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Linjie Zhao

Publications and source records attributed to Linjie Zhao.

At least 19 recordsLinked to original sources
arXiv2026

Scaling limit of additive functionals for reversible non-gradient exclusion process: critical cases

For the reversible speed-change exclusion process $(\eta_t)_{t \geq 0}$ in $\mathbb{Z}^d$, we study the scaling limit of additive functionals ${\Gamma_t(f) = \int_0^t f(\eta_s)\, \mathrm{d} s}$. Concerning the local centered function $f$, the previous work [Commun. Math. Phys. 104, 1-19, 1986] by Kipnis and Varadhan and [Comm. Pure Appl. Math., 66: 649-677, 2013] by Gon{\c{c}}alves and Jara respectively covered the cases $d \geq 3$ and $d=1$. The present paper completes the missing part $d=2$, and also develops the theory for functions with higher degree. The novelty is a quantitative homogenization of the resolvent, which allows to overcome the obstacle of correlation function in non-gradient models.

math.PR↗
arXiv2026

Nonequilibrium fluctuations and moderate deviations for the occupation time of the SSEP with Glauber dynamics

We study the symmetric simple exclusion process with Glauber dynamics. When the process starts from a nonequilibrium measure, we prove central limit theorems for the occupation time in dimension two, and sample path moderate deviation principles in dimension one. For the fluctuations, we use the martingale method and the sharp relative entropy method from [Jara and Menezes, arXiv:1810.09526]. For the moderate deviations, the main idea is to relate the occupation time to the density fluctuation field by using the logarithmic Sobolev inequality from the Glauber dynamics.

math.PR↗
arXiv2025

Moderate deviation principles for the current and the tagged particle in the WASEP

We study the weakly asymmetric simple exclusion process in one dimension. We prove sample path moderate deviation principles for the current and the tagged particle when the process starts from one of its stationary measures. We simplify the proof in our previous works [Xue and Zhao, Electronic Journal of Probability, 2024] and [Xue and Zhao, Stochastic Processes and their Applications, 2023], where the same problem was investigated in the symmetric simple exclusion process.

math.PR↗
arXiv2025

Moderate deviations for the facilitated exclusion process in equilibrium

We derive the moderate deviation principles for the fluctuation fields of the facilitated exclusion process (FEP) in one dimension when the process starts from its stationary measure, both in the symmetric and asymmetric cases. The main step is to prove a super-exponential version of the Boltzmann-Gibbs principle, which relies on the logarithmic Sobolev inequality for the FEP.

math.PR↗
arXiv2024

Stationary fluctuations for the WASEP with long jumps and infinitely extended reservoirs

We study a weakly asymmetric exclusion process with long jumps and with infinitely many extended reservoirs. We prove that the stationary fluctuations of the process are governed by the generalized Ornstein-Uhlenbeck process or the stochastic Burgers equation with Dirichlet boundary conditions depending on the strength of the asymmetry of the dynamics.

math.PR↗
arXiv2024

Moderate deviation principles for a reaction diffusion model in non-equilibrium

We study moderate deviations from hydrodynamic limits of a reaction diffusion model. The process is defined as the superposition of the symmetric exclusion process with a Glauber dynamics. When the process starts from a product measure with a constant density, which is a non-equilibrium measure for the process, we prove that the re-scaled density fluctuation field satisfies the moderate deviation principle. Our proof relies on the so-called main lemma developed by Jara and Menezes in [arXiv:1810.09526, 2018].

math.PR↗
arXiv2024

Moderate deviation principles for the WASEP

We study the weakly asymmetric simple exclusion process on the integer lattice. Under suitable constraints on the strength of the weak asymmetry of the dynamics, we prove moderate deviation principles for the fluctuation fields when the process starts from stationary measures. As an application, we obtain sample path moderate deviation principles for the occupation time of the process in one dimension.

math.PR↗
arXiv2024

Hydrodynamics for asymmetric simple exclusion on a finite segment with Glauber-type source

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a relaxation term. Different from the usual boundary conditions introduced in [Bardos, Roux, and Nedelec, (1979), Comm. Part. Diff. Equ], discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme.

math.PR↗
arXiv2023

Stationary fluctuations for the facilitated exclusion process

We derive the stationary fluctuations for the Facilitated Exclusion Process (FEP) in one dimension in the symmetric, weakly asymmetric and asymmetric cases. Our proof relies on the mapping between the FEP and the zero-range process, and extends the strategy in \cite{erignoux2022mapping}, where hydrodynamic limits were derived for the FEP, to its stationary fluctuations. Our results thus exploit works on the zero-range process's fluctuations \cite{gonccalves2010equilibrium,gonccalves2015stochastic}, but we also provide a direct proof in the symmetric case, for which we derive a sharp estimate on the equivalence of ensembles for the FEP's stationary states.

math.PR↗
arXiv2023

Stationary fluctuations for a multi-species zero range process with long jumps

We consider stationary fluctuations for the multi-species zero range process with long jumps in one dimension, where the underlying transition probability kernel is $p(x) = c_+ |x|^{-1-\alpha}$ if $x > 0$ and $= c_-|x|^{-1-\alpha}$ if $x < 0$. Above, $c_{\pm} \geq 0, \alpha > 0$ are parameters. We prove that for $0 < \alpha < 3/2$, the density fluctuation fields converge to the stationary solution of a coupled fractional Ornstein-Uhlenbeck process, and for $\alpha=3/2$, the limit points are concentrated on stationary energy solutions to a coupled fractional Burgers equation.

math.PR↗
arXiv2023

The voter model with a slow membrane

We introduce the voter model on the infinite lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The model is defined as follows: a voter adopts one of its neighbors' opinion at rate one except for neighbors crossing the hyperplane $\{x:x_1 = 1/2\}$, where the rate is $αN^{-β}$. Above, $α>0,\,β\geq 0$ are two parameters and $N$ is the scaling parameter. The hydrodynamic equation turns out to be heat equation with various boundary conditions depending on the value of $β$. For the nonequilibrium fluctuations, the limit is described by generalized Ornstein-Uhlenbeck process with certain boundary condition corresponding to the hydrodynamic equation.

math.PR↗
arXiv2022

Equilibrium Perturbations for Asymmetric Zero Range Process under Diffusive Scaling in Dimensions $d \geq 2$

We consider the asymmetric zero range process in dimensions $d \geq 2$. Assume the initial density profile is a perturbation of the constant density, which has order $N^{-α}$, $α\in (0,1)$, and is constant along the drift direction. Here, $N$ is the scaling parameter. We show that under some constraints on the jump rate of the zero range process, the perturbed quantity macroscopically obeys the heat equation under diffusive scaling.

math.PR↗
arXiv2022

Equilibrium perturbations for stochastic interacting systems

We consider the equilibrium perturbations for two stochastic systems: the $d$-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order $N^{-\alpha}$ to the equilibrium profile and speed up the process by $N^{1+\kappa}$ for parameters $0<\kappa\le\alpha$. Under some additional constraints on $\kappa$ and $\alpha$, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.

math.PR↗
arXiv2022

Moderate Deviations for the current and Tagged Particle in Symmetric Simple Exclusion Processes

We prove moderate deviation principles for the tagged particle position and current in one-dimensional symmetric simple exclusion processes. There is at most one particle per site. A particle jumps to one of its two neighbors at rate $1/2$, and the jump is suppressed if there is already one at the target site. We distinguish one particular particle which is called the tagged particle. We first establish a variational formula for the moderate deviation rate functions of the tagged particle positions based on moderate deviation principles from hydrodynamic limit proved by Gao and Quastel \cite{gao2003moderate}. Then we construct a minimizer of the variational formula and obtain explicit expressions for the moderate deviation rate functions.

math.PR↗
arXiv2022

Mapping hydrodynamics for the facilitated exclusion and zero-range processes

We derive the hydrodynamic limit for two degenerate lattice gases, the \emph{facilitated exclusion process} (FEP) and the \emph{facilitated zero-range process} (FZRP), both in the symmetric and the asymmetric case. For both processes, the hydrodynamic limit in the symmetric case takes the form of a diffusive Stefan problem, whereas the asymmetric case is characterized by a hyperbolic Stefan problem. Although the FZRP is attractive, a property that we extensively use to derive its hydrodynamic limits in both cases, the FEP is not. To derive the hydrodynamic limit for the latter, we exploit that of the zero-range process, together with a classical mapping between exclusion and zero-range processes, both at the microscopic and macroscopic level. Due to the degeneracy of both processes, the asymmetric case is a new result, but our work also provides a simpler proof than the one that was previously proposed for the FEP in the symmetric case in \cite{blondel2021stefan}.

math.PR↗