y,z\geq 10\log L\end{array}$}}\sinh\frac{y}{2}\sinh\frac{z}{2}dydz$
	$\displaystyle=\left(e^{\frac{L}{2}}+O\left(L^{5}\right)\right)\times\left(\frac{1}{2}Le^{\frac{L}{2}}+O\left(e^{\frac{L}{2}}\log L\right)\right)$
	$\displaystyle=\frac{1}{2}Le^{L}+O\left(e^{L}\log L\right).$

From (44), (49) and the assumption that $L=O\left(\log g\right)$ , we obtain

	$\displaystyle\ \ \ \ \mathbb{E}_{\textnormal{WP}}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]$
	$\displaystyle=\frac{1}{\pi^{2}g}\cdot\left(\frac{1}{2}Le^{L}+O\left(e^{L}\log L\right)\right)\cdot\left(1+O\left(\frac{1}{g}\right)\right)\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)$
	$\displaystyle=\frac{1}{2\pi^{2}g}Le^{L}\left(1+O\left(\frac{\log L}{L}\right)\right)$

as desired. ∎

4.2. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[B(X,L)\right]$

For $B(X,L)$ , we will show that

\mathbb{E}_{\rm WP}^{g}\left[B(X,L_{g})\right]\sim\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2},\quad\textit{as $g\to\infty$},

where $L_{g}=\log g-\log\log g+\omega(g)$ and $\omega(g)=o(\log g)$ . More precisely,

Proposition 21.

Assume $L>1$ and $L=O(\log g)$ , then

(53)

\displaystyle\mathbb{E}_{\textnormal{WP}}^{g}\left[B(X,L)\right]=\frac{1}{4\pi^{4}g^{2}}L^{2}e^{2L}\left(1+O\left(\frac{\log L}{L}\right)\right),

where the implied constant is independent of $g$ and $L$ .

Proof.

Assume $(\Gamma_{1},\Gamma_{2})\in\mathcal{B}(X,L)$ and $\Gamma_{1}=(\alpha_{1},\alpha_{2},\alpha_{3})$ , $\Gamma_{2}=(\beta_{1},\beta_{2},\beta_{3})$ . From the definition of $\mathcal{B}(X,L)$ , it is not hard to check that

X\setminus\left(\Gamma_{1}\cup\Gamma_{2}\right)=P(\Gamma_{1})\cup P(\Gamma_{2})\cup Y,\text{ where }Y\simeq S_{g-4,6}.

Define a function $\phi_{L,2}:\mathbb{R}^{3}_{+}\times\mathbb{R}^{3}_{+}\to\mathbb{R}_{\geq 0}$ as follows:

\phi_{L,2}(u,v):=\phi_{L}(u)\cdot\phi_{L}(v)

where $\phi_{L}$ is defined in the proof of Proposition 20. Assume $(\gamma_{1},\gamma_{2},\cdots,\gamma_{6})$ is an ordered simple closed multi-curve in $S_{g}$ such that

X\setminus\mathop{\cup}\limits_{i=1}^{6}\gamma_{i}\simeq S_{0,3}^{1}\cup S_{0,3}^{2}\cup S_{g-4,6}

where the boundary of $S_{0,3}^{1}$ consists of $\gamma_{i}(1\leq i\leq 3)$ and the boundary of $S_{0,3}^{2}$ consists of $\gamma_{j}(4\leq j\leq 6)$ . By Part $(2)$ of Theorem 6 we have

(54)

\frac{V_{g-4,6}}{V_{g}}=\frac{1}{64\pi^{4}g^{2}}\left(1+O\left(\frac{1}{g}\right)\right).

By Theorem 8 we have

(55)

\displaystyle V_{g-4,6}(x_{1},...,x_{6})=V_{g-4,6}\cdot\prod\limits_{i=1}^{6}\frac{2\sinh\frac{x_{i}}{2}}{x_{i}}\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)

where the implied constant is independent of $L$ and $g$ . Applying Mirzakhani’s integration formula Theorem 4 to ordered simple closed multi-curve $(\gamma_{1},\gamma_{2},\cdots,\gamma_{6})$ and function $\phi_{L,2}$ , together with (49), (54) and (55), we have

(56)		$\displaystyle\ \ \ \ \mathbb{E}_{\textnormal{WP}}^{g}\left[B(X,L)\right]$
	$\displaystyle=\frac{1}{V_{g}}\int_{D_{L}\times D_{L}}V_{g-4,6}(x_{1},...,x_{6})\prod\limits_{i=1}^{6}x_{i}dx_{1}...dx_{6}$
	$\displaystyle=\frac{V_{g-4,6}}{V_{g}}\left(\int_{D_{L}}8\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}dxdydz\right)^{2}\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)$
	$\displaystyle=\frac{1}{64\pi^{4}g^{2}}\left(1+O\left(\frac{1}{g}\right)\right)\cdot\left(4Le^{L}+O\left(e^{L}\log L\right)\right)^{2}\cdot\left(1+O\left(\frac{L^{2}}{g}\right)\right)$
	$\displaystyle=\frac{1}{4\pi^{4}g^{2}}L^{2}e^{2L}\left(1+O\left(\frac{\log L}{L}\right)\right)$

where the implied constant is independent of $L$ and $g$ . ∎

4.3. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[C(X,L)\right]$

For $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L)$ , $\Gamma_{1}\cup\Gamma_{2}$ is not the union of disjoint simple closed geodesics but with intersections. This is the hard case. We will need the following construction as in [MP19, WX22b, NWX23] to deform $\Gamma_{1}\cup\Gamma_{2}$ .

Construction.

Fix a closed hyperbolic surface $X\in\mathcal{M}_{g}$ and let $X_{1},X_{2}$ be two distinct connected, precompact subsurfaces of $X$ with geodesic boundaries, such that $X_{1}\cap X_{2}\neq\emptyset$ and neither of them contains the other. Then the union $X_{1}\cup X_{2}$ is a subsurface whose boundary consists of only piecewise geodesics. We can construct from it a new subsurface, with geodesic boundary, by deforming each of its boundary components $\xi\subset\partial\left(X_{1}\cup X_{2}\right)$ as follows:

(1)

if $\xi$ is homotopically nontrivial, we deform $X_{1}\cup X_{2}$ by shrinking $\xi$ to the unique simple closed geodesic homotopic to it;
(2)

if $\xi$ is homotopically trivial, we fill into $X_{1}\cup X_{2}$ the disc bounded by $\xi$ .

Denote by $S(X_{1},X_{2})$ the subsurface with geodesic boundary constructed from $X_{1}$ and $X_{2}$ as above. For any $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L)$ , denote by $S(\Gamma_{1},\Gamma_{2})=S\left(P(\Gamma_{1}),P(\Gamma_{2})\right)$ the subsurface of $X$ constructed from $P(\Gamma_{1})$ and $P(\Gamma_{2})$ .

By the construction, it is clear that

(57)

\ell(\partial S(X_{1},X_{2}))\leq\ell(\partial X_{1})+\ell(\partial X_{2}),

and by isoperimetric inequality(see e.g. [Bus10, Section 8.1] or [WX22c]) we have

(58)

\mathop{\rm Area}(S(X_{1},X_{2}))\leq\mathop{\rm Area}(X_{1})+\mathop{\rm Area}(X_{2})+\ell(\partial X_{1})+\ell(\partial X_{2}).

Recall that by the definition of $\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ , we know that for any $\Gamma\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ ,

X\setminus\Gamma\simeq S_{0,3}\cup S_{g-2,3}.

Thus we may divide $\mathcal{C}(X,L)$ into following pairwisely disjoint three parts:

(59)

\mathcal{C}(X,L)=\mathcal{C}_{0,4}(X,L)\cup\mathcal{C}_{1,2}(X,L)\cup\mathcal{C}_{\geq 3}(X,L)

where

	$\displaystyle\mathcal{C}_{0,4}(X,L)=\left\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L);\ S(\Gamma_{1},\Gamma_{2})\simeq S_{0,4}\right\},$
	$\displaystyle\mathcal{C}_{1,2}(X,L)=\left\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L);\ S(\Gamma_{1},\Gamma_{2})\simeq S_{1,2}\right\},$
	$\displaystyle\mathcal{C}_{\geq 3}(X,L)=\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L);\ \|\chi(S(\Gamma_{1},\Gamma_{2}))\|\geq 3\}.$

Assume $\Gamma_{1}=(\gamma_{11},\gamma_{12},\gamma_{13})$ and $\Gamma_{2}=(\gamma_{21},\gamma_{22},\gamma_{23})$ . As in Figure 4:

(1)

in the first picture, the simple closed geodesic $\gamma_{12}$ coincides with $\gamma_{22}$ . We have $S(\Gamma_{1},\Gamma_{2})\simeq S_{0,4}$ of geodesic boundaries $\gamma_{11},\gamma_{12}=\gamma_{22},\gamma_{21}$ and $\beta$ . Hence $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}(X,L)$ ;
(2)

in the second picture, we have $S(\Gamma_{1},\Gamma_{2})\simeq S_{1,2}$ of geodesic boundaries $\gamma_{11}$ and $\gamma_{21}$ . Hence $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L)$ ;
(3)

in the third picture, we have $S(\Gamma_{1},\Gamma_{2})\simeq S_{0,5}$ of geodesic boundaries $\gamma_{11},\gamma_{21},\gamma_{23}$ where $\gamma_{21}$ and $\gamma_{23}$ appear twice in the boundary of $S(\Gamma_{1},\Gamma_{2})$ . Hence $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{\geq 3}(X,L)$ .

For $L>0$ and $X\in\mathcal{M}_{g}$ , define

\textnormal{Sub}_{L}(X)\overset{\textnormal{def}}{=}\left\{\begin{matrix}Y\subset X\text{ is a connected subsurface of geodesic boundary}\\ \text{such that }\ell(\partial Y)\leq 2L\text{ and }\textnormal{Area}(Y)\leq 4L+4\pi\end{matrix}\right\}.

Lemma 22.

For any $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L)$ , there exists a triple $(\alpha_{1},\alpha_{2},Y)$ and a universal constant $c>0$ such that

(1)

$Y=S(\Gamma_{1},\Gamma_{2})\in\textnormal{Sub}_{L}(X)$ ;
(2)

$\alpha_{i}$ is a figure-eight closed geodesic contained in $P(\Gamma_{i})$ for $i=1,2$ ;
(3)

$(\alpha_{1},\alpha_{2})$ is a filling 2-tuple in $Y$ and $\ell(\alpha_{1}),\ell(\alpha_{2})\leq L+c$ .

Proof.

For Part (1), it follows from (57) and (58) that for any $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L)$ ,

S(\Gamma_{1},\Gamma_{2})\in\textnormal{Sub}_{L}(X).

Part (2) is clear.

For Part (3), we first assume $\Gamma_{i}=(\gamma_{i1},\gamma_{i2},\gamma_{i3})(i=1,2)$ . For $i=1,2$ , let $\alpha_{i}$ be the figure-eight closed geodesic contained in $P(\Gamma_{i})$ winding around $\gamma_{i2}$ and $\gamma_{i3}$ . Then from (17) we have

(60)

\displaystyle\cosh\frac{\ell(\alpha_{i})}{2}=\cosh\frac{\ell(\gamma_{i1})}{2}+2\cosh\frac{\ell(\gamma_{i2})}{2}\cosh\frac{\ell(\gamma_{i3})}{2},\ i=1,2.

From the assumption that $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}(X,L)$ , we have

(61)

\displaystyle\ell(\gamma_{i1})\leq L\text{ and }\ell(\gamma_{i2})+\ell(\gamma_{i3})\leq L,\ i=1,2.

From (60) and (61), one may check that there exists a universal constant $c>0$ such that $\ell(\alpha_{1}),\ell(\alpha_{2})\leq L+c$ . Now we show that $(\alpha_{1},\alpha_{2})$ is a filling 2-tuple in $S(\Gamma_{1},\Gamma_{2})$ . Suppose not, then there exists a simple closed geodesic $\beta$ in $Y$ such that $\beta\cap(\alpha_{1}\cup\alpha_{2})=\emptyset.$ Since $\alpha_{i}$ fills $P(\Gamma_{i})(i=1,2)$ , it follows that $\beta\cap(P(\Gamma_{1})\cup P(\Gamma_{2}))=\emptyset$ . Then by the construction of $S(\Gamma_{1},\Gamma_{2})$ we have $\beta\cap S(\Gamma_{1},\Gamma_{2})=\emptyset$ , which is a contradiction.

The proof is complete. ∎

Similar to [WX22b], we set the following assumption.

Assumption $(\star)$ . Let $Y_{0}\in\textnormal{Sub}_{L}(X)$ satisfying

(1)

$Y_{0}$ is homeomorphic to to $S_{g_{0},k}$ for some $g_{0}\geq 0$ and $k>0$ with $m=|\chi(Y_{0})|=2g_{0}-2+k\geq 1$ ;
(2)

the boundary $\partial Y_{0}$ is a simple closed multi-geodesics in $X$ consisting of $k$ simple clsoed geodesics which has $n_{0}$ pairs of simple closed geodesics for some $n_{0}\geq 0$ such that each pair corresponds to a single simple closed geodesic in $X$ ;
(3)

the interior of its complement $X\setminus S_{g_{0},k}$ consists of $q$ components $S_{g_{1},n_{1}},...,S_{g_{q},n_{q}}$ for some $q\geq 1$ where $\sum_{i=1}^{q}n_{i}=k-2n_{0}$ .

Our aim is to bound $\mathbb{E}_{\textnormal{WP}}^{g}\left[C(X,L)\right]$ , from (59) it suffices to bound the three terms $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{\geq 3}(X,L)\right],\ \mathbb{E}_{\textnormal{WP}}^{g}\left[C_{0,4}(X,L)\right]$ and $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{1,2}(X,L)\right]$ separately.

4.3.1. Bounds for $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{\geq 3}(X,L)\right]$

We first bound $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{\geq 3}(X,L)\right]$ through using the method in [WX22b].

Proposition 23.

Assume $L>1$ and $L\prec\log g$ , then for any fixed small $\epsilon>0$ ,

\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{\geq 3}(X,L)\right]\prec\left(L^{67}e^{2L+\epsilon L}\frac{1}{g^{3}}+\frac{L^{3}e^{8L}}{g^{11}}\right).

Proof.

For $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{\geq 3}(X,L)$ , by Lemma 22, there exists a filling 2-tuple $(\alpha_{1},\alpha_{2})$ in $Y$ with total length $\leq 2L+2c$ , and $\alpha_{i}$ is a filling figure-eight closed geodesic in a unique pair of pants for both $i=1,2$ respectively. Consider the alternatives of three geodesics in elements in $\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ , there are at most $36$ pairs $(\Gamma_{1},\Gamma_{2})$ corresponding to the same triples $(\alpha_{1},\alpha_{2},Y)$ . It follows that

C_{\geq 3}(X,L)\leq\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ 3\leq|\chi(Y)|\end{subarray}}36\cdot N_{2}^{\text{fill}}(Y,2L+2c)

where $N_{2}^{\text{fill}}(Y,2L+2c)$ is defined in Subsection 2.5.2. Therefore we have

(62)

\mathbb{E}_{\rm WP}^{g}\left[C_{\geq 3}(X,L)\right]\leq\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ 3\leq|\chi(Y)|\end{subarray}}36N_{2}^{\text{fill}}(Y,2L+2c)\cdot 1_{[0,2L]}(\ell(\partial Y))dX.

Now we divide the summation above into following two parts: the first part consists of all subsurfaces $Y\in\textnormal{Sub}_{L}(X)$ such that $3\leq|\chi(Y)|\leq 10$ ; the second part consists of all subsurfaces $Y\in\textnormal{Sub}_{L}(X)$ such that $10<|\chi(Y)|\leq\left[\frac{4L+4\pi}{2\pi}\right]$ .

For the first part, assume $Y\simeq S_{g_{0},k}\in\textnormal{Sub}_{L}(X)$ satisfies Assumption $(\star)$ with an additional assumption that

(63)

\displaystyle 3\leq m=2g_{0}-2+k\leq 10.

From [WX22b, Proposition 34] and Theorem 11, we have that for any fixed $0<\epsilon<\frac{1}{2}$ ,

(64)		$\displaystyle\ \ \ \ \frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ Y\simeq S_{g_{0},k}\end{subarray}}36N_{2}^{\text{fill}}(Y,2L+2c)\cdot 1_{[0,2L]}(\ell(\partial Y))dX$
	$\displaystyle\prec\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ Y\simeq S_{g_{0},k}\end{subarray}}Le^{2L-\frac{1-\epsilon}{2}\ell(\partial Y)}\cdot 1_{[0,2L]}(\ell(\partial Y))dX\quad(\textit{\rm{by Theorem \ref{thm count fill k-tuple}}})$
	$\displaystyle=Le^{\frac{3}{2}L}\times\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ Y\simeq S_{g_{0},k}\end{subarray}}e^{\frac{1}{2}L-\frac{1-\epsilon}{2}\ell(\partial Y)}\cdot 1_{[0,2L]}(\ell(\partial Y))dX$
	$\displaystyle\prec Le^{\frac{3}{2}L}\times L^{66}e^{\frac{1}{2}L+\epsilon L}\frac{1}{g^{m}}\quad(\textit{\rm{by\cite[cite]{[\@@bibref{}{WX22-GAFA}{}{}, Proposition 34 ]}}})$
	$\displaystyle=L^{67}e^{2L+\epsilon L}\frac{1}{g^{m}}.$

Since there are at most finite pairs $(g_{0},k)$ satisfying the assumption (63), take summation over all possible subsurfaces $Y$ for inequality (64), we have

(65)

\displaystyle\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ 3\leq|\chi(Y)|\leq 10\end{subarray}}36N_{2}^{\text{fill}}(Y,2L+2c)\cdot 1_{[0,2L]}(\ell(\partial Y))dX\prec L^{67}e^{2L+\epsilon L}\frac{1}{g^{3}}.

For the second part, firstly by (58) and the Gauss-Bonnet formula we know that $|\chi(Y)|\prec L$ . Since $\ell(\partial Y)\leq 2L$ , by Theorem 10 we have

(66)

\displaystyle N_{2}^{\text{fill}}(Y,2L+2c)

\displaystyle\prec\left(\left|\chi(Y)\right|e^{2L}\right)^{2}\prec L^{2}e^{\frac{9}{2}L-\frac{1}{4}\ell(\partial Y)}.

From (66) and [WX22b, Proposition 33], we have

(67)		$\displaystyle\ \ \ \ \frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ 11\leq\|\chi(Y)\|\leq\left[\frac{4L+4\pi}{2\pi}\right]\end{subarray}}36N_{2}^{\text{fill}}(Y,2L+2c)\cdot 1_{[0,2L]}(\ell(\partial Y))dX$
	$\displaystyle\prec\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ 11\leq\|\chi(Y)\|\leq\left[\frac{4L+4\pi}{2\pi}\right]\end{subarray}}L^{2}e^{\frac{9}{2}L-\frac{1}{4}\ell(\partial Y)}\cdot 1_{[0,2L]}(\ell(\partial Y))dX$
	$\displaystyle\prec L^{2}e^{\frac{9}{2}L}\times\frac{1}{V_{g}}\int_{\mathcal{M}_{g}}\sum_{\begin{subarray}{c}Y\in\operatorname{Sub}_{L}(\mathrm{X});\\ 11\leq\|\chi(Y)\|\leq\left[\frac{4L+4\pi}{2\pi}\right]\end{subarray}}e^{-\frac{1}{4}\ell(\partial Y)}\cdot 1_{[0,2L]}(\ell(\partial Y))dX$
	$\displaystyle\prec L^{2}e^{\frac{9}{2}L}\times Le^{\frac{7}{2}L}\frac{1}{g^{11}}\quad(\textit{\rm{by\cite[cite]{[\@@bibref{}{WX22-GAFA}{}{}, Proposition 33]}}})$
	$\displaystyle=\frac{L^{3}e^{8L}}{g^{11}}.$

Then combining (62), (65) and (67), we complete the proof. ∎

4.3.2. Bounds for $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{1,2}(X,L)\right]$

One may be aware of that the method in Proposition 23 cannot afford desired estimations for $\mathbb{E}_{\rm WP}^{g}\left[C_{1,2}(X,L)\right]$ and $\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}(X,L)\right]$ . Our aim for $\mathbb{E}_{\rm WP}^{g}\left[C_{1,2}(X,L)\right]$ is as follows.

Proposition 24.

For $L>1$ and large $g$ ,

\mathbb{E}_{\rm WP}^{g}\left[C_{1,2}(X,L)\right]\prec\frac{e^{2L}}{g^{2}}.

The estimations for pairs $(\Gamma_{1},\Gamma_{2})$ with $S(\Gamma_{1},\Gamma_{2})=Y$ in Lemma 22 are not good enough. We need to accurately classify the relative position of $(\Gamma_{1},\Gamma_{2})$ in $Y$ . We begin with the following bounds.

Lemma 25.

For $X\in\mathcal{M}_{g}$ and $L>1$ ,

(68)		$\displaystyle\#\Big{\{}(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L),\textit{ either }\Gamma_{1}\textit{ or }\Gamma_{2}\textit{ contains }\partial{S(\Gamma_{1},\Gamma_{2})}\Big{\}}$
	$\displaystyle\prec$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\gamma_{3})}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\gamma_{3}))$
	$\displaystyle\cdot$	$\displaystyle\left(\frac{\ell(\gamma_{1})}{\mathcal{R}(\ell(\gamma_{1}),\ell(\gamma_{2}),L)}+\frac{\ell(\gamma_{1})}{\mathcal{D}(\ell(\gamma_{1}),2L,0)}\right),$

where $(\gamma_{1},\gamma_{2},\gamma_{3})$ are taken over all triples of simple closed geodesics on $X$ satisfying that $\gamma_{1}\cup\gamma_{2}$ cuts off a subsurface $Y\simeq S_{1,2}$ in $X$ and $\gamma_{3}$ separates $Y$ into $S_{1,1}\cup S_{0,3}.$

Proof.

For any $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L)$ such that either $\Gamma_{1}$ or $\Gamma_{2}$ contains $\partial{S(\Gamma_{1},\Gamma_{2})}$ , WLOG, one may assume that $Y={S(\Gamma_{1},\Gamma_{2})}\simeq S_{1,2}$ and $\Gamma_{1}$ contains $\partial Y=\gamma_{1}\cup\gamma_{2}$ . Denote the rest simple closed geodesic in $\Gamma_{1}$ by $\gamma_{3}$ . Consider the map

\pi:(\Gamma_{1},\Gamma_{2})\mapsto(\gamma_{1},\gamma_{2},\gamma_{3})

where the union $\gamma_{1}\cup\gamma_{2}$ cuts off $Y\simeq S_{1,2}$ in $X$ and $\gamma_{3}$ separates $Y$ into $S_{1,1}\cup S_{0,3}$ with length

(69)

\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\gamma_{3})\in[10\log L,L].

Now we count all $\Gamma_{2}$ ’s satisfying $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L)$ and $P(\Gamma_{2})\subset Y$ . Since $\Gamma_{1},\Gamma_{2}\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ and $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L)$ , it is clear that $\Gamma_{2}$ must contain at least one of $\gamma_{1}$ and $\gamma_{2}$ .

Case-1: $\Gamma_{2}$ contains $\gamma_{1}\cup\gamma_{2}$ (see Figure 5 for an illustation). For this case, the remaining simple closed geodesic $\tilde{\gamma}$ in $\Gamma_{2}$ is of length $\leq L$ and bounds a $S_{0,3}$ in $Y$ along with $\gamma_{1}\cup\gamma_{2}$ as in Figure 5. Then it follows by (19) that the number of such $\tilde{\gamma}$ ’s is at most

\frac{\ell(\gamma_{1})}{\mathcal{R}(\ell(\gamma_{1}),\ell(\gamma_{2}),L)}.

Case-2: $\Gamma_{2}$ contains only one of $\gamma_{1},\gamma_{2}$ (see Figure 6 for an illustation). WLOG, one may assume that $\Gamma_{2}$ contains only $\gamma_{1}$ . Then the rest two simple closed geodesics $\alpha,\beta$ in $\Gamma_{2}$ , along with $\gamma_{1}$ , will bound a $S_{0,3}$ in $Y$ of total length $\ell(\alpha)+\ell(\beta)\leq 2L$ as in Figure 6. Then it follows by (20) that the number of such pairs of $(\alpha,\beta)$ ’s is at most

\frac{\ell(\gamma_{1})}{\mathcal{D}(\ell(\gamma_{1}),2L,0)}.

Combine these two cases, we have

\#\pi^{-1}(\gamma_{1},\gamma_{2},\gamma_{3})\leq 200\cdot\left(\frac{\ell(\gamma_{1})}{\mathcal{R}(\ell(\gamma_{1}),\ell(\gamma_{2}),L)}+\frac{\ell(\gamma_{1})}{\mathcal{D}(\ell(\gamma_{1}),2L,0)}\right).

Then the conclusion follows by taking a summation over all possible $(\gamma_{1},\gamma_{2},\gamma_{3})$ ’s satisfying (69). This completes the proof. ∎

Remark.

The coefficient $200$ in the proof of Lemma 25 comes from the symmetry of the three boundary components of a pair of pants, the symmetry of $\Gamma_{1}$ and $\Gamma_{2}$ , and is not essential. What we need is a universal positive constant.

Lemma 26.

For $X\in\mathcal{M}_{g}$ and $L>1$ ,

(70)		$\displaystyle\#\Big{\{}(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L),\ \Gamma_{1}\textit{ only contains one}$
		$\displaystyle\textit{ component of }\partial S(\Gamma_{1},\Gamma_{2}),\,\Gamma_{2}\textit{ only contains the other }\Big{\}}$
	$\displaystyle\prec$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))$
		$\displaystyle\cdot 1_{[0,L]}(\ell(\alpha_{1}))\cdot 1_{[0,L]}(\ell(\beta_{1}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)},$

where $(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})$ are taken over all quadruples satisfying that $\gamma_{1}\cup\gamma_{2}$ cuts off a subsurface $Y\simeq S_{1,2}$ in $X$ and $\alpha_{1}\cup\beta_{1}$ separates $Y$ into $S_{0,3}\cup S_{0,3}$ with $\gamma_{1},\gamma_{2}$ belonging to the boundaries of the two different $S_{0,3}$ ’s.

Proof.

For any $(\Gamma_{1},\Gamma_{2})$ belonging to the set in the left side of (70), WLOG, one may assume that $Y=S(\Gamma_{1},\Gamma_{2})$ , $\partial Y=\gamma_{1}\cup\gamma_{2}$ , $\Gamma_{1}$ only contains $\gamma_{1}$ and $\Gamma_{2}$ only contains $\gamma_{2}$ . The rest two simple closed geodesics $\alpha_{1},\beta_{1}$ in $\Gamma_{1}$ will separate $Y$ into $S_{0,3}\cup S_{0,3}$ , i.e. two copies of $S_{0,3}^{\prime}s$ . Consider the map

\pi:(\Gamma_{1},\Gamma_{2})\mapsto(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1}),

where the union $\gamma_{1}\cup\gamma_{2}$ cuts off $Y\simeq S_{1,2}$ in $X$ , and $\alpha_{1}\cup\beta_{1}$ separates $Y$ into $S_{0,3}\cup S_{0,3}$ such that $\gamma_{1},\gamma_{2}$ belong to the boundaries of two different $S_{0,3}$ ’s (see Figure 7 for an illustation). Moreover, their lengths satisfy

(71)

\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\alpha_{1}),\ell(\beta_{1})\in[10\log L,L].

Then we count all $\Gamma_{2}$ ’s such that $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L)$ with $\gamma_{1}\notin\Gamma_{2},\gamma_{2}\in\Gamma_{2}$ and $P(\Gamma_{2})\subset Y$ . Such a $\Gamma_{2}$ is uniquely determined by the rest two simple closed geodesics $\alpha_{2},\beta_{2}$ in $Y$ whose union separates $Y$ into $S_{0,3}\cup S_{0,3}$ with $\gamma_{1},\gamma_{2}$ in two different $S_{0,3}$ ’s, of length $\ell(\alpha_{2}),\ell(\beta_{2})\leq L$ as shown in Figure 7.

It is clear that

\ell(\alpha_{2})+\ell(\beta_{2})\leq 2L.

Then it follows by (20) that there are at most

\frac{\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}

such pairs of $(\alpha_{2},\beta_{2})$ ’s. This implies that

\#\pi^{-1}(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})\leq\frac{200\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}.

Sum it over all possible $(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})$ ’s satisfying (71), we complete the proof. ∎

Lemma 27.

For $X\in\mathcal{M}_{g}$ and $L>1$ ,

(72)		$\displaystyle\#\Big{\{}(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L),\textit{ both }\Gamma_{1}\textit{ and }\Gamma_{2}$
		$\displaystyle\textit{ only contain the same one component of }\partial{S(\Gamma_{1},\Gamma_{2})}\Big{\}}$
	$\displaystyle\prec$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[0,4L]}(2\ell(\gamma_{1})+\ell(\gamma_{2}))$
		$\displaystyle\cdot 1_{[0,L]}(\ell(\alpha_{1}))\cdot 1_{[0,L]}(\ell(\beta_{1}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)},$

Proof.

For any $(\Gamma_{1},\Gamma_{2})$ belonging to the set in the left side of (72), WLOG, one may assume that $Y={S(\Gamma_{1},\Gamma_{2})}$ , $\partial Y=\gamma_{1}\cup\gamma_{2}$ , both $\Gamma_{1}$ and $\Gamma_{2}$ contain $\gamma_{1}$ and do not contain $\gamma_{2}$ . Assume that $\Gamma_{1}$ contains $\gamma_{1},\alpha_{1},\beta_{1}$ and $\Gamma_{2}$ contains $\gamma_{1},\alpha_{2},\beta_{2}$ . In this situation, we warn here that $\ell(\gamma_{2})$ may exceed $L$ . Since $P(\Gamma_{1})\cup P(\Gamma_{2})$ fills $Y$ , there is a connected component $C$ of $Y\setminus P(\Gamma_{1})\cup P(\Gamma_{2})$ such that $C$ is topologically a cylinder and $\gamma_{2}$ is a connected component of $\partial C$ . The other connected component of $\partial C$ , denoted by $\eta$ , is a closed piecewisely smooth geodesic loop, freely homotopical to $\gamma_{2}$ . Each geodesic arcs in $\eta$ are different parts of arcs in $\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}$ as shown in Figure 8. Firstly it is clear that

(73)

\ell(\gamma_{2})\leq\ell(\eta)\leq\ell(\alpha_{1})+\ell(\beta_{1})+\ell(\alpha_{2})+\ell(\beta_{2}).

Since $\Gamma_{1},\Gamma_{2}\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ , we have

(74)

\left\{\begin{aligned} \ell(\gamma_{1})+\ell(\alpha_{1})+&\ell(\beta_{1})\leq 2L\\ \ell(\gamma_{1})+\ell(\alpha_{2})+&\ell(\beta_{2})\leq 2L\\ \ell(\gamma_{1})\geq&10\log L\\ \end{aligned}\right..

It follows from (73) and (74) that

(75)

\ell(\gamma_{2})\leq 4L-2\ell(\gamma_{1})\leq 4L-20\log L.

Consider the map

\pi:(\Gamma_{1},\Gamma_{2})\mapsto(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})

for $(\Gamma_{1},\Gamma_{2})$ belonging to the set in the left side of (72). Here $\gamma_{1}\cup\gamma_{2}$ cuts off $Y\simeq S_{1,2}$ in $X$ and $\alpha_{1}\cup\beta_{1}$ separates $Y$ into $S_{0,3}\cup S_{0,3}$ such that $\gamma_{1},\gamma_{2}$ belong to boundaries of two different $S_{0,3}$ ’s. Moreover, their lengths satisfy

(76)

\ell(\gamma_{1}),\ell(\alpha_{1}),\ell(\beta_{1})\in[10\log L,L],\ 2\ell(\gamma_{1})+\ell(\gamma_{2})\leq 4L.

Then we count all possible $\Gamma_{2}$ ’s such that $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{1,2}(X,L)$ , $\gamma_{1}\in\Gamma_{2},\gamma_{2}\notin\Gamma_{2}$ and $P(\Gamma_{2})\subset Y$ . Since $\ell(\alpha_{2})+\ell(\beta_{2})\leq 2L$ , it follows by (20) that there are at most

\frac{\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}

such pairs of $(\alpha_{2},\beta_{2})$ ’s. This implies that

\#\pi^{-1}(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})\leq\frac{200\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}.

Sum it over all possible $(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})$ ’s satisfying (76), we complete the proof. ∎

Now we are ready to Proposition 24.

Proof of Proposition 24.

Following Lemma 25, Lemma 26 and Lemma 27, for $L>1$ we have

(77)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\left[C_{1,2}(X,L)\right]$
	$\displaystyle\prec$	$\displaystyle\mathbb{E}_{\rm WP}^{g}\Bigg{[}\sum_{(\gamma_{1},\gamma_{2},\gamma_{3})}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\gamma_{3}))$
	$\displaystyle\cdot$	$\displaystyle\left(\frac{\ell(\gamma_{1})}{\mathcal{R}(\ell(\gamma_{1}),\ell(\gamma_{2}),L)}+\frac{\ell(\gamma_{1})}{\mathcal{D}(\ell(\gamma_{1}),2L,0)}\right)$
	$\displaystyle+$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})}\Big{(}1_{[10\log L,L]}(\ell(\gamma_{1})\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,L]}(\ell(\alpha_{1}))\cdot 1_{[0,L]}(\ell(\beta_{1}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}$
	$\displaystyle+$	$\displaystyle 1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[0,4L]}(2\ell(\gamma_{1})+\ell(\gamma_{2}))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,L]}(\ell(\alpha_{1}))\cdot 1_{[0,L]}(\ell(\beta_{1}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}\Big{)}\Bigg{]},$

where $\gamma_{1}\cup\gamma_{2}$ cuts off $Y\simeq S_{1,2}$ in $X$ , $\gamma_{3}$ separates $Y$ into $S_{1,1}\cup S_{0,3}$ , and $\alpha_{1}\cup\beta_{1}$ separates $Y$ into $S_{0,3}\cup S_{0,3}$ with $\gamma_{1},\gamma_{2}$ in boundaries of two different $S_{0,3}$ ’s.

For $Y\simeq S_{1,2}$ in $X$ , the completion subsurface $X\setminus\overline{Y}$ can be of type either $S_{g-2,2}$ or $S_{g_{1},1}\cup S_{g_{2},1}$ with $g_{1}+g_{2}=g-1$ . By Mirzakhani’s integration formula, i.e. Theorem 4, Theorem 5, Theorem 8, and Theorem 13, we have that for $L>1$ ,

(78)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\Bigg{[}\sum_{(\gamma_{1},\gamma_{2},\gamma_{3})}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\gamma_{3}))$
	$\displaystyle\cdot$	$\displaystyle\left(\frac{\ell(\gamma_{1})}{\mathcal{R}(\ell(\gamma_{1}),\ell(\gamma_{2}),L)}+\frac{\ell(\gamma_{1})}{\mathcal{D}(\ell(\gamma_{1}),2L,0)}\right)\Bigg{]}$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{0\leq x,y,z\leq L}\Big{[}(1+x)(1+e^{\frac{L-x-y}{2}})+(1+x)(1+e^{\frac{2L-x}{2}})\Big{]}V_{1,1}(z)$
	$\displaystyle\cdot$	$\displaystyle V_{0,3}(x,y,z)\left(V_{g-2,2}(x,y)+\sum_{(g_{1},g_{2})}V_{g_{1},1}(x)V_{g_{2},1}(y)\right)xyz\cdot dxdydz$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{0\leq x,y,z\leq L}\Big{[}(1+x)(1+e^{\frac{L-x-y}{2}})+(1+x)(1+e^{\frac{2L-x}{2}})\Big{]}$
	$\displaystyle\cdot$	$\displaystyle(1+z^{2})\left(V_{g-2,2}+\sum_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}\right)\sinh\frac{x}{2}\sinh\frac{y}{2}\cdot z\cdot dxdydz$
	$\displaystyle\prec$	$\displaystyle\frac{V_{g-2,2}+\sum\limits_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}}{V_{g}}\cdot\left(L^{5}e^{L}+L^{7}e^{\frac{L}{2}}+L^{6}e^{\frac{3}{2}L}\right),$

where $(g_{1},g_{2})$ are taken over all possible $1\leq g_{1},g_{2}$ and $g_{1}+g_{2}=g-1$ .

Similarly, by Mirzakhani’s integration formula, i.e. Theorem 4, Theorem 5, Theorem 8, and Theorem 13, we have that for $L>1$ ,

(79)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\Big{[}\sum_{(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,L]}(\ell(\alpha_{1}))\cdot 1_{[0,L]}(\ell(\beta_{1}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}\Big{]}$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{0\leq x,y,z,w\leq L}(1+y)(1+e^{\frac{2L-y}{2}})V_{0,3}(x,z,w)V_{0,3}(y,z,w)$
	$\displaystyle\cdot$	$\displaystyle\left(V_{g-2,2}(x,y)+\sum_{(g_{1},g_{2})}V_{g_{1},1}(x)V_{g_{2},1}(y)\right)xyzw\cdot dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{0\leq x,y,z,w\leq L}(1+y)(1+e^{\frac{2L-y}{2}})\left(V_{g-2,2}+\sum_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}\right)$
	$\displaystyle\cdot$	$\displaystyle\sinh\frac{x}{2}\sinh\frac{y}{2}\cdot zw\cdot dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{V_{g-2,2}+\sum\limits_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}}{V_{g}}\cdot\left(L^{5}e^{L}+L^{6}e^{\frac{3}{2}L}\right),$

where $(g_{1},g_{2})$ are taken over all possible $1\leq g_{1},g_{2}$ and $g_{1}+g_{2}=g-1$ .

For the rest term in the right side of (77), if we set

\textbf{cond}=\left\{\begin{aligned} &10\log L\leq x\leq L\\ &0\leq y\leq 4L-2x\\ &0\leq z,w\leq L\\ \end{aligned}\right.,

then it follows by Mirzakhani’s integration formula, i.e. Theorem 4, Theorem 5, Theorem 8, and Theorem 13 that for $L>1$ ,

(80)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\Big{[}\sum_{(\gamma_{1},\gamma_{2},\alpha_{1},\beta_{1})}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[0,4L]}(2\ell(\gamma_{1})+\ell(\gamma_{2}))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,L]}(\ell(\alpha_{1}))\cdot 1_{[0,L]}(\ell(\beta_{1}))\cdot\frac{200\ell(\gamma_{2})}{\mathcal{D}(\ell(\gamma_{2}),2L,0)}\Big{]}$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{\textbf{cond}}(1+y)(1+e^{\frac{2L-y}{2}})V_{0,3}(x,z,w)V_{0,3}(y,z,w)$
	$\displaystyle\cdot$	$\displaystyle\left(V_{g-2,2}(x,y)+\sum\limits_{(g_{1},g_{2})}V_{g_{1},1}(x)V_{g_{2},1}(y)\right)xyzw\cdot dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{\textbf{cond}}(1+y)(1+e^{\frac{2L-y}{2}})$
	$\displaystyle\cdot$	$\displaystyle\left(V_{g-2,2}+\sum_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}\right)\sinh\frac{x}{2}\sinh\frac{y}{2}\cdot zw\cdot dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{V_{g-2,2}+\sum_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}}{V_{g}}\cdot L^{5}\cdot\int_{\tiny{\begin{aligned} &10\log L\leq x\leq L\\ &0\leq y\leq 4L-2x\\ \end{aligned}}}(1+e^{\frac{2L-y}{2}})e^{\frac{x+y}{2}}dxdy$
	$\displaystyle\prec$	$\displaystyle\frac{V_{g-2,2}+\sum\limits_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}}{V_{g}}\cdot L^{5}\cdot\left(e^{2L-5\log L}+Le^{\frac{3}{2}L}\right),$

where $(g_{1},g_{2})$ are taken over all possible $1\leq g_{1},g_{2}$ and $g_{1}+g_{2}=g-1$ .

By Theorem 6 and Theorem 7, we have

(81)

\frac{V_{g-2,2}+\sum\limits_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}}{V_{g}}\prec\frac{1}{V_{g}}\left(V_{g-2,2}+\frac{W_{2g-4}}{g}\right)\prec\frac{1}{g^{2}}.

Then combining (77), (78), (79), (80) and (81) we get

		$\displaystyle\mathbb{E}_{\rm WP}^{g}\left[C_{1,2}(X,L)\right]\prec\frac{V_{g-2,2}+\sum\limits_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}}{V_{g}}\cdot(L^{5}e^{L}+L^{7}e^{\frac{L}{2}}+L^{6}e^{\frac{3L}{2}}+e^{2L})$
	$\displaystyle\prec$	$\displaystyle\frac{V_{g-2,2}+\frac{W_{2g-4}}{g}}{V_{g}}\cdot(L^{5}e^{L}+L^{7}e^{\frac{L}{2}}+L^{6}e^{\frac{3L}{2}}+e^{2L})\prec\frac{e^{2L}}{g^{2}}$

as desired. ∎

4.3.3. Bounds for $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{0,4}(X,L)\right]$

Our aim for $\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}(X,L)\right]$ is as follows. The proof is similar as the one in bounding $\mathbb{E}_{\rm WP}^{g}\left[C_{1,2}(X,L)\right]$ .

Proposition 28.

For $L>1$ and large $g$ ,

\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}(X,L)\right]\prec\frac{Le^{2L}}{g^{2}}.

When $S(\Gamma_{1},\Gamma_{2})\simeq S_{0,4}$ , two boundary geodesics of $S(\Gamma_{1},\Gamma_{2})$ may be the same closed geodesic in $X$ , in this case, the completion $\overline{S(\Gamma_{1},\Gamma_{2})}\simeq S_{1,2}$ ; otherwise $\overline{S(\Gamma_{1},\Gamma_{2})}\simeq S(\Gamma_{1},\Gamma_{2})\simeq S_{0,4}$ . Moreover, each of $\Gamma_{1}$ and $\Gamma_{2}$ has exactly two closed geodesics contained in the boundary of $S(\Gamma_{1},\Gamma_{2})$ . Now we define

	$\displaystyle\mathcal{C}_{0,4}^{0}(X,L):=\left\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}(X,L),\ \overline{S(\Gamma_{1},\Gamma_{2})}\simeq S_{0,4}\right\},$
	$\displaystyle\mathcal{C}_{0,4}^{1}(X,L):=\left\{(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}(X,L),\ \overline{S(\Gamma_{1},\Gamma_{2})}\simeq S_{1,2}\right\},$

and set

C_{0,4}^{0}(X,L)=\#\mathcal{C}_{0,4}^{0}(X,L),\ C_{0,4}^{1}(X,L)=\#\mathcal{C}_{0,4}^{1}(X,L).

For $(\Gamma_{1},\Gamma_{2})$ in $\mathcal{C}_{0,4}^{1}(X,L),$ view $S(\Gamma_{1},\Gamma_{2})$ as the result surface of cutting $\overline{S(\Gamma_{1},\Gamma_{2})}$ along a non-separating simple closed geodesic. Then the number of elements in $\mathcal{C}_{0,4}^{1}(X,L)$ has same estimations as in Lemma 25, Lemma 26 and Lemma 27. Therefore, the proof of Proposition 24 yields that

Proposition 29.

For $L>1$ and large $g$ ,

\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}^{1}(X,L)\right]\prec\frac{e^{2L}}{g^{2}}.

Now we consider $\mathcal{C}_{0,4}^{0}(X,L)$ . Again we need to accurately classify elements in it according to the relative position of $(\Gamma_{1},\Gamma_{2})$ in $S(\Gamma_{1},\Gamma_{2})\simeq S_{0,4}.$ The first one is as follows.

Lemma 30.

For $X\in\mathcal{M}_{g}$ and $L>1$ , we have

(82)		$\displaystyle\#\Big{\{}(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}^{0}(X,L):\Gamma_{1}\cup\Gamma_{2}\textit{ contains }\partial S(\Gamma_{1},\Gamma_{2})\Big{\}}$
	$\displaystyle\prec$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}1_{[0,L]}(\ell(\gamma_{1}))\cdot 1_{[0,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\gamma_{3}))\cdot 1_{[0,L]}(\ell(\gamma_{4}))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,L]}(\ell(\eta))\cdot 1_{[0,2L-10\log L]}(\ell(\gamma_{1})+\ell(\gamma_{2}))\cdot 1_{[0,2L-10\log L]}(\ell(\gamma_{3})+\ell(\gamma_{4}))$
	$\displaystyle\cdot$	$\displaystyle\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)},$

where $(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)$ are taken over all quintuples satisfying that $\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}\cup\gamma_{4}$ cuts off a subsurface $Y\simeq S_{0,4}$ in $X$ and $\eta$ bounds a $S_{0,3}$ in $Y$ along with $\gamma_{1},\gamma_{2}$ .

Proof.

For any $(\Gamma_{1},\Gamma_{2})$ belonging to the set in the left side of (82), WLOG, one may assume that $Y=S(\Gamma_{1},\Gamma_{2})$ , $\partial Y=\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\},$ and $\Gamma_{1}$ contains $\gamma_{1},\gamma_{2}$ . Then it follows that $\Gamma_{2}$ contains $\gamma_{3},\gamma_{4}$ . Assume the rest simple closed geodesic in $\Gamma_{1}$ is $\eta$ and the rest simple closed geodesic in $\Gamma_{2}$ is $\xi$ as shown in Figure 9. Consider the map

\pi:(\Gamma_{1},\Gamma_{2})\mapsto(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)

where $\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}\cup\gamma_{4}$ cuts off a subsurface $Y\simeq S_{0,4}$ in $X$ and $\eta$ bounds a $S_{0,3}$ along with $\gamma_{1},\gamma_{2}$ in $Y$ . Moreover their lengths satisfy

(83)

\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\gamma_{3}),\ell(\gamma_{4}),\ell(\eta)\in[10\log L,L]

and

(84)

\ell(\gamma_{1})+\ell(\gamma_{2})\leq 2L-10\log L,\ell(\gamma_{3})+\ell(\gamma_{4})\leq 2L-10\log L.

Then we count all possible $\Gamma_{2}$ ’s such that $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}^{0}(X,L)$ , $P(\Gamma_{2})\subset Y$ and $\{\gamma_{3},\gamma_{4}\}\subset\Gamma_{2}$ . We only need to count all possible $\xi$ ’s of length $\leq L$ , each of which bounds a $S_{0,3}$ in $Y$ along with $\gamma_{3},\gamma_{4}.$ It follows by (18) that there are at most

\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}

such $\xi$ ’s. So we have that

\#\pi^{-1}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)\leq\frac{200\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}.

Then the proof is completed by taking a summation over all possible quintuples $(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)$ ’s satisfying (83) and (84). ∎

Lemma 31.

For $X\in\mathcal{M}_{g}$ and $L>1$ , we have

(85)		$\displaystyle\#\Big{\{}(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}^{0}(X,L):\Gamma_{1}\cup\Gamma_{2}\textit{ contains }$
		$\displaystyle\textit{ exactly }3\textit{ boundary geodesics of }S(\Gamma_{1},\Gamma_{2})\Big{\}}$
	$\displaystyle\prec$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[0,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\gamma_{3}))\cdot 1_{[0,L]}(\ell(\eta))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,4L]}(2\ell(\gamma_{1})+\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{R}(\ell(\gamma_{2}),\ell(\gamma_{4}),L)},$

Proof.

For any $(\Gamma_{1},\Gamma_{2})$ belonging to the set in the left side of (85), WLOG, one may assume that $Y=S(\Gamma_{1},\Gamma_{2})$ , $\partial Y=\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\},$ $\Gamma_{1}$ contains $\gamma_{1},\gamma_{2}$ and $\Gamma_{2}$ contains $\gamma_{1},\gamma_{3}$ . Assume that the rest simple closed geodesic in $\Gamma_{1}$ is $\eta$ and the rest simple closed geodesic in $\Gamma_{2}$ is $\xi$ as shown in Figure 10.

In this case, $\ell(\gamma_{4})$ may exceed $L$ . However, since $P(\Gamma_{1})\cup P(\Gamma_{2})$ fills $Y$ , there is a connected component $C$ of $Y\setminus P(\Gamma_{1})\cup P(\Gamma_{2})$ such that $C$ is topologically a cylinder and $\gamma_{4}$ is a connected component of $\partial C$ . The other connected component of $\partial C$ , is the union of some geodesic arcs on $\eta,\xi.$ It follows that

(86)

\ell(\gamma_{4})\leq\ell(\eta)+\ell(\xi).

Since $\Gamma_{1},\Gamma_{2}\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L),$ then we have

(87)

\left\{\begin{aligned} \ell(\gamma_{1})+\ell(\gamma_{2})&+\ell(\eta)\leq 2L\\ \ell(\gamma_{1})+\ell(\gamma_{3})&+\ell(\xi)\leq 2L\\ \ell(\gamma_{1})&\geq 10\log L\\ \end{aligned}\right..

It follows from (86) and (87) that

(88)

\ell(\gamma_{1})+\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4})\leq 4L-\ell(\gamma_{1})\leq 4L-10\log L.

Consider the map

\pi:(\Gamma_{1},\Gamma_{2})\mapsto(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)

where $\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}\cup\gamma_{4}$ cuts off a $Y\simeq S_{0,4}$ in $X$ and $\eta$ cuts off a $S_{0,3}$ in $Y$ along with $\gamma_{1}\cup\gamma_{2}$ . Moreover their lengths satisfy

(89)			$\displaystyle\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\gamma_{3}),\ell(\eta)\in[10\log L,L],$
(89)			$\displaystyle 2\ell(\gamma_{1})+\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4})\leq 4L.$

Then we count all possible $\Gamma_{2}$ ’s such that $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}^{0}(X,L),\ P(\Gamma_{2})\subset Y$ and $\gamma_{1},\gamma_{3}\in\Gamma_{2}$ . We only need to count all possible $\xi$ ’s of length $\leq L$ , which bounds a $S_{0,3}$ in $Y$ along with $\gamma_{1},\gamma_{3}.$ It follows by (18) that there are at most

\frac{\ell(\gamma_{2})}{\mathcal{R}(\ell(\gamma_{2}),\ell(\gamma_{4}),L)}

such $\xi$ ’s. So we have that

\#\pi^{-1}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)\leq\frac{200\ell(\gamma_{2})}{\mathcal{R}(\ell(\gamma_{2}),\ell(\gamma_{4}),L)}.

Then the proof is completed by taking a summation over all possible quintuples $(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)$ ’s satisfying (89). ∎

Lemma 32.

For $X\in\mathcal{M}_{g}$ and $L>1$ , we have

(90)		$\displaystyle\#\Big{\{}(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}^{0}(X,L):\Gamma_{1}\cup\Gamma_{2}\textit{ contains }$
		$\displaystyle\textit{ exactly }2\textit{ boundary geodesics of }S(\Gamma_{1},\Gamma_{2})\Big{\}}$
	$\displaystyle\prec$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\eta))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,4L]}(2\ell(\gamma_{1})+2\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4}))\cdot\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)},$

Proof.

For any $(\Gamma_{1},\Gamma_{2})$ belonging to the set in the left side of (90), WLOG, one may assume that $Y=S(\Gamma_{1},\Gamma_{2})$ , $\partial Y=\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\}$ and both $\Gamma_{1},\Gamma_{2}$ contain $\gamma_{1},\gamma_{2}$ . Assume the rest simple closed geodesic in $\Gamma_{1}$ is $\eta$ and the rest simple closed geodesic in $\Gamma_{2}$ is $\xi$ . Then both $\eta$ and $\xi$ will bound a $S_{0,3}$ in $Y$ along with $\gamma_{1}\cup\gamma_{2}$ as shown in Figure 11.

In this case, both $\ell(\gamma_{3})$ and $\ell(\gamma_{4})$ may exceed $L$ . Since $P(\Gamma_{1})\cup P(\Gamma_{2})$ fills $Y$ , there are two connected components $C_{1},C_{2}$ of $Y\setminus P(\Gamma_{1})\cup P(\Gamma_{2})$ such that both $C_{1},C_{2}$ are topologically cylinders, $\gamma_{3}$ is a connected component of $\partial C_{1}$ and $\gamma_{4}$ is a connected component of $\partial C_{2}$ . The other connected components of $\partial C_{1}$ and $\partial C_{2}$ , are the union of different geodesic arcs on $\eta,\xi.$ It is clear that

(91)

\ell(\gamma_{3})+\ell(\gamma_{4})\leq\ell(\eta)+\ell(\xi).

Since $\Gamma_{1},\Gamma_{2}\in\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L),$ we have

(92)

\left\{\begin{aligned} \ell(\gamma_{1})+\ell(\gamma_{2})&+\ell(\eta)\leq 2L\\ \ell(\gamma_{1})+\ell(\gamma_{2})&+\ell(\xi)\leq 2L\\ \ell(\gamma_{1})&\geq 10\log L\\ \ell(\gamma_{2})&\geq 10\log L\\ \end{aligned}\right..

Then It follows from (91) and (92) that

\ell(\gamma_{1})+\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4})\leq 4L-\ell(\gamma_{1})-\ell(\gamma_{2})\leq 4L-20\log L.

Consider the map

\pi:(\Gamma_{1},\Gamma_{2})\mapsto(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)

(93)			$\displaystyle\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\eta)\in[10\log L,L],$
(93)			$\displaystyle 2\ell(\gamma_{1})+2\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4})\leq 4L.$

Then we count all possible $\Gamma_{2}$ ’s such that $(\Gamma_{1},\Gamma_{2})\in\mathcal{C}_{0,4}^{0}(X,L),P(\Gamma_{2})\subset Y,\gamma_{1},\gamma_{2}\in\Gamma_{2}$ . We only need to count all possible $\xi$ ’s of length $\leq L$ , each of which bounds a $S_{0,3}$ in $Y$ along with $\gamma_{1},\gamma_{2}.$ It follows by (18) that there are at most

\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}

such $\xi$ ’s. So we have

\#\pi^{-1}(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)\leq\frac{200\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}.

Sum it over all possible quintuples $(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)$ ’s satisfying (93), we complete the proof. ∎

For $Y\simeq S_{0,4}$ , the complement subsurface $X\setminus\overline{Y}$ can be one of five types: (1) $S_{g-3,4}$ ; (2) $S_{g_{1},1}\cup S_{g_{2},3}$ with $g_{1}\geq 1$ and $g_{1}+g_{2}=g-2$ ; (3) $S_{g_{1},1}\cup S_{g_{2},1}\cup S_{g_{3},2}$ with $g_{1},g_{2},g_{3}\geq 1$ and $g_{1}+g_{2}+g_{3}=g-1$ ; (4) $S_{g_{1},1}\cup S_{g_{2},1}\cup S_{g_{3},1}\cup S_{g_{4},1}$ with $g_{1},g_{2},g_{3},g_{4}\geq 1$ and $g_{1}+g_{2}+g_{3}+g_{4}=g$ ; (5) $S_{g_{1},2}\cup S_{g_{2},2}$ with $g_{1},g_{2}\geq 1$ and $g_{1}+g_{2}=g-2$ . Set

\mathop{\rm Vol_{\rm WP}}\big{(}\mathcal{M}\left(S_{g}\setminus\overline{Y},\ell(\gamma_{1})=x,\ell(\gamma_{2})=y,\ell(\gamma_{3})=z,\ell(\gamma_{4})=w\right)\big{)}

to be the Weil-Petersson volume of the moduli space of Riemann surfaces each of which is homeomorphic to $S_{g}\setminus\overline{Y}$ with geodesic boundaries of lengths $\ell(\gamma_{1})=x,\ell(\gamma_{2})=y,\ell(\gamma_{3})=z,\ell(\gamma_{4})=w$ . Define

		$\displaystyle V^{\Sigma}(S_{g}\setminus\overline{Y},x,y,z,w)=\sum_{\textit{all types of }S_{g}\setminus\overline{Y}}\mathop{\rm Vol_{\rm WP}}\Big{(}\mathcal{M}\big{(}S_{g}\setminus\overline{Y},$
		$\displaystyle\ell(\gamma_{1})=x,\ell(\gamma_{2})=y,\ell(\gamma_{3})=z,\ell(\gamma_{4})=w\big{)}\Big{)}$

and

V^{\Sigma}(S_{g}\setminus\overline{Y})=V^{\Sigma}(S_{g}\setminus\overline{Y},0,0,0,0)=\sum_{\textit{all types of }S_{g}\setminus\overline{Y}}\mathop{\rm Vol_{\rm WP}}\big{(}\mathcal{M}\left(S_{g}\setminus\overline{Y}\right)\big{)}.

Then by Theorem 6 and Theorem 7 we have

(94)		$\displaystyle V^{\Sigma}(S_{g}\setminus\overline{Y})$
	$\displaystyle\prec$	$\displaystyle V_{g-3,4}+\sum_{g_{1}+g_{2}=g-2}V_{g_{1},1}V_{g_{2},3}+\sum_{g_{1}+g_{2}+g_{3}=g-1}V_{g_{1},1}V_{g_{2},1}V_{g_{3},2}$
	$\displaystyle+$	$\displaystyle\sum_{g_{1}+g_{2}+g_{3}+g_{4}=g}V_{g_{1},1}V_{g_{2},1}V_{g_{3},1}V_{g_{4},1}+\sum_{g_{1}+g_{2}=g-2}V_{g_{1},2}V_{g_{2},2}$
	$\displaystyle\prec$	$\displaystyle W_{2g-4}\left(1+\frac{1}{g}+\frac{1}{g^{2}}+\frac{1}{g^{3}}\right)\prec\frac{V_{g}}{g^{2}}.$

Now we are ready to bound $\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}^{0}(X,L)\right]$ .

Proposition 33.

For $L>1$ and large $g$ ,

\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}^{0}(X,L)\right]\prec\frac{Le^{2L}}{g^{2}}.

Proof.

Following Lemma 30, Lemma 31 and Lemma 32, we have that for $L>1$ ,

(95)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\Big{[}C_{0,4}^{0}(X,L)\Big{]}$
	$\displaystyle\leq$	$\displaystyle\mathbb{E}_{\rm WP}^{g}\Bigg{[}\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}\Big{(}1_{[0,L]}(\ell(\gamma_{1}))\cdot 1_{[0,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\gamma_{3}))\cdot 1_{[0,L]}(\ell(\gamma_{4}))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,L]}(\ell(\eta))\cdot 1_{[0,2L-10\log L]}(\ell(\gamma_{1})+\ell(\gamma_{2}))\cdot 1_{[0,2L-10\log L]}(\ell(\gamma_{3})+\ell(\gamma_{4}))$
	$\displaystyle\cdot$	$\displaystyle\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}$
	$\displaystyle+$	$\displaystyle 1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[0,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\gamma_{3}))\cdot 1_{[0,L]}(\ell(\eta))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,4L]}(2\ell(\gamma_{1})+\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{R}(\ell(\gamma_{2}),\ell(\gamma_{4}),L)}$
	$\displaystyle+$	$\displaystyle 1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[10\log L,L]}(\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\eta))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,4L]}(2\ell(\gamma_{1})+2\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4}))\cdot\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}\Big{)}\Bigg{]},$

\textbf{cond}_{1}=\left\{\begin{aligned} &0\leq x,y,z,w,v\leq L\\ &x+y\leq 2L-10\log L\\ &z+w\leq 2L-10\log L\\ \end{aligned}\right..

Then by Mirzakhani’s integration formula Theorem 4, Theorem 5, Theorem 8 and Theorem 13, we have that for $L>1$ ,

(96)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\Bigg{[}\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}\Big{(}1_{[0,L]^{5}}\left(\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\gamma_{3}),\ell(\gamma_{4}),\ell(\eta)\right)$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,2L-10\log L]^{2}}\left(\ell(\gamma_{1})+\ell(\gamma_{2}),\ell(\gamma_{3})+\ell(\gamma_{4})\right)\cdot\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}\Big{)}\Bigg{]}$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{\textbf{cond}_{1}}(1+z)(1+e^{\frac{L-z-w}{2}})V_{0,3}(x,y,v)V_{0,3}(z,w,v)$
	$\displaystyle\cdot$	$\displaystyle V^{\Sigma}(S_{g}\setminus\overline{Y},x,y,z,w)\cdot xyzwv\cdot dxdydzdwdv$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{\textbf{cond}_{1}}(1+z)(1+e^{\frac{L-z-w}{2}})\cdot V^{\Sigma}(S_{g}\setminus\overline{Y})$
	$\displaystyle\cdot$	$\displaystyle\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}\sinh\frac{w}{2}\cdot v\cdot dxdydzdwdv$
	$\displaystyle\prec$	$\displaystyle\frac{V^{\Sigma}(S_{g}\setminus\overline{Y})}{V_{g}}\cdot L^{3}\cdot\int_{\tiny\begin{aligned} x+y&\leq 2L-10\log L\\ z+w&\leq 2L-10\log L\\ \end{aligned}}(1+e^{\frac{L-z-w}{2}})e^{\frac{x+y+z+w}{2}}dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{V^{\Sigma}(S_{g}\setminus\overline{Y})}{V_{g}}\cdot L^{3}\cdot(L^{2}e^{2L-10\log L}+L^{3}e^{\frac{3L}{2}-5\log L}).$

Set

\textbf{cond}_{2}=\left\{\begin{aligned} &0\leq y,z,v\leq L,\quad 10\log L\leq x\leq L\\ &2x+y+z+w\leq 4L\\ \end{aligned}\right.

and

\textbf{cond}_{3}=\left\{\begin{aligned} &0\leq v\leq L,\quad 10\log L\leq x,y\leq L\\ &2x+2y+z+w\leq 4L\\ \end{aligned}\right..

Similarly, by Mirzakhani’s integration formula Theorem 4, Theorem 5, Theorem 8 and Theorem 13, we have

(97)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\Bigg{[}\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}\Big{(}1_{[10\log L,L]}(\ell(\gamma_{1}))\cdot 1_{[0,L]^{3}}(\ell(\gamma_{2}),\ell(\gamma_{3}),\ell(\eta))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,4L]}(2\ell(\gamma_{1})+\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4}))\cdot\frac{\ell(\gamma_{2})}{\mathcal{R}(\ell(\gamma_{2}),\ell(\gamma_{4}),L)}\Big{)}\Bigg{]}$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{\textbf{cond}_{2}}(1+y)(1+e^{\frac{L-y-w}{2}})\cdot V^{\Sigma}(S_{g}\setminus\overline{Y})$
	$\displaystyle\cdot$	$\displaystyle\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}\sinh\frac{w}{2}\cdot v\cdot dxdydzdwdv$
	$\displaystyle\prec$	$\displaystyle\frac{V^{\Sigma}(S_{g}\setminus\overline{Y})}{V_{g}}\cdot L^{3}\cdot\int_{\tiny\begin{aligned} x,y,z&\leq L\\ x+y+z+w&\leq 4L-10\log L\\ \end{aligned}}(1+e^{\frac{L-y-w}{2}})e^{\frac{x+y+z+w}{2}}dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{V^{\Sigma}(S_{g}\setminus\overline{Y})}{V_{g}}\cdot L^{3}\cdot(L^{3}e^{2L-5\log L}+L^{3}e^{\frac{3L}{2}})$

and

(98)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\Bigg{[}1_{[10\log L,L]^{2}}(\ell(\gamma_{1}),\ell(\gamma_{2}))\cdot 1_{[0,L]}(\ell(\eta))$
	$\displaystyle\cdot$	$\displaystyle 1_{[0,4L]}(2\ell(\gamma_{1})+2\ell(\gamma_{2})+\ell(\gamma_{3})+\ell(\gamma_{4}))\cdot\frac{\ell(\gamma_{3})}{\mathcal{R}(\ell(\gamma_{3}),\ell(\gamma_{4}),L)}\Big{)}\Bigg{]}$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{\textbf{cond}_{3}}(1+z)(1+e^{\frac{L-z-w}{2}})\cdot V^{\Sigma}(S_{g}\setminus\overline{Y},x,y,z,w)$
	$\displaystyle\cdot$	$\displaystyle\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}\sinh\frac{w}{2}\cdot v\cdot dxdydzdwdv$
	$\displaystyle\prec$	$\displaystyle\frac{V^{\Sigma}(S_{g}\setminus\overline{Y})}{V_{g}}\cdot L^{3}\cdot\int_{\tiny\begin{aligned} x,y&\leq L\\ x+y+z+w&\leq 4L-20\log L\\ \end{aligned}}(1+e^{\frac{L-z-w}{2}})e^{\frac{x+y+z+w}{2}}dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{V^{\Sigma}(S_{g}\setminus\overline{Y})}{V_{g}}\cdot L^{3}\cdot(L^{3}e^{2L-10\log L}+L^{3}e^{\frac{3L}{2}}).$

Then combining (95), (96), (97) and (98) we have for $L>1$ ,

(99)

\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}^{0}(X,L)\right]\prec\frac{V^{\Sigma}(S_{g}\setminus\overline{Y})}{V_{g}}\cdot Le^{2L}.

Therefore, by (94) and (99) we obtain

\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}^{0}(X,L)\right]=O\left(\frac{Le^{2L}}{g^{2}}\right)

as desired. ∎

Now we are ready to prove Proposition 28.

Proof of Proposition 28.

Since $\mathcal{C}_{0,4}(X,L)=\mathcal{C}_{0,4}^{0}(X,L)\cup\mathcal{C}_{0,4}^{1}(X,L)$ , it follows by Proposition 29 and Proposition 33 that

\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}(X,L)\right]\leq\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}^{1}(X,L)\right]+\mathbb{E}_{\rm WP}^{g}\left[C_{0,4}^{0}(X,L)\right]\prec\frac{Le^{2L}}{g^{2}}.

This completes the proof. ∎

4.4. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[D(X,L)\right]$

For this part, we always assume that $g>2$ . We will show that as $g\to\infty$ ,

\mathbb{E}_{\rm WP}^{g}\left[D(X,L_{g})\right]=o\left(\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}\right).

More precisely,

Proposition 34.

For $L>1$ , we have

\mathbb{E}_{\rm WP}^{g}\left[D(X,L)\right]\prec\frac{e^{2L}}{g^{2}L^{6}}.

Proof.

For $(\Gamma_{1},\Gamma_{2})\in\mathcal{D}(X,L)$ , the two pairs of pants $P(\Gamma_{1})$ and $P(\Gamma_{2})$ will share one or two simple closed geodesic boundary components. For the first case, assume that $\Gamma_{1}=(\gamma_{1},\gamma_{2},\eta)$ and $\Gamma_{2}=(\gamma_{3},\gamma_{4},\eta)$ . For the second case, assume that $\Gamma_{1}=(\gamma_{1},\alpha,\beta)$ and $\Gamma_{2}=(\gamma_{2},\alpha,\beta)$ . By the definition of $\mathcal{N}_{(0,3),\star}^{(g-2,3)}(X,L)$ , any two simple closed geodesics in $\Gamma_{1}$ have total length $\leq 2L-10\log L$ . So does $\Gamma_{2}$ . We have

(100)		$\displaystyle D(X,L)$
	$\displaystyle\prec$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}1_{[10\log L,L]}(\ell(\eta))\cdot 1_{[0,2L-10\log L]^{2}}\Big{(}\ell(\gamma_{1})+\ell(\gamma_{2}),\ell(\gamma_{3})+\ell(\gamma_{4})\Big{)}$
	$\displaystyle+$	$\displaystyle\sum_{(\gamma_{1},\gamma_{2},\alpha,\beta)}1_{[0,L]^{4}}\Big{(}\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\alpha),\ell(\beta)\Big{)}.$

Here $(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)$ are taken over all quintuples satisfying that $\gamma_{1}\cup\gamma_{2}\cup\gamma_{3}\cup\gamma_{4}$ cuts off a subsurface $Y\simeq S_{0,4}$ in $X$ and $\eta$ bounds a $S_{0,3}$ in $Y$ along with $\gamma_{1},\gamma_{2}$ ; and while $(\gamma_{1},\gamma_{2},\alpha,\beta)$ are taken over all quadruples satisfying that $\gamma_{1}\cup\gamma_{2}$ cuts off a subsurface $Y\simeq S_{1,2}$ in $X$ and $\alpha\cup\beta$ separates $Y$ into $S_{0,3}\cup S_{0,3}$ with $\gamma_{1},\gamma_{2}$ belonging to the boundaries of the two different $S_{0,3}$ ’s. Set

\textbf{cond}_{4}=\left\{\begin{matrix}&0\leq v\leq L\\ &x+y\leq 2L-10\log L\\ &z+w\leq 2L-10\log L\\ \end{matrix}.\right.

By Mirzakhani’s integration formula Theorem 4, Theorem 5, Theorem 8 and (94), for $L>1$ we have

(101)			$\displaystyle\mathbb{E}_{\rm WP}^{g}\Bigg{[}\sum_{(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\eta)}1_{[10\log L,L]}(\ell(\eta))$
			$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\times 1_{[0,2L-10\log L]^{2}}\Big{(}\ell(\gamma_{1})+\ell(\gamma_{2}),\ell(\gamma_{3})+\ell(\gamma_{4})\Big{)}\Bigg{]}$
			$\displaystyle\prec\frac{1}{V_{g}}\int_{\textbf{cond}_{4}}V_{0,3}(x,y,v)V_{0,3}(z,w,v)$
			$\displaystyle\cdot\sum_{\textit{type of }S_{g}\setminus\overline{Y},Y\simeq S_{0,4}}\mathop{\rm Vol}\left(\mathcal{M}(S_{g}\setminus\overline{Y},\ell(\gamma_{1})=x,\ell(\gamma_{2})=y,\ell(\gamma_{3})=z,\ell(\gamma_{4})=w)\right)$
			$\displaystyle\cdot xyzwv\cdot dxdydzdwdv$
			$\displaystyle\prec\frac{1}{g^{2}}\int_{\textbf{cond}_{4}}\sinh\frac{x}{2}\sinh\frac{y}{2}\sinh\frac{z}{2}\sinh\frac{w}{2}\cdot v\cdot dxdydzdwdv$
			$\displaystyle\prec\frac{e^{2L}}{g^{2}L^{6}}.$

Where in the last inequality we apply

	$\displaystyle\int_{x>0,y>0,x+y\leq 2L-10\log L}\sinh\frac{x}{2}\sinh\frac{y}{2}dxdy$
	$\displaystyle\prec\int_{x>0,y>0,x+y\leq 2L-10\log L}e^{\frac{x+y}{2}}dxdy$
	$\displaystyle\prec\int_{0}^{2L-10\log L}\left(e^{\frac{x}{2}}\int_{0}^{2L-10\log L-x}e^{\frac{y}{2}}dy\right)dx\asymp\frac{e^{L}}{L^{4}}.$

For the remaining term, it follows by Mirzakhani’s integration formula Theorem 4, Theorem 5, Theorem 8 and (81) that for $L>1$ ,

(102)		$\displaystyle\mathbb{E}_{\rm WP}^{g}\left[\sum_{(\gamma_{1},\gamma_{2},\alpha,\beta)}1_{[0,L]^{4}}\Big{(}\ell(\gamma_{1}),\ell(\gamma_{2}),\ell(\alpha),\ell(\beta)\Big{)}\right]$
	$\displaystyle\prec$	$\displaystyle\frac{1}{V_{g}}\int_{[0,L]^{4}}V_{0,3}(x,z,w)V_{0,3}(y,z,w)$
	$\displaystyle\cdot$	$\displaystyle\left(V_{g-2,2}(x,y)+\sum_{(g_{1},g_{2})}V_{g_{1},1}(x)V_{g_{2},1}(y)\right)\cdot xyzw\cdot dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{V_{g-2,2}+\sum_{(g_{1},g_{2})}V_{g_{1},1}V_{g_{2},1}}{V_{g}}\int_{[0,L]^{4}}\sinh\frac{x}{2}\sinh\frac{y}{2}\cdot zw\cdot dxdydzdw$
	$\displaystyle\prec$	$\displaystyle\frac{L^{4}e^{L}}{g^{2}}.$

Combine (100), (101) and (102), we have

\mathbb{E}_{\rm WP}^{g}\left[D(X,L)\right]\prec\frac{1}{g^{2}}\left(\frac{e^{2L}}{L^{6}}+L^{4}e^{L}\right)\prec\frac{e^{2L}}{g^{2}L^{6}}.

This completes the proof. ∎

4.5. Finish of the proof

Now we are ready to complete the proof of Theorem 19.

Proof of Theorem 19.

Take $L=L_{g}=\log g-\log\log g+\omega(g)>1$ with $\omega(g)=o(\log\log g)$ . By (37) and (38) we have

(103)		$\displaystyle\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{(0,3),\star}^{(g-2,3)}(X,L_{g})=0\right)$
	$\displaystyle\leq$	$\displaystyle\frac{\left\|\mathbb{E}_{\rm WP}^{g}\left[B(X,L_{g})\right]-\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}\right\|}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}}$
	$\displaystyle+$	$\displaystyle\frac{\mathbb{E}_{\rm WP}^{g}\left[A(X,L_{g})\right]+\mathbb{E}_{\rm WP}^{g}\left[C(X,L_{g})\right]+\mathbb{E}_{\rm WP}^{g}\left[D(X,L_{g})\right]}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}}.$

By Proposition 20 and Proposition 21 we have

(104)

\frac{\left|\mathbb{E}_{\rm WP}^{g}\left[B(X,L_{g})\right]-\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}\right|}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}}=O\left(\frac{\log L_{g}}{L_{g}}\right).

By (39) and Proposition 20, for $L=L_{g}=\log g-\log\log g+\omega(g)$ we have

(105)

\frac{\mathbb{E}_{\rm WP}^{g}\left[A(X,L_{g})\right]}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}}\prec\frac{1}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]}=O\left(\frac{1}{e^{\omega(g)}}\right).

By Proposition 20, Proposition 23, Proposition 24 and Proposition 28, fix $0<\epsilon<\frac{1}{2}$ , we have

(106)

\frac{\mathbb{E}_{\rm WP}^{g}\left[C(X,L_{g})\right]}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}}\prec\left(\frac{L_{g}^{65}e^{\epsilon L_{g}}}{g}+\frac{L_{g}e^{6L_{g}}}{g^{9}}+\frac{1+L_{g}}{L_{g}^{2}}\right)=O\left(\frac{1}{L_{g}}\right).

By Proposition 20 and Proposition 34 we have

(107)

\frac{\mathbb{E}_{\rm WP}^{g}\left[D(X,L_{g})\right]}{\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L_{g})\right]^{2}}=O\left(\frac{1}{L_{g}^{8}}\right).

Therefore if $\lim\limits_{g\to\infty}\omega(g)=\infty$ , by (103), (104), (105), (106) and (107), we have

\lim_{g\to\infty}\mathop{\rm Prob}\nolimits_{\rm WP}^{g}\left(X\in\mathcal{M}_{g};\ N_{(0,3),\star}^{(g-2,3)}(X,L_{g})=0\right)=0.

It follows that (35) holds by (36). This finishes the proof of Theorem 19. ∎

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Non-simple systoles on random hyperbolic surfaces for large genus

Abstract.

1. Introduction

Theorem 1.

Remark.

Theorem 2.

Proof.

Remark.

2. Preliminaries

2.1. Moduli space and Weil-Petersson metric

Theorem 3 (Wolpert).

2.2. Mirzakhani’s integration formula

Theorem 4 (Mirzakhani’s integration formula).

Remark.

2.3. Weil-Petersson volumes

Theorem 5 ([Mir07a, Theorem 1.1]).

Theorem 6.

Theorem 7 ([NWX23, Lemma 24]).

Theorem 8 ([NWX23, Lemma 20]).

2.4. Figure-eight closed geodesics

Remark.

Lemma 9.

Outline of the proof of Lemma 9.

Remark.

2.5. Three countings on closed geodesics

2.5.1. On closed hyperbolic surfaces

Theorem 10.

2.5.2. On compact hyperbolic surfaces with geodesic boundaries

Definition.

Theorem 11 ([WX22b, Theorem 4] or [WX22a, Theorem 18]).

2.5.3. On S1,2S_{1,2} and S0,4S_{0,4}

Theorem 12 ([Mir07a, Theorem 1.3]).

Lemma 13 ([NWX23, Lemma 27]).

Theorem 14.

Proof.

3. Lower bound

Proposition 15.

Theorem 16.

Proof.

Lemma 17.

Proof.

Remark.

Lemma 18.

Proof.

Proof of Proposition 15.

4. Upper bound

Theorem 19.

Definition.

4.1. Estimations of 𝔼WPg​[N(0,3),⋆(g−2,3)​(X,L)]\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]

Proposition 20.

Proof.

4.2. Estimations of 𝔼WPg​[B​(X,L)]\mathbb{E}_{\rm WP}^{g}\left[B(X,L)\right]

Proposition 21.

Proof.

4.3. Estimations of 𝔼WPg​[C​(X,L)]\mathbb{E}_{\rm WP}^{g}\left[C(X,L)\right]

Construction.

Lemma 22.

Proof.

4.3.1. Bounds for 𝔼WPg​[C≥3​(X,L)]\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{\geq 3}(X,L)\right]

Proposition 23.

Proof.

4.3.2. Bounds for 𝔼WPg​[C1,2​(X,L)]\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{1,2}(X,L)\right]

Proposition 24.

Lemma 25.

Proof.

Remark.

Lemma 26.

Proof.

Lemma 27.

Proof.

Proof of Proposition 24.

4.3.3. Bounds for 𝔼WPg​[C0,4​(X,L)]\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{0,4}(X,L)\right]

Proposition 28.

Proposition 29.

Lemma 30.

Proof.

Lemma 31.

Proof.

Lemma 32.

Proof.

2.5.3. On $S_{1,2}$ and $S_{0,4}$

4.1. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[N_{(0,3),\star}^{(g-2,3)}(X,L)\right]$

4.2. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[B(X,L)\right]$

4.3. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[C(X,L)\right]$

4.3.1. Bounds for $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{\geq 3}(X,L)\right]$

4.3.2. Bounds for $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{1,2}(X,L)\right]$

4.3.3. Bounds for $\mathbb{E}_{\textnormal{WP}}^{g}\left[C_{0,4}(X,L)\right]$

4.4. Estimations of $\mathbb{E}_{\rm WP}^{g}\left[D(X,L)\right]$