相关论文: Constructing fractional Gaussian fields from long-…

In a recent work Levine et al. (2015) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility…

We study the scaling limit of a divisible sandpile model associated to a truncated $\alpha$-stable random walk. We prove that the limiting distribution is related to an obstacle problem for a truncated fractional Laplacian. We also provide,…

This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian $(-\Delta)^{-s}$ (where, in particular, we include the case $s >1$). We define a lattice…

Gaussian fields $(g_x)$ on $\mathbb{Z}_q^d$ are constructed from a class of reversible long range random walks $(X_t)_{t\in \mathbb{N}}$ on $\mathbb{Z}_q^d$ in arXiv:2510.22554. The construction is from taking the covariance function of…

We consider the fractional $\Phi^3_d$-measure on the $d$-dimensional torus, with Gaussian free field having inverse covariance $(1-\Delta)^\alpha$, and show a phase transition at $d=3\alpha$. More precisely, in a regular regime $d<3\alpha$,…

We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to the case of the free field conditioned on…

For $d \geq 3$ we obtain an approximation of the zero-average Gaussian free field on the discrete $d$-dimensional torus of large side length $N$ by the Gaussian free field on $\mathbb Z^d$, valid in boxes of roughly side length $N -…

We study the fractional Laplacian $(-\Delta)^{\sigma/2}$ on the $n$-dimensional torus $\mathbb{T}^n$, $n\geq1$. First, we present a general extension problem that describes \textit{any} fractional power $L^\gamma$, $\gamma>0$, where $L$ is…

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free…

We establish a coupling between the $\mathcal{P}(\phi)_2$ measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result…

We construct a natural discrete random field on $\mathbb{Z}^{d}$, $d\geq 5$ that converges weakly to the bi-Laplacian Gaussian field in the scaling limit. The construction is based on assigning i.i.d. Bernoulli random variables on each…

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpinski gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the…

The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an…

The Feynman-Kac formula for the exterior-value problem for the fractional Laplacian leads to a walk-outside-spheres algorithm via sampling alpha-stable Levy processes on their exit from maximally inscribed balls and sampling their…

Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of…

Fractional Gaussian fields provide a rich class of spatial models and have a long history of applications in multiple branches of science. However, estimation and inference for fractional Gaussian fields present significant challenges. This…

We show that the dissipative Abelian sandpile on a graph L can be related to a random walk on a graph which consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that…

In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in $U_{\epsilon}=U/\epsilon\cap \mathbb{Z}^d$, $U\subset \mathbb{R}^d$ and $d\geq 2$. The covariance structure of the…

This is the second in a series of two works which study the discrete Gaussian free field on the binary tree when all leaves are conditioned to be positive. In the first work ("Gaussian free field on the tree subject to a hard wall I:…

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the…