arXiv subjects
Publications and source records attributed to Tiecheng Xu.
We study the symmetric simple exclusion process in two or higher dimensions. We prove the invariance principles for the occupation time when the process starts from nonequilibrium measures. Our proof combines the martingale method and correlation estimates for the exclusion process.
In \cite{fgn1}, the hydrodynamic limit in the diffusive scaling of the symmetric simple exclusion process with a finite number of slow bonds of strength $n^{-β}$ has been studied. Here $n$ is the scaling parameter and $β>0$ is fixed. As shown in \cite{fgn1}, when $β>1$, such a limit is given by the heat equation with Neumann boundary conditions. In this work, we find more non-trivial super-diffusive scaling limits for this dynamics. Assume that there are $k$ equally spaced slow bonds in the system. If $k$ is fixed and the time scale is $k^2n^θ$, with $θ\in (2,1+β)$, the density is asymptotically constant in each of the $k$ boxes, and equal to the initial expected mass in that box, i.e., there is no time evolution. If $k$ is fixed and the time scale is $k^2n^{1+β}$, then the density is also spatially constant in each box, but evolves in time according to the discrete heat equation. Finally, if the time scale is $k^2n^{1+β}$ and, additionally, the number of boxes $k$ increases to infinity, then the system converges to the continuous heat equation on the torus, with no boundary conditions.
We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=Δu(t,x) +ξ_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$. Here, the $ξ$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and $Δ$ represents the discrete Laplacian. We focus on the case where $ξ$ is given by a rescaled symmetric simple exclusion process which converges to an Ornstein--Uhlenbeck process. By scaling the Laplacian diffusively and considering the equation on a torus, we demonstrate that in dimension $d=2$, when a suitably renormalized version of the above equation is considered, the sequence of solutions converges in law. This resolves an open problem from~\cite{EH23}, where a similar result was shown in the three-dimensional case. The novel contribution in the present work is the establishment of a gradient bound on the transition probability of a fixed but arbitrary number of labelled exclusion particles.
Consider the weakly asymmetric simple exclusion processes on the one-dimensional torus. We study the non-equilibrium fluctuation of a class of additive functionals, and show that its scaling limit is a Gaussian process. The proof is mainly based on the results obtained and techniques developed by Jara and Menezes [Non-equiliburim fluctuations of interacting particle systems, arXiv:1810.09526].
We provide a full description for the joint fluctuations of current and occupation time in the one-dimensional nonequilibrium simple symmetric exclusion process, furnishing explicit formulas for the covariances of the limiting Gaussian process. The main novelties consist of a proof of the tightness of the nonequilibrium current based on new correlation estimates, refined estimates on the discrete gradient of the transition probabilities of the SSEP, and a nonequilibrium Kipnis-Varadhan Lemma based on a Fourier approach.
We study the hydrodynamic limit of SSEP with slow boundaries on hypercubes in dimension at least two. The hydrodynamic limit equation is shown to be a heat equation with three different types of boundary conditions according to the slowness of the boundary dynamics.The proof is based on Yau's relative entropy method.
For $α\geq 1$, let $g:\mathbb N\to\mathbb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k)=(k/k-1)^α$, $k\geq 2$. Consider the symmetric nearest neighbour zero range process on the discrete torus $\mathbb T_L$ in which a particle jumps from a site, occupied by $k$ particles, to one of its neighbors with rate $g(k)$. Armendáriz and Loulakis\cite{al09} proved a strong form of the equivalence of ensembles for the invariant measure of the supercritical zero range process with $α>2$. We generalize their result to all $α\geq 1$.