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In this paper we investigate the scaling limit of the range (the set of visited vertices) for a class of critical lattice models, starting from a single initial particle at the origin. We give conditions on the random sets and an associated ancestral relation under which, conditional on longterm survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of exiting a large ball, that is for the extrinsic one-arm probabililty. We show that these conditions are satisfied by the voter model in dimensions $d ge 2$ and critical sufficiently spread out lattice trees in dimensions $d > 8$. The latter result also has important consequences for the behaviour of random walks on lattice trees in high dimensions. We conjecture that our conditions are also satisfied by other models (at criticality above the critical dimension) such as sufficiently spread out oriented percolation and contact processes in dimensions d > 4. This version of the paper contains details not present in the submitted version.
Consider the centered Gaussian field on the lattice $mathbb{Z}^d,$ $d$ large enough, with covariances given by the inverse of $sum_{j=k}^K q_j(-Delta)^j,$ where $Delta$ is the discrete Laplacian and $q_j in mathbb{R},kleq jleq K,$ the $q_j$ satisfyin
In this article, we try to give an answer to the simple question: ``textit{What is the critical growth rate of the dimension $p$ as a function of the sample size $n$ for which the Central Limit Theorem holds uniformly over the collection of $p$-dimen
We consider long-range self-avoiding walk, percolation and the Ising model on $mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)asymp|x|^{-d-alpha}$ with $alpha>0$. The upper-critical dimension $d_{mathrm{c}}$ is $
The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively.
We consider a damped/driven nonlinear Schrodinger equation in an $n$-cube $K^{n}subsetmathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions [ u_t- uDelta u+i|u|^2u=sqrt{ u}eta(t,x),quad xin K^{n},quad u|_{partial K^{n}}=0, quad u>0, ]