A distance exponent for Liouville quantum gravity

Let $\gamma\in(0,2)$ and let $D\subset\mathbbm{C}$ be a simply connected domain. Heuristically speaking, a $\gamma$-Liouville quantum gravity (LQG) surface is the surface parametrized by $D$ whose Riemannian metric tensor is given by

where $h$ is some variant of the Gaussian free field [She07] on $D$, $dx^{2}+dy^{2}$ is the Euclidean metric tensor, and $d_{\gamma}$ is the so-called dimension of $\gamma$-LQG. LQG is a natural model of a continuum random surface. One reason for this is that LQG is the conjectured scaling limit of various random planar map models, the most natural discrete random surfaces. The case when $\gamma=\sqrt{8/3}$ corresponds to pure gravity, which is the scaling limit of uniform random planar maps. Other values of $\gamma$ arise from random planar maps weighted by the partition function of some statistical mechanics model, e.g., the uniform spanning tree ($\gamma=\sqrt{2}$), the Ising model ($\gamma=\sqrt{3}$), or a bipolar orientation ($\gamma=\sqrt{4/3}$). For many such models, it is expected that the scaling limit of the statistical mechanics model on the planar map is described by an $\operatorname{SLE}_{\kappa}$-type curve [Sch00] or a family of such curves, independent from the LQG surface, for $\kappa=\gamma^{2}$ or $\kappa=16/\gamma^{2}$.

Since $h$ is a distribution, or generalized function, and is not well-defined pointwise, the formula (1.1) does not make rigorous sense. However, one can rigorously construct the volume form associated with the metric (1.1), which should be a regularized version of $e^{\gamma h(z)}\,dz$, where $dz$ is the Euclidean volume form. This was accomplished in [DS11], where it was shown that several different regularization procedures for $e^{\gamma h(z)}\,dz$ converge to the same limiting measure $\mu_{h}$, the $\gamma$-quantum area measure induced by $h$. See also [RV14a] and the references therein for a more general theory of regularized random measures. The procedure used in [DS11] also allows one to define a length measure $\nu_{h}$ on certain curves in $D\cup\partial D$ (including $\partial D$ and independent SLE_κ-type curves for $\kappa=\gamma^{2}$).

A major problem in the study of LQG is to make sense of (1.1) as a random metric (distance function). This has recently been accomplished in the special case when $\gamma=\sqrt{8/3}$ by Miller and Sheffield in the series of works [MS16f, MS15c, MS15a, MS15b, MS16b, MS16c], using a random growth process called quantum Loewner evolution. For certain special types of quantum surfaces defined in [DMS14], the resulting metric space is isometric to a certain Brownian surface, a random metric space which locally looks like the Brownian map [Le 13, Mie13] and which arises as the scaling limit of certain uniform random planar maps. For example, the quantum sphere is isometric to the Brownian map and the $\sqrt{8/3}$-quantum cone is isometric to the Brownian plane [CL14]. In the case when $\gamma\not=\sqrt{8/3}$, the problem of constructing a LQG metric remains open.

Another major problem is to determine the Hausdorff dimension of the $\gamma$-LQG metric, assuming that it exists. In the case when $\gamma=\sqrt{8/3}$ it is known that this dimension is 4 [Le 07]. For general $\gamma$ it is predicted by Watabiki [Wat93] that the dimension $d_{\gamma}$ of $\gamma$-LQG is a.s. given by

There have been several works which support the Watabiki prediction. The authors of [AB14] perform numerical simulations using the discrete GFF which agree with the formula (1.2). In [MS16f, Section 3.3], the authors give an alternative non-rigorous derivation of (1.2) using so-called quantum Loewner evolution processes. The works [MRVZ16, AK16] prove upper and lower bounds for the Liouville heat kernel. If one assumes a certain relationship between two exponents (which the authors of the mentioned papers are not able to verify), these estimates suggest upper and lower bounds for the LQG dimension; this will be discussed further in Section 1.6. There is also a related quantity for LQG, called the spectral dimension, which is expected to be equal to 2 for all values of $\gamma$ [ANR⁺98]. This prediction is confirmed in the context of the Liouville heat kernel in [RV14b, AK16] and for a class of random planar maps in [GM17b].

In contrast to the above results, the recent work [DG16b] proves estimates for several natural approximations of the $\gamma$-LQG metric (different from the approximations considered in this paper) for small values of $\gamma$ which contradict the Watabiki prediction; c.f. Section 1.6.

If the dimension of LQG is $d_{\gamma}$, it is expected that the diameter (with respect to the graph distance) of a random planar map with $n$ edges which converges in the scaling limit to LQG is typically of order $n^{1/d_{\gamma}+o_{n}(1)}$. Hence computing the dimension of LQG is expected to be equivalent to computing the tail exponent for the diameter of a random planar map in the LQG universality class.

The goal of this article is to present some small progress toward the above two problems. Miller and Sheffield’s approach in the case $\gamma=\sqrt{8/3}$ does not have a direct generalization to other values of $\gamma$, since it relies on special symmetries which are specific to $\gamma=\sqrt{8/3}$. Instead, we will use a different approach based on the peanosphere construction of [DMS14], which applies for all $\gamma\in(0,2)$ and which we will now describe.

A peanosphere is a random pair $(M,\eta)$ consisting of a topological space $M$ and a space-filling curve $\eta$ on $M$ (with a specified parametrization) which is constructed from a correlated two-sided two-dimensional Brownian motion (see Figure 4). A peanosphere has a natural volume measure, which is defined by the condition that $\eta$ traces one unit of mass in one unit of time.

It is shown in [DMS14, Theorems 1.13 and 1.14] that there is a canonical (up to rotation) way to embed a peanosphere into $\mathbbm{C}$ in such a way that the following is true. The peanosphere volume measure is mapped to the $\gamma$-quantum area measure corresponding to a particular type of $\gamma$-LQG surface $(\mathbbm{C},h,0,\infty)$ called a $\gamma$-quantum cone [DMS14, Definition 4.9]. Note that here $h$ is a variant of the GFF on $\mathbbm{C}$. The curve $\eta$ is mapped to a space-filling variant¹¹1Our $\kappa$ corresponds to the parameter $\kappa^{\prime}$ in [MS17, DMS14]. of SLE_κ, $\kappa=16/\gamma^{2}>4$ from $\infty$ to $\infty$ [MS17, Sections 1.2.3 and 4.3] which is sampled independently from $h$ and then parametrized in such a way that $\eta(0)=0$ and the $\gamma$-quantum area $\mu_{h}(\eta([t_{1},t_{2}]))$ is equal to $t_{2}-t_{1}$ for each $t_{1}<t_{2}$. The correlation of the peanosphere Brownian motion is given by $-\cos(\pi\gamma^{2}/4)$ (for $\gamma<\sqrt{2}$, this is proven in [GHMS17]). The two coordinates of this Brownian motion give the net change in the quantum lengths of the left and right sides of $\eta$ relative to time 0, respectively, so are denoted by $L_{t}$ and $R_{t}$ for $t\in\mathbbm{R}$. We write $Z_{t}=(L_{t},R_{t})$ for the peanosphere Brownian motion. See Section 2.1.2 and the references therein for more background on the above objects.

In this article, we will study a family of planar maps $\{\mathcal{G}^{\epsilon}\}_{\epsilon>0}$, called the LQG structure graphs(also called mated-CRT maps) associated with the pair $(h,\eta)$, which we expect converges in the scaling limit in the Gromov-Hausdorff sense (when equipped with their graph distances) to an LQG metric induced by $h$.

The vertices of the structure graph $\mathcal{G}^{\epsilon}$ are the elements of $\epsilon\mathbbm{Z}$. Two such vertices $x_{1}$ and $x_{2}$ are connected by an edge if the corresponding cells $\eta([x_{1}-\epsilon,x_{1}])$ and $\eta([x_{2}-\epsilon,x_{2}])$ intersect along a non-trivial boundary arc (i.e., a connected set with more than one point).²²2It is easy to see that $\mathcal{G}^{\epsilon}$ is a planar map with the embedding given by mapping each vertex to the corresponding cell. In fact, $\mathcal{G}^{\epsilon}$ is a triangulation provided we draw two edges instead of one between the cells $\eta([x_{1}-\epsilon,x_{1}])$ and $\eta([x_{2}-\epsilon,x_{2}])$ whenever $|x_{1}-x_{2}|>\epsilon$ and these cells intersect along a non-trivial arc of each of their left and right boundaries (equivalently, both conditions in (1.2) hold); see, the introduction of [GMS17b] for more details. In this paper we only care about graph distances in $\mathcal{G}^{\epsilon}$, so these facts will not be relevant for us. Note that this means that $x\in\epsilon\mathbbm{Z}$ corresponds to the time interval $[x-\epsilon,x]$ for $\eta$. See Figure 1 for an illustration of the above definition.

The peanoshpere construction of [DMS14] implies that a.s. $\mathcal{G}^{\epsilon}$ can equivalently be defined as the graph with vertex set $\epsilon\mathbbm{Z}$, with $x_{1},x_{2}\in\epsilon\mathbbm{Z}$ with $x_{1}<x_{2}$ are connected by an edge if and only if either

See Figure 2, right, for an illustration of this adjacency condition. We note that Brownian scaling shows that the law of $\mathcal{G}^{\epsilon}$ (as a graph) does not depend on $\epsilon$, but it is convenient to view the family of graphs $\{\mathcal{G}^{\epsilon}\}_{\epsilon>0}$ as being coupled together with the same Brownian motion.

The formula (1.2) tells us that $\mathcal{G}^{\epsilon}$ is a discretization of the mating of the two continuum random trees constructed from the Brownian motions $L$ and $R$ (which is the reason for the term “mated-CRT map”). In particular, the LQG structure graphs can be defined using only Brownian motion, without any reference to LQG or SLE. In fact, all of the arguments of this paper except for the ones in Section 3 can be phrased entirely in terms of Brownian motion.

One of the main reasons for our interest in the graphs $\mathcal{G}^{\epsilon}$ is the following conjecture.

The recent work [GMS17b] shows that random walk on $\mathcal{G}^{\epsilon}$ converges to Brownian motion modulo time parametrization, which partially resolves the last part of Conjecture 1.2.

One a priori reason to expect that the LQG structure graphs should yield a metric on LQG in the scaling limit comes from comparison to discrete models. Indeed, many natural combinatorial random planar map models which belong to the $\gamma$-LQG universality class for some $\gamma\in(0,2)$ can be encoded by means of a two-dimensional random walk via a discrete analogue of the above encoding of the LQG structure graph in terms of a correlated two-dimensional Brownian motion. Planar maps for which this is the case include the uniform infinite planar triangulation (UIPT) [Ber07a, BHS18] ($\gamma=\sqrt{8/3}$) spanning tree-decorated planar maps [Mul67, Ber07b, She16b] ($\gamma=\sqrt{2}$), bipolar-oriented planar maps [KMSW15] ($\gamma=\sqrt{4/3}$), and Schnyder wood-decorated planar maps [LSW17] ($\gamma=1$). In the subsequent work [GHS17], we use this relationship to transfer the estimates proven here for LQG structure graphs to these discrete models (see also Section 1.6.3).

In [MS15b, Section 1.1], the structure graphs are suggested as a possible approach for constructing a metric on LQG. These graphs can also be viewed as a potential approach to [DMS14, Question 13.1]—which asks for a direct construction of a metric on the peanosphere—since they depend only on the Brownian motion $Z$.

In the present paper, we will prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius $n$ in $\mathcal{G}^{\epsilon}$ (Theorem 1.10), which we expect should scale like $n^{d_{\gamma}+o_{n}(1)}$, where $d_{\gamma}$ is the dimension of $\gamma$-LQG. Hence our bounds can be interpreted as upper and lower bound for $d_{\gamma}$. As we will explain in Section 1.6 below, these bounds for $d_{\gamma}$ are consistent with both the Watabiki prediction (1.2) and the estimates of [DG16b] and sharper than what can be obtained from [MRVZ16, AK16] (although there is presently no direct rigorous connection between our results and those of [MRVZ16, AK16, DG16b]).

We also prove that there is an exponent $\chi=\chi(\gamma)>0$ such that the expected graph distance between two typical vertices of the sub-graph of $\mathcal{G}^{\epsilon}$ corresponding to the SLE segment $\eta([-1,1])$ is of order $\epsilon^{-\chi+o_{\epsilon}(1)}$ and the distance between two such vertices is extremely unlikely to be greater than $\epsilon^{-\chi+o_{\epsilon}(1)}$ (Theorems 1.12 and 1.15).

Although $\gamma=\sqrt{8/3}$ is special from the perspective of [MS15b, MS16b, MS16c], none of our results are specific to the case $\gamma=\sqrt{8/3}$. We will, however, obtain stronger estimates than the ones in this paper for $\gamma=\sqrt{8/3}$ (which give, in particular, the correct exponent for the cardinality of a metric ball) in [GHS17] by comparing $\mathcal{G}^{\epsilon}$ to a uniform random triangulation.

Here we record some basic notations which we will use throughout this paper.

Unless otherwise stated, all implicit constants in $\asymp,\preceq$, and $\succeq$ and $O_{x}(\cdot)$ and $o_{x}(\cdot)$ errors involved in the proof of a result are required to depend only on the auxiliary parameters that the implicit constants in the statement of the result are allowed to depend on.

We next introduce some notation concerning graphs.

For many of the statements in this paper, the choice of graph $G$ in Definition 1.9 will be important. Note that if $G^{\prime}$ is a subgraph of $G$, then $\operatorname{dist}(x,y;G^{\prime})\geq\operatorname{dist}(x,y;G)$ for each $x,y\in\mathcal{V}(G)$.

Throughout this subsection we assume we are in the setting of Section 1.2, so that $\{\mathcal{G}^{\epsilon}\}_{\epsilon>0}$ is the family of LQG structure graphs constructed from an SLE_κ-decorated $\gamma$-quantum cone for $\gamma\in(0,2)$ and $\kappa=16/\gamma^{2}>4$. We make frequent use of Definition 1.9. Our first main result gives upper and lower bounds for the scaling dimension of $\mathcal{G}^{\epsilon}$.

See Figure 3, left panel, for a graph of the bounds appearing in Theorem 1.10. By Brownian scaling, the laws of the graphs $\mathcal{G}^{1}$ and $\mathcal{G}^{\epsilon}$ for $\epsilon>0$ agree, so the statement of Theorem 1.10 is also true with $\mathcal{G}^{\epsilon}$ in place of $\mathcal{G}^{1}$.

By Conjecture 1.2 we expect that the graph distance of $\mathcal{G}^{\epsilon}$ (appropriately re-scaled) is a good approximation for the $\gamma$-LQG metric when $\epsilon$ is small. Hence it should be the case that in fact $\#\mathcal{B}_{n}\mathopen{}\mathclose{{\left(0;\mathcal{G}^{\epsilon}}}\right)=n^{d_{\gamma}+o_{n}(1)}$ with high probability, where $d_{\gamma}$ is the dimension of $\gamma$-LQG. Therefore Theorem 1.10 can be interpreted as giving upper and lower bounds for $d_{\gamma}$, namely $d_{-}\leq d_{\gamma}\leq d_{+}$. These bounds are consistent with the Watabiki prediction (1.2). In the special case when $\gamma=\sqrt{8/3}$, we know that $d_{\gamma}=4$, so in this case we expect that $\#\mathcal{B}_{n}(0;\mathcal{G}^{1})=n^{4+o_{n}(1)}$ with high probability when $n$ is large. This will be proven in [GHS17] by comparing $\mathcal{G}^{\epsilon}$ to a uniform triangulation.

Our next main result is the existence of an exponent for distances in the LQG structure graphs. More precisely, we will consider distances in the sub-graphs of $\mathcal{G}^{\epsilon}$ defined as follows.

The exponent $\chi$ will be proven to exist via a sub-additivity argument. The reason for the quantity $1-2/\gamma^{2}$ appearing in (1.6) is as follows. In the case when $\gamma>\sqrt{2}$, the left and right outer boundaries of $\eta^{\prime}([0,1])$ touch each other to form “bottlenecks” (c.f. Figure 2). These bottlenecks correspond to simultaneous running infima of $L$ and $R$ relative to time 0 (which a.s. do not exist for a non-positively correlated Brownian motion). The dimension of the time set for $Z$ such that this is the case is $1-2/\gamma^{2}$ [Eva85], so we expect that there are typically of order $\epsilon^{-(1-2/\gamma^{2})}$ cells $\eta^{\prime}([x-\epsilon,x])$ for $x\in\epsilon\mathbbm{Z}$ which intersect a bottleneck. Any path from $\epsilon$ to 1 in $\mathcal{G}^{\epsilon}|_{(0,1]}$ must pass through each of these cells.

If we allow for paths between vertices of $\mathcal{G}^{\epsilon}|_{(0,1]}$ which traverse vertices in all of $\mathcal{G}^{\epsilon}$ (rather than just in $\mathcal{G}^{\epsilon}|_{(0,1]}$) then bottlenecks do not pose a problem. Hence we still expect that (in the notation of Definition 1.9)

where $d_{\gamma}$ is the dimension of $\gamma$-LQG; and that $\chi=1/d_{\gamma}$ when there are either few or no bottlenecks. There are no bottlenecks for $\gamma\leq\sqrt{2}$, and according to the Watabiki prediction (1.2), we have $1/d_{\gamma}\leq 1-2/\gamma^{2}$ when $\gamma\geq\sqrt{8/3}$. This leads to the following conjecture.

We do not have a prediction for the value of $\gamma_{*}$ in Conjecture 1.13.

Our next result estimates the probability that distances between vertices of $\mathcal{G}^{\epsilon}|_{(0,1]}$ are of order $\epsilon^{-\chi+o_{\epsilon}(1)}$. We get an upper bound which holds except on an event of probability decaying faster than any power of $\epsilon$ and a lower bound which holds on an event of probability decaying slower than any power of $\epsilon$.

Theorem 1.15 implies that for each $p>0$,

Note that Theorem 1.15 does not state that the diameter of $\mathcal{G}^{\epsilon}|_{(0,1]}$ is at least $\epsilon^{-\chi+o_{\epsilon}(1)}$ with uniformly positive probability. We expect, but do not prove, that this holds with probability tending to 1 as $\epsilon\rightarrow 0$.

Here we give a moderately detailed overview of the remainder of the paper. In Section 2, we will review the definitions of LQG, space-filling SLE, and the peanosphere construction; and prove some basic facts about the structure graphs and about correlated two-dimensional Brownian motion.

In Section 3, we will use estimates for SLE-decorated LQG to prove quantitative estimates for distances in the $\gamma$-LQG structure graph, which will eventually lead to a proof of Theorem 1.10. These estimates are the only arguments in this paper which require non-trivial facts about LQG and SLE. In particular, to prove the upper bound in Theorem 1.10, we will prove bounds for the Euclidean diameter of structure graph cells and thereby a lower bound for the minimal number of cells in a path in $\mathcal{G}^{\epsilon}$ from 0 to the boundary of $\eta([-1,1])$. The lower bound in Theorem 1.10 will be deduced from a KPZ-type formula (Proposition 3.5) which gives us an upper bound for the number of cells needed to cover a line segment.

In Sections 4 and 5, we will prove the existence of the limit $\chi$ in (1.5) for $\epsilon$ restricted to powers of 2 using a subadditivity argument. We will also show that the diameter of $\mathcal{G}^{2^{-n}}|_{(0,1]}$ is very unlikely to be larger than $2^{-n(\chi+o_{n}(1))}$ (Proposition 5.2). The key idea of the proof is that if we set $D_{n}:=\operatorname{diam}(\mathcal{G}^{2^{-n}}|_{(0,1]})$ then $\mathbbm{E}[D_{n}]$ is approximately sub-multiplicative in the sense $\mathbbm{E}[D_{n+m}]\lesssim\mathbbm{E}[D_{n}]\mathbbm{E}[D_{m}]$ for a certain range of values of $n,m\in\mathbbm{N}$. This gives the existence of $\chi$ due to a variant of Fekete’s subaddivity lemma (Lemma 5.3).

Roughly speaking, to prove the sub-multiplicativity relation for $D_{n}$ we start with a path $P$ in $\mathcal{G}^{2^{-n}}|_{(0,1]}$ of length $|P|=D_{n}$ and divide each of the cells hit by $P$ into $2^{m}$ sub-cells. Concatenating paths within each of these sub-divided cells gives us a path in $\mathcal{G}^{2^{-n-m}}|_{(0,1]}$ whose length is at most the sum of $D_{n}$ terms with the same law as $D_{m}$. The estimates of Section 4 are needed to allow us to compare the conditional law of the sub-divided cells given $\mathcal{G}^{2^{-n}}|_{(0,1]}$ to their unconditional law.

In Section 6, we will deduce Theorems 1.12 and 1.15 from the results of Section 5.

Appendix A contains the proofs of some technical lemmas.

In this subsection we give a brief review of some basic properties of Liouville quantum gravity, space-filling SLE, and the peanosphere construction. Although these objects are the main motivation of this work, their non-trivial properties will only be used explicitly in Section 3. The proofs in the other sections can be phrased in terms of Brownian motion by means of the peanosphere construction. We refer to the cited references for further background.

Throughout this subsection, we fix $\gamma\in(0,2)$ and use the notation of Section 1.2, so in particular $(\mathbbm{C},h,0,\infty)$ is a $\gamma$-quantum cone, $\eta$ is a whole-plane space-filling SLE_κ for $\kappa=16/\gamma^{2}$ independent from $\gamma$ and parametrized by $\mu_{h}$-length, $Z=(L,R)$ is the correlated Brownian motion from [DMS14, Theorem 1.13], and $\{\mathcal{G}^{\epsilon}\}_{\epsilon>0}$ are the associated $\epsilon$-structure graphs, with vertex set $\mathcal{V}(\mathcal{G}^{\epsilon})=\epsilon\mathbbm{Z}$.