Local central limit theorem for gradient field models

In this paper we study a two dimensional gradient interface field with a nearest neighbor potential $V$. Explicitly, let $Q_{N}:=\left[-N,N\right]^{2}\cap\mathbb{Z}^{2}$ and let the boundary $\partial Q_{N}$ consist of the vertices in $Q_{N}$ that are connected to $\mathbb{Z}^{2}\setminus Q_{N}$ by an edge. The gradient field on $Q_{N}$ with zero boundary condition is a random field denoted by $\phi^{Q_{N},0}$, whose distribution is given by the Gibbs measure

(1.1)

$$d\mu_{N}=Z_{N}^{-1}\exp\left[-\sum_{v\in Q_{N}}\sum_{i=1}^{2}V\left(\nabla_{i}\phi\left(v\right)\right)\right]\prod_{v\in Q_{N}\backslash\partial Q_{N}}d\phi\left(v\right)\prod_{v\in\partial Q_{N}}\delta_{0}\left(\phi\left(v\right)\right),$$

where $\nabla_{i}\phi\left(v\right)=\phi\left(v+e_{i}\right)-\phi\left(v\right)$, $e_{1}=(1,0)$ and $e_{2}=(0,1)$, and we set $\phi\left(v\right)=0$ for all $v\in\mathbb{Z}^{2}\setminus Q_{N}$. Here $Z_{N}$ is the normalizing constant ensuring that $\mu_{N}$ is a probability measure, i.e. $\mu_{N}(\mathbb{R}^{|Q_{N}|})=1$. We denote expectation and variance with respect to $\mu_{N}$ by $\langle\cdot\rangle_{\mu_{N}}$ and $\operatorname{var}_{\mu_{N}}$, respectively.

We assume the interaction potential $V:\mathbb{R}\to\mathbb{R}$ in (1.1) satisfies the following:

1. (i)

   Symmetry: for every $t\in\mathbb{R}$, we have $V(t)=V(-t)$.

2. (ii)

   Uniform convexity: for every $t\in\mathbb{R}$, we have $0<\lambda\leq V^{\prime\prime}(t)\leq\Lambda<\infty$.

3. (iii)

   Regularity: $V\in C^{2,1}(\mathbb{R})$. In other words, $\mathsf{V}^{\prime\prime}$ is Lipschitz continuous with Lipschitz constant $L$.

The Gibbs measure (1.1) was introduced in the 1970s by Brascamp, Lieb and Lebowitz [11], in the name of anharmonic crystals. Since then, numerous efforts have been made to study the large-scale (macroscopic) statistical behavior of the field $\nabla\phi$. Notable progress was made by Naddaf and Spencer [22], who studied the infinite volume limit of the Gibbs measure (1.1) (the infinite volume Gibbs states were rigorously characterized by Funaki and Spohn [15]), and proved a central limit for (rescaled) linear functions of $\nabla\phi$. More precisely, they consider, for $R\geq 1$, the random variable

(1.2)

$$F_{R}(\nabla\phi):=R^{-\frac{d}{2}}\sum_{x\in\mathbb{Z}^{d}}\sum_{i=1}^{d}f_{i}\left(\frac{x}{R}\right)\left(\phi(x+e_{i})-\phi(x)\right),$$

where for each $i\in\{1,\ldots,d\}$, take $f_{i}:{\mathbb{R}^{d}}\to\mathbb{R}$ to be a compactly supported, smooth, deterministic function, and proved that the random variable $F_{R}$ converges in law to a normal random variable. Later, the result of [22] has been generalized to dynamical settings [16] and also to finite volume measures such that the boundary condition has at most poly-logarithmic fluctuations [20]. We also mention the related work [14] which characterizes the Wulff shape and the large deviation principle for macroscopic profiles of $\nabla\phi$, and results that are related to the extension of the Gibbs measure (1.1) to non-uniformly convex settings [25, 9, 1, 21, 18, 6, 7], the study of the maximum of the random field (1.1) [8, 26], and to spin models which is related to a Gibbs measure with convex interaction by a duality transform [13].

A natural question arises is whether the Gaussian limit established in [22] holds on much smaller scales, given that the microscopic interaction is not Gaussian. Under an additional assumption on the ellipticity contrast for $V$, namely $\Lambda<2\lambda$, Conlon and Spencer proved a stronger result [12], which states that for the infinite volume gradient Gibbs measure $\mu$,

$$\left|\log\left\langle\exp(t(\phi(0)-\phi(x)))\right\rangle_{\mu}-\frac{t^{2}}{2}\operatorname{var}_{\mu}(\phi(0)-\phi(x))\right|\leq Ct^{3}\sup_{x\in\mathbb{R}}|V^{\prime\prime\prime}(x)|.$$

The result suggests that under these assumptions of $V$, the pointwise distribution of $\phi(x)-\phi(0)$ is close to a Gaussian, and one has to move to a large deviation regime to see non-Gaussian tails.

In this paper we are able to bring down the scale to $R=1$, and prove a local central limit theorem of the gradient Gibbs measure (1.1), under the assumptions on the potential $V$ given at the beginning of the paper.

We remark here the same proof (with a slightly modification of the multiscale argument in Section 5 ) also gives the local CLT at other locations inside the bulk. Namely, for any $x_{N}\in Q_{N}$ such that the graph distance between $x_{N}$ and $\partial Q_{N}$ tends to infinity, the density of $\frac{\phi(x_{N})}{\sqrt{\log\text{dist}(x_{N},\partial Q_{N})}}$ converges uniformly in $\mathbb{R}$ to the same Gaussian limit. The same proof also works in the inifnite volume setting, gives the local CLT for $\frac{\phi(0)-\phi(x)}{(\log|x|)^{\frac{1}{2}}}$ as $|x|\to\infty$.

Notice that if $\phi(0)$ can be written as $X_{1}+\cdots X_{\log N}$, where $X_{i}$ are i.i.d random variables, then the Berry-Esseen Theorem gives $|g_{N}(x)-\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}|\leq\frac{C\mathbb{E}[|X_{1}|^{3}]}{\sqrt{\log N}}$. We will see in Section 5 that the analogues of $X_{i}$ are increments of harmonic averages of $\phi$, which are not independent but have certain decoupling properties (thanks to [20]), and Theorem 1.1 gives a Berry-Esseen type estimate for the density function. An immediate consequence of Theorem 1.1 is that the distribution of $\phi(0)$ is sufficiently spreadout in $[-\sqrt{\log N},\sqrt{\log N}]$. Indeed, given any $a\in[-\sqrt{\log N},\sqrt{\log N}]$, we apply Theorem 1.1 to obtain

$$\int_{\frac{a}{\sqrt{\log N}}}^{\frac{a+1}{\sqrt{\log N}}}g_{N}(x)\,dx-\int_{\frac{a}{\sqrt{\log N}}}^{\frac{a+1}{\sqrt{\log N}}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}\,dx=O(\frac{1}{\log N})$$

Thus

$$\mathbb{P}(\phi(0)\in[a,a+1])=\frac{1}{\sqrt{2\pi\log N}}e^{-\frac{1}{2\mathbf{g}}\frac{a^{2}}{\log N}}+O(\frac{1}{\log N})$$

Why do we expect a local CLT at scale $O(1)$? The argument to prove a macroscopic CLT in [22] was based on a beautiful observation that the scaling limit can be derived from an elliptic homogenization problem via the Helffer-Sjöstrand representation [17]. With Armstrong [5], we extend and quantify the homogenization argument by Naddaf and Spencer, based on the quantitative theory for homogenization developed by Armstrong, Kuusi and Mourrat [3, 4]. In particular, we obtained the convergence of the Hessian of the surface tension with an algebraic rate, resolve an open question posed by Funaki and Spohn [15] regarding the $C^{2}$ regularity of surface tension, and the fluctuation-dissipation conjecture of [16]. Following the approach of [5] and [4], we are able to obtain a quantitative homogenization of the Helffer-Sjöstrand PDE, thus estimate the covariance structure of $\mu_{N}$, as $N$ gets large, with a high precision (we also refer to [2] for some further applications of the quantitative homgenization ideas to the $\nabla\phi$-model). To obtain a Gaussian limit of $\phi(0)/\sqrt{\log N}$, we apply the harmonic approximation result by Miller [20], which enables us to write $\phi(0)$ as the sum of $\log N$ increments, with certain decoupling properties. These are the two main ingredients behind the proof of Theorem 1.1.

We remark here that the key estimate for proving Theorem 1.1 is the characteristic function asymptotics for $\left\langle\exp\left(i\frac{t}{\sqrt{\log N}}\phi(0)\right)\right\rangle_{\mu_{N}}$, given in Lemma 2.1. If one has the convergence stated in Lemma 2.1 for $t\in\mathbb{R}$, without quantifying the rate of convergence (namely, a CLT for $\phi(0)/\sqrt{\log N}$), then it implies a local CLT without any rate of convergence. The rate of convergence for the local CLT depdends on the convergence rate and the validity of the $t$ region such that Lemma 2.1 holds. In this paper we present a proof with a convergence rate using quantitative homogenization, and explain briefly in the next section how it can be simplified if one only aims for a qualitative local CLT.

The proof of Theorem 1.1 presented in this paper implies the asymptotics for the characteristic function for the distribution of $\phi(0)$, namely there exists some $\mathbf{g}=\mathbf{g}(V)>0$, such that

(1.3)

$$\left\langle\exp\left(is\phi(0)\right)\right\rangle_{\mu_{N}}\approx e^{-\frac{s^{2}}{2}\mathbf{g}\log N},$$

as long as $s=O((\log N)^{-\frac{1}{4}})$. Using the same major ingredients, but with a more elaborate multiscale argument, we may improve the (1.3) so that it holds for $s$ within a small neighborhood of the origin, with radius independent of $N$. It would be very interesting to extend the characteristic function estimates (1.3) beyond $|s|=1$. For this, we include an open question posed by Tom Spencer.

Open Question: Consider the lattice dipole gas model, which is a special case of the gradient Gibbs measure (1.1) with $V(x)=\frac{x^{2}}{2}+z\cos x$, with $|z|<1$. Prove that the leading term in the asymptotic expansion of the characteristic function

(1.4)

$$\left\langle\exp\left(i\pi(\phi(0)-\phi(x))\right)\right\rangle_{\mu_{dipole}}\approx e^{-\frac{\pi^{2}}{2}\mathbf{g}\log|x|},$$

for some $\mathbf{g}>0$. Estimates of type (1.4) plays an important role in the study of the (lattice) Coulomb gas and of the Coulomb gas representation of the low temperature Abelian spin models.

We summarize the characteristic function estimates needed for proving Theorem 1.1 in the next section, and introduce some notations in Section 3. In Section 4 we derive a central limit theorem for the linear statistics of $\nabla\phi$ with an algebraic rate of convergence, which quantifies the result of [22], following the argument of [5] and [4]. In Section 5 we recall the harmonic approximation result in [20], and use a multiscale argument to obtain the precise characteristic function asymptotic of $\phi(0)$. Finally, we gave an upper bound for the large $s$ characteristic function in Section 6, based on a Mermin-Wagner type argument, and finishes the proof of Theorem 1.1 .

Theorem 1.1 follows from the quantitative estimates of the characteristic function below. The first lemma gives the precise estimate for the characteristic function $\langle e^{is\phi(0)}\rangle_{\mu_{N}}$ for $s=o((\log N)^{-\frac{1}{4}})$.

We also need (non-optimal) decay estimate of the characteristic function for large $s$, summarized in the two lemmas below.

Before proving these lemmas, we now explain how they imply Theorem 1.1.

Lemma 2.1 and 2.3 will be proved in Section 5.3, whereas Lemma 2.5 will be proved in Section 6.

Given a set $U\subseteq Q_{R}$, we let $\mathcal{E}(U)$ denote the set of directed edges on $U$ and $U^{\circ}$ the interior of $U$. Define $\Omega_{0}(U)$ to be the set of functions $\phi:U\to\mathbb{R}$ such that $\phi=0$ on $\partial U$. Given $e=(x,y)\in\mathcal{E}(U)$ and $\phi\in\mathbb{R}^{U}$, we define $\nabla\phi(e):=\phi(y)-\phi(x)$. The formal adjoint $\nabla^{*}$ of $\nabla$, which is the discrete version of the negative of the divergence operator, is defined for functions $\mathbf{g}:\mathcal{E}(U)\to\mathbb{R}$ by

(3.1)

$$\left(\nabla^{*}\mathbf{g}\right)(x):=\sum_{e\ni x}\mathbf{g}(e),\quad x\in U^{\circ}.$$

The average of a function $f:U\to\mathbb{R}$ on $U$ is denoted as $(f)_{U}:=\frac{1}{|U|}\sum_{x\in U}f(x)$.

We define, for each $x\in U$, the basis element $\omega_{x}\in\Omega_{0}(U)$ by

$$\omega_{x}(y):=\left\{\begin{aligned} &1&\mbox{if}\ x=y,\\ &0&\mbox{if}\ x\neq y,\end{aligned}\right.$$

and the differential operator $\partial_{x}$ by

(3.2)

$$\partial_{x}u(\phi):=\lim_{h\to 0}\frac{1}{h}\left(u(\phi+h\omega_{x})-u(\phi)\right).$$

Define $L^{p}(\mu)$ to be the set of measurable functions $u:\Omega_{0}(U)\to\mathbb{R}$ such that

$$\left\|u\right\|_{L^{p}(\mu)}:=\left(\int_{\Omega}\left|u(\phi)\right|^{p}\,d\mu(\phi)\right)^{\frac{1}{p}}<+\infty.$$

We define $H^{1}(\mu)$ to be

$$\left\|u\right\|_{H^{1}(\mu)}:=\left(\left\|u\right\|_{L^{2}(\mu)}^{2}+\sum_{x\in U^{\circ}}\left\|\partial_{x}u\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}}.$$

We let $H^{-1}(\mu)$ denote the dual space of $H^{1}(\mu)$, that is, the closure of $C^{\infty}(\Omega_{0}(U))$ functions under the norm

$$\left\|w\right\|_{H^{-1}(\mu)}:=\sup\left\{\int_{\Omega}u(\phi)w(\phi)\,d\mu(\phi)\,:\,u\in H^{1}(\mu),\ \left\|u\right\|_{H^{1}(\mu)}\leq 1\right\}.$$

We define the space $L^{2}(U,\mu)=L^{2}(U;L^{2}(\mu))$ to be the set of measurable functions $u:U\times\Omega_{0}(U)\to\mathbb{R}$ with respect to the norm

$$\left\|u\right\|_{L^{2}(U,\mu)}:=\left(\sum_{x\in U}\left\|u(x,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}}.$$

We also define $H^{1}(U,\mu)$ by the norm

$$\left\|u\right\|_{H^{1}(U,\mu)}:=\left(\sum_{x\in U}\left\|u(x,\cdot)\right\|_{H^{1}(\mu)}^{2}+\sum_{e\in\mathcal{E}(U)}\left\|\nabla u(e,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}}$$

The subset $H^{1}_{0}(U,\mu)\subseteq H^{1}(U,\mu)$ consists of those functions $u\in H^{1}(U,\mu)$ which satisfy $u(x,\phi)=0$ for every $\partial U\times\Omega_{0}(U)$.

We define $H^{-1}(U,\mu)$ to be the dual space of $H^{1}_{0}(U,\mu)$. That is, $H^{-1}(U,\mu)$ is the closure of smooth functions with respect to the norm

$$\left\|w\right\|_{H^{-1}(U,\mu)}:=\sup\left\{\sum_{x\in U}\int_{\Omega_{0}(U)}u(x,\phi)w(x,\phi)\,d\mu(\phi)\,:\,u\in H^{1}_{0}(U,\mu),\ \left\|u\right\|_{H^{1}(U,\mu)}\leq 1\right\}.$$

It is sometimes convenient to work with the volume-normalized versions of the $L^{2}$ and Sobolev norms, defined by

$$\left\|u\right\|_{\underline{L}^{2}(U,\mu)}:=\left(\frac{1}{|U|}\sum_{x\in U}\left\|u(x,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}},$$

$$\left\|u\right\|_{\underline{H}^{1}(U,\mu)}:=\left(\frac{1}{|U|}\sum_{x\in U}\left\|u(x,\cdot)\right\|_{H^{1}(\mu)}^{2}+\frac{1}{|U|}\sum_{e\in\mathcal{E}(U)}\left\|\nabla u(e,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}},$$

$$\left\|w\right\|_{\underline{H}^{-1}(U,\mu)}\\ :=\sup\left\{\frac{1}{|U|}\sum_{x\in U}\int_{\Omega_{0}(U)}u(x,\phi)w(x,\phi)\,d\mu(\phi)\,:\,u\in H^{1}_{0}(U,\mu),\ \left\|u\right\|_{\underline{H}^{1}(U,\mu)}\leq 1\right\}.$$

We notice that the formal adjoint of $\partial_{x}$ with respect to $\mu_{N}$, which we denote as $\partial_{x}^{*}$, is given by

$$\partial_{x}^{*}w:=-\partial_{x}w+\sum_{y\sim x}V^{\prime}(\phi(y)-\phi(x)-\xi\cdot(y-x))w(\phi).$$

This can be easily checked by the identity for all $u,v\in H^{1}(\mu_{N})$ that

$$\left\langle(\partial_{x}u)v\right\rangle_{\mu_{N}}=\left\langle u(\partial^{*}_{x}v)\right\rangle_{\mu_{N}}.$$

We also have the commutator identity

(3.3)

$$\left[\partial_{x},\partial_{y}^{*}\right]=-\mathds{1}_{\{x\sim y\}}V^{\prime\prime}\big{(}\phi(y)-\phi(x)-\xi\cdot(y-x)\big{)}+\mathds{1}_{\{x=y\}}\sum_{e\ni x}V^{\prime\prime}\left(\nabla\phi(e)-\nabla\ell_{\xi}(e)\right).$$

Define the Witten Laplacian $\mathcal{L}_{\mu_{N}}$ as

$$\mathcal{L}_{\mu_{N}}F=-\sum_{x\in Q_{N}^{\circ}}\partial_{x}^{*}\partial_{x}F,$$

For every cube $Q\subseteq\mathbb{Z}^{d}$ and $u,v\in H^{1}(Q,\mu)$, we define

(3.4)

$$\mathsf{B}_{\mu,Q}\left[u,v\right]:=\frac{1}{|Q|}\sum_{y\in Q_{N}^{\circ}}\sum_{x\in Q^{\circ}}\left(\partial_{y}u(x,\cdot),\partial_{y}v(x,\cdot)\right)_{\mu}+\frac{1}{|Q|}\sum_{e\in\mathcal{E}(Q)}\left\langle\nabla u(e,\cdot)V^{\prime\prime}(e,\cdot)\nabla v(e,\cdot)\right\rangle_{\mu}$$

and

$\displaystyle\mathsf{E}_{\mu,Q,\mathbf{f}}\left[u\right]$

$\displaystyle:=\frac{1}{2}\mathsf{B}_{\mu,Q}\left[u,u\right]-\frac{1}{|Q|}\sum_{e\in\mathcal{E}(Q)}\left\langle\mathbf{f}(e,\phi)\nabla u(e,\cdot)\right\rangle_{\mu}.$

For $D\subseteq Q_{R}$, and $f:\partial D\to\mathbb{R}$, define the $\nabla\phi$ measure on $D$ with Dirichlet boundary condition $f$ by

(3.5)

$$d\mu_{D}^{f}=Z_{D}^{-1}\exp\left[-\sum_{x\in D}\sum_{i=1}^{2}V\left(\nabla_{i}\phi\left(x\right)\right)\right]\prod_{x\in D\backslash\partial D}d\phi\left(x\right)\prod_{x\in\partial D}\delta_{0}\left(\phi\left(x\right)-f\left(x\right)\right).$$

Here $Z_{D}$ is the normalizing constant ensuring that $\mu_{D}^{f}$ is a probability measure. We denote expectation and variance with respect to $\mu_{D}^{f}$ by $\mathbb{E}^{D,f}$ and $\operatorname{var}_{D,f}$, respectively.

We finally present the Brascamp-Lieb inequality [10, 22], which states that the variance of observables with respect to a log-concave measure is dominated by that of a Gaussian measure. We denote the Green function for the discrete Laplacian with zero Dirichlet boundary conditions in $Q_{L}$ by $G_{Q_{L}}(x,y)$.

We sometimes denote by $\mu_{G,D}^{f}$ the finite volume Gaussian measure in $D$ (i.e., the special case of (3.5) with $V(x)=\frac{1}{2}x^{2}$). We denote the corresponding expectation and variance by $\mathbb{E}^{G,D,f}$ and $\operatorname{var}_{G,D,f}$ respectively. When $f=0$ we will omit its appearence on the supercripts.

A main ingredient for the refined estimate of $\Psi_{N}$, defined in (2.6) is the following convergence of the variance of the linear statisitcs of $\nabla\phi$, with an algebraic rate.

Theorem 4.1 follows from homogenization of an elliptic PDE based on the convergence results of [5], as we explain below. The starting observation is the variational characterization of the variance, known as the Helffer-Sjöstrand representation (see [22, 5]), which gives

(4.1)

$$\operatorname{var}_{\mu_{R}}[R^{-d/2}\sum_{e\in\mathcal{E}(Q_{R})}\nabla\phi(e)\cdot f_{R}\left(e\right)]=-2\inf_{w\in H^{1}_{0}(Q_{R},\mu_{R})}\mathsf{E}_{\mu_{R},Q_{R},f_{R}}\left[w\right],$$

where we let $\mathsf{E}_{\mu,U,f}\left[\cdot\right]$ denote the energy functional

	$\displaystyle\mathsf{E}_{\mu,U,f}\left[w\right]$	$\displaystyle:=\frac{1}{2}\sum_{y\in Q}\sum_{x\in U^{\circ}}\left\langle(\partial_{y}w(x,\cdot))^{2}\right\rangle_{\mu}+\frac{1}{2}\sum_{e\in\mathcal{E}(U)}\left\langle V^{\prime\prime}(e)(\nabla w(e,\cdot))^{2}\right\rangle_{\mu}$
		$\displaystyle\qquad-\sum_{x\in U^{\circ}}\left\langle f(x,\cdot)w(x,\cdot)\right\rangle_{\mu}.$

The minimizer of (4.1) can be written as $R^{-\frac{d}{2}}u_{R}$, where $u_{R}$ solves the Helffer-Sjöstrand PDE

(4.2)

$$\left\{\begin{aligned} &-\mathcal{L}_{\mu}u_{R}+\nabla^{*}\cdot V^{\prime\prime}\nabla u_{R}=\nabla^{*}\cdot f_{R}\quad&\mbox{in}&\ Q_{R}\times\Omega_{0}(Q_{R})\\ &u_{R}=0&\mbox{on}&\ \partial Q_{R}\times\Omega_{0}(Q_{R}),\end{aligned}\right.$$

and by testing (4.2) with $u_{R}$ and integration by parts, we may rewrite the energy functional $\mathsf{E}_{\mu_{R},Q_{R},f_{R}}\left[u_{R}\right]$, and thus (4.1) as [22, 5]

$$\operatorname{var}_{\mu_{R}}[R^{-d/2}\sum_{x\in Q_{R}}\nabla\phi(x)\cdot f_{R}\left(x\right)]=\sum_{x\in Q_{R}}R^{-d}\left\langle f_{R}\left(x\right)\nabla u_{R}(x)\right\rangle_{\mu_{R}}$$

Therefore Theorem 4.1 follows from the quantitative homogenization of the Hellfer-Sjöstrand equation (4.2), presented below.

Applying Theorem 4.4 and rescale the domain by $R$, we have

$$\left|\operatorname{var}_{\mu_{R}}[R^{-d/2}\sum_{x\in Q_{R}}\nabla\phi(x)\cdot f_{R}\left(x\right)]-\sum_{x\in Q_{R}}R^{-d}f_{R}\left(x\right)\nabla u\left(\frac{x}{R}\right)\right|\\ \leq\|\nabla^{*}\cdot f_{R}\|_{L^{\infty}}\|u_{R}-u\left(\frac{\cdot}{R}\right)\|_{\underline{L}^{2}(Q_{R},\mu_{R})}\leq CR^{-\beta}\|\nabla^{*}\cdot f_{R}\|_{L^{\infty}}$$