The Gaussian free field (GFF) is the natural equilibrium measure of an Ornstein-Uhlenbeck type dynamics, for which we can study the long time behavior. This short post explores this idea, on the discrete one-dimensional GFF. The stochastic differential equation of the process is a discrete version of the stochastic heat equation (SHE).
A discrete pinned stochastic heat equation (SHE). Let $\Delta\in\mathcal{M}_n$ be the discrete Laplacian on $\Lambda=\{1,\ldots,n\}$, with zero Dirichlet conditions, namely \[ (\Delta u)_x=u_{x-1}+u_{x+1}-2u_x,\quad x\in\Lambda, \] for all $u\in\mathbb{R}^\Lambda\equiv\mathbb{R}^{|\Lambda|}=\mathbb{R}^n$, with the convention $u_0=u_{n+1}=0$. Here $u$ is seen as a function from $\Lambda$ to $\mathbb{R}$, extended by zero at the (outer) boundary $\partial\Lambda=\{0,n+1\}$. It turns out that the $n\times n$ matrix $\Delta$ is tridiagonal and Toeplitz symmetric \[ \Delta=J-2I \quad\text{where}\quad J_{xy}=\mathbf{1}_{|x-y|=1},\quad x,y\in\Lambda. \]
We consider now the Markov diffusion process ${(U_t)}_{t\in\mathbb{R}_+}$, $U_t=(U_{t,x}:x\in\Lambda)$, with state space $\mathbb{R}^n$, solution of the stochastic differential equation \[ \mathrm{d}U_t=\Delta U_t\mathrm{d}t+\sqrt{2}\mathrm{d}B_t,\quad t\in\mathbb{R}_+,\quad U_0=u_{\mathrm{in}}\in\mathbb{R}^\Lambda, \] where ${(B_t)}_{t\in\mathbb{R}_+}$ is a standard Brownian motion on $\mathbb{R}^n$. In other words ${(U_t)}_{t\in\mathbb{R}_+}$ is the solution of the system of stochastic differential equations \[ \mathrm{d}U_{t,x}=(U_{t,x-1}+U_{t,x+1}-2U_{t,x})\mathrm{d}t+\sqrt{2}\mathrm{d}B_{t,x},\quad t\in\mathbb{R}_+, x\in\Lambda, \quad U_{t,0}=U_{t,n+1}=0. \] This is a discrete in space version of the stochastic heat equation (SHE) \[ \partial_tu(t,x)=\partial_{xx}u(t,x)+\sqrt{2}\xi(t,x),\quad t\in\mathbb{R}_+, x\in [0,1], \] where $\xi$ is a space-time white noise, and more generally \[ \partial_tu(t,x)=\Delta u(t,x)+\sqrt{2}\xi(t,x),\quad t\in\mathbb{R}_+, x\in [0,1]^d \] see for instance [W]. The stochastic process ${(U_t)}_{t\in\mathbb{R}_+}$ is nothing but an overdamped Langevin process with potential $V(u)=\frac{1}{2}\langle(-\Delta)u,u\rangle$. It is actually Gaussian, known as the Ornstein-Uhlenbeck process. The Mehler formula for ${(U_t)}_{t\in\mathbb{R}_+}$ reads \[ U_t=\mathrm{e}^{t\Delta}u_{\mathrm{in}}+\sqrt{2}\int_0^t\mathrm{e}^{(t-s)\Delta}\mathrm{d}B_s \sim\mathcal{N}\Bigl(\mathrm{e}^{t\Delta}u_{\mathrm{in}},(-\Delta)^{-1}(I-\mathrm{e}^{2t\Delta})\Bigr). \] In particular, using the fact that $-\Delta$ is positive-definite, the long time behavior writes \[ U_t \xrightarrow[t\to\infty]{\mathrm{law}} \mu_\Delta \quad\text{where}\quad\mu_\Delta=\mathcal{N}(0,(-\Delta)^{-1}). \] There is an explicit formula for $(-\Delta)^{-1}$, but we do not use it in the sequel: \[ (-\Delta)^{-1}_{xy} =\min(x,y)-\frac{xy}{n+1}. \] We recognize the 1D Green function plus a vanishing boundary term as $n\to\infty$.
Gaussian free field (GFF). The law $\mu_\Delta=\mathcal{N}(0,(-\Delta)^{-1})$ is known as the Gaussian free field on $\Lambda$. This normal on $\mathbb{R}^\Lambda$ has density proportional to \[ u\in\mathbb{R}^\Lambda\mapsto\mathrm{e}^{-\frac{1}{2}\langle(-\Delta)u,u\rangle}. \] The eigenvalues/eigenvectors of $-\Delta$, namely $-\Delta\varphi_{n,k}=\lambda_{n,k}\varphi_{n,k}$, are given by \begin{align*} \varphi_{n,k}(x) &=\sqrt{\frac{2}{n+1}}\sin\left(\frac{k\pi x}{n+1}\right)\\ \lambda_{n,k} &= 2-2\cos\left(\frac{k\pi}{n+1}\right) =4\sin^2\left(\frac{k\pi}{2(n+1)}\right), \qquad k\in\{1,\dots,n\}. \end{align*} See for instance [S]. In particular $\lambda_{n,1}\leq\cdots\leq\lambda_{n,n}$ and the curvature is \[ \lambda_{n,1} = 2-2\cos\left(\frac{\pi}{n+1}\right) \underset{n\to\infty}{\sim}\frac{\pi^2}{(n+1)^2}\xrightarrow[n\to\infty]{}0. \] The process ${(U_t)}_{t\in\mathbb{R}_+}$ is rigid: curvature and spectral gap are equal to $\lambda_{n,1}$. In order to make them of order $1$ as $n\to\infty$, we could consider the stochastic differential equation \[ \mathrm{d}U_t'=(n+1)^2\Delta U_t'\mathrm{d}t+\sqrt{2(n+1)^2}\mathrm{d}B_t, \quad U_0'=u_{\mathrm{in}}. \] The invariant law of ${(U'_t)}_{t\in\mathbb{R}_+}$ is still $\mu_\Delta=\mathcal{N}(0,(-\Delta)^{-1})$, but the spectral gap is now $\sim\pi^2$. Actually the process ${(U'_t)}_{t\in\mathbb{R}_+}$ has the law of the time change ${(U_{(n+1)^2t})}_{t\in\mathbb{R}_+}$. But regarding the long time behavior and the cutoff phenomenon, the overdamped Langevin processes ${(U_t)}_{t\in\mathbb{R}_+}$ and ${(U_t')}_{t\in\mathbb{R}_+}$ are outside the scope of [CF], because the spectral gap goes to zero or because the diffusion coefficient goes to infinity.
Long time behavior and diffusive behavior instead of cutoff phenomenon. Following [BCL] we can take advantage of the Gaussian nature of the distributions and use the available explicit formulas for various distances and divergences. Let us take a look for instance at the Kullback-Leibler divergence or relative entropy. We set \[ D_t:=\mathrm{H}(\mathrm{Law}(U_t)\mid\mu_\Delta) \quad\text{where}\quad \mathrm{H}(\nu\mid\mu):=\int\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\log\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\mathrm{d}\mu. \] We have the explicit formula \begin{align*} \mathrm{H}(\mathcal{N}(m_1,K_1)\mid\mathcal{N}(m_2,K_2)) &=\frac{1}{2}K_2^{-1}(m_1-m_2)\cdot(m_1-m_2)\\ &\quad +\frac{1}{2}\mathrm{Tr}\bigl(K_2^{-1}K_1-I\bigr)-\frac{1}{2}\log\det(K_2^{-1}K_1). \end{align*} and in view of the Mehler formula, we would like to use it with \begin{align*} \mathcal{N}(m_1,K_1) &=\mathcal{N}(\mathrm{e}^{t\Delta}u_{\mathrm{in}},(-\Delta)^{-1}(I-\mathrm{e}^{2t\Delta}))\sim U_t\\ \mathcal{N}(m_2,K_2) &=\mathcal{N}(0,(-\Delta)^{-1}) \end{align*} which gives \begin{align*} K_2^{-1}(m_1-m_2)\cdot(m_1-m_2) &=u_{\mathrm{in}}\cdot(-\Delta)\mathrm{e}^{2t\Delta}u_{\mathrm{in}}\\ \mathrm{Tr}(K_2^{-1}K_1-I) &=\mathrm{Tr}((-\Delta)(-\Delta)^{-1}(I-\mathrm{e}^{2t\Delta})-I) =-\mathrm{Tr}(\mathrm{e}^{2t\Delta})\\ \log\det(K_2^{-1}K_1) &=\log\det(I-\mathrm{e}^{2t\Delta}). \end{align*} Therefore, using the spectral decomposition $-\Delta\varphi_{n,k}=\lambda_{n,k}\varphi_{n,k}$ and $u_{\mathrm{in}}=\sum_{k=1}^n\alpha_{n,k}\varphi_{n,k}$, \[ D_t=M_t+C_t,\quad t > 0, \quad\text{with}\quad \begin{cases} M_t=\frac{1}{2}\sum_{k=1}^n\lambda_{n,k}r_{n,k}(t)\alpha_{n,k}^2\\ C_t=\frac{1}{2}\sum_{k=1}^ng(r_{n,k}(t)) \end{cases} \] where $r_{n,k}(t)=\mathrm{e}^{-2t\lambda_{n,k}}$ and $g(u)=-u-\log(1-u)$ are both non-negative quantities. In this decomposition, $M_t$ is the mean term that depends on $u_{\mathrm{in}}$, while $C_t$ is the covariance term which is universal in the sense that it does not depend on $u_{\mathrm{in}}$.
In order to explore the long time behavior while stabilizing the spectral gap, we take \[ t_n=(n+1)^2s \] for a fixed $s > 0$. For each fixed $k\geq1$, using $\lim_{n\to\infty}(n+1)^2\lambda_{n,k}=(\pi k)^2$ we obtain \[ r_{n,k}(t_n)=\mathrm{e}^{-2t_n\lambda_{n,k}} \xrightarrow[n\to\infty]{}\mathrm{e}^{-2\pi^2k^2s}. \] Moreover the mean term is negligible for $|u_{\mathrm{in}}|\le c\sqrt{n}$. Indeed, \[ 0\leq M_{t_n} \leq \frac12 |u_{\mathrm{in}}|^2\max_{1\le k\le n}\lambda_{n,k}\mathrm{e}^{-2t_n\lambda_{n,k}} =\frac{|u_{\mathrm{in}}|^2}{2(n+1)^2}\max_{1\le k\le n}a_{n,k}\mathrm{e}^{-2sa_{n,k}}, \] where $a_{n,k}:=(n+1)^2\lambda_{n,k}\geq0$. Since $\max_{a\geq0}(a\mathrm{e}^{-2sa})=\frac{1}{2\mathrm{e}s}$, we get, \[ 0\leq M_{t_n}\leq \frac{|u_{\mathrm{in}}|^2}{2(n+1)^2}\frac{1}{2\mathrm{e}s} \leq \frac{c^2n}{4\mathrm{e}s(n+1)^2}\xrightarrow[n\to\infty]{}0 \] uniformly over $|u_{\mathrm{in}}|\leq c\sqrt n$. For the covariance term, using that $g(u)=-u-\log(1-u)\sim \frac{u^2}{2}$ as $u\searrow0$, the series below is finite for every $s > 0$, and by dominated convergence \[ C_{t_n}\xrightarrow[n\to\infty]{} C(s) :=\frac12\sum_{k=1}^{\infty} g\bigl(\mathrm{e}^{-2\pi^2k^2s}\bigr)\in(0,\infty). \] Therefore, we have obtained that for $t_n\sim n^2s$, $s > 0$, and $|u_{\mathrm{in}}|\leq c\sqrt{n}$, $c > 0$, we have \[ D_{t_n}=M_{t_n}+C_{t_n}\xrightarrow[n\to\infty]{} C(s), \] and the limit is universal, in other words independent of $u_{\mathrm{in}}$ in the regime $|u_{\mathrm{in}}|\le c\sqrt n$. The function $s\mapsto C(s)$ is decreasing and continuous on $(0,\infty)$, with $C(s)\to+\infty$ as $s\searrow0$ and $C(s)\to0$ as $s\to\infty$. In particular, on the diffusive time scale $t\asymp n^2$ the relaxation in relative entropy has a nontrivial profile rather than a cutoff.
Higher dimensions and continuous setting. The analysis above works essentially the same way for a $d$-dimensional domain \[ \Lambda:=[a,b]^d\cap\mathbb{Z}^d, \] with $a < b$ and $d\geq1$. We identify the functions or fields $\Lambda\mapsto\mathbb{R}$ with $\mathbb{R}^{\Lambda}$ and then with $\mathbb{R}^n$ where $n=|\Lambda|\sim(b-a)^d$. The discrete Laplacian is defined here as \[ \Delta=J-2dI,\quad\text{with}\quad J_{xy}=\mathbf{1}_{|x-y|_1=1},\quad x,y\in\Lambda. \] The GFF is here again $\mathcal{N}(0,(-\Delta)^{-1})$. It is a centered normal on $\mathbb{R}^\Lambda$ with covariance given by the Green function of the Laplacian with zero Dirichlet boundary conditions. In other words, at each point of the space domain $\Lambda\subset\mathbb{Z}^d$, we put a Gaussian real random variable, and these are correlated in such a way that their covariance matrix is $(-\Delta)^{-1}$. This describes a field of Gaussian real random variables.
The extension to continuous setting can be done at the cost of analytic technicalities. More precisely, the continuous GFF on $[0,1]$ becomes a normal distribution on continuous functions defined on $[0,1]$ pinned at $0$ and $1$, which can be identified with a Brownian Bridge (BB). The continuous GFF on $[0,1]^d$, $d > 1$, is no longer a random function, it is a random distribution, which can be constructed like BB by regularizing random series in $L^2$ and using the Kolmogorov-Chentsov continuity theorem.
Further reading.
- [W] John B. Walsh
An introduction to stochastic partial differential equations
École d'Été de Probabilités de Saint-Flour XIV 1984
Lecture Notes in Mathematics 1180, Springer (1986) - [S] Daniel A. Spielman
Spectral and Algebraic Graph Theory
Book draft - Yale University lecture notes (2025) - [BCL] Jeanne Boursier, Djalil Chafaï, and Cyril Labbé
Universal cutoff for Dyson Ornstein Uhlenbeck process
Probability Theory and Related Fields 185 1-2 449-512 (2023) - [CF] D. Chafaï and M. Fathi
On cutoff via rigidity for high dimensional curved diffusions
Comptes Rendus Mathématique 363 1103--1121 (2025) - On this blog
Spectral gap concentration in high dimension
(2025-02-07)