Alexander Vandenberg-Rodes
Using the recently discovered strong negative dependence properties of the symmetric exclusion process, we derive general conditions for when the normalized current of particles between regions converges to the Gaussian distribution. The main novelty is that the results do not assume any translation invariance, and hold for most initial configurations.
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We prove Gaussian tail estimates for the transition probability of $n$ particles evolving as symmetric exclusion processes on $\bb Z^d$, improving results obtained in \cite{l}. We derive from this result a non-equilibrium Boltzmann-Gibbs principle for the symmetric simple exclusion process in dimension 1 starting from a product measure with slowly varying parameter.
For the exclusion process with symmetric kernel p(x,y)=p(y,x), the set of invariant measures has been completely studied. This paper gives results concerning the invariant measures for exclusion processes where p(x,y)=p(y,x) except for finitely many (x,y).
We prove hydrostatics of boundary driven gradient exclusion processes, Fick's law and we present a simple proof of the dynamical large deviations principle which holds in any dimension
We prove a nonequilibirum central limit theorem for the position of a tagged particle in the one-dimensional nearest-neighbor symmetric simple exclusion process under diffusive scaling starting from a Bernoulli product measure associated to a smooth profile $ρ_0:\bb R\to [0,1]$.
Non-trivial linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale renormalization methods of Kesten and Sidoravicius.
We prove an energy estimate for the polar empirical measure of the two-dimensional symmetric simple exclusion process. We deduce from this estimate and from results in reference [2] large deviations principles for the polar empirical measure and for the occupation time of the origin.
Dynamical large deviations for boundary driven gradient symmetric exclusion processes in mild contact with reservoirs
研究一维梯度对称排斥过程在温和边界接触下的动力学大偏差，证明其流体动力学极限满足非线性Robin边界抛物方程，并建立大偏差原理。
We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.
We consider a one-dimensional symmetric simple exclusion process in contact with slowed reservoirs: at the left (resp. right) boundary, particles are either created or removed at rates given by $α/n$ or $(1-α)/n$ (resp. $β/n$ or $(1-β)/n$) where $α, β>0$ and $n$ is a scaling parameter. We obtain the non-equilibrium fluctuations and consequently the non-equilibrium stationary fluctuations.
We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes, a Brownian motion and a Gaussian process with stationary increments.
We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.
A well-known result with respect to the one dimensional nearest-neighbor symmetric simple exclusion process is the convergence to fractional Brownian motion with Hurst parameter 1/4, in the sense of finite-dimensional distributions, of the subdiffusively rescaled current across the origin, and the subdiffusively rescaled tagged particle position. The purpose of this note is to improve this convergence to a functional central limit theorem, with respect to the uniform topology, and so complete the solution to a conjecture in the literature with respect to simple exclusion processes.
We consider the one dimensional symmetric simple exclusion process (SSEP) with additional births and deaths restricted to a subset of configurations where there is a leftmost hole and a rightmost particle. At a fixed rate birth of particles occur at the position of the leftmost hole and at the same rate, independently, the rightmost particle dies. We prove convergence to a hydrodynamic limit and discuss its relation with a free boundary problem.
We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly non-uniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for…
We consider a finite range symmetric exclusion process on the integer lattice in any dimension. We interpret it as a non-elliptic time-dependent random conductance model by setting conductances equal to one over the edges with end points occupied by particles of the exclusion process and to zero elsewhere. We prove a law of large number and a central limit theorem for the random walk driven by such a dynamical field of conductances by using the Kipnis-Varhadan martingale approximation. Unlike the tagged particle in the exclusion process, which is in some sense similar to this model, this rando…
We study the hydrodynamic limit for a model of symmetric exclusion processes with heavy-tailed long jumps and in contact with infinitely extended reservoirs. We show how the corresponding hydrodynamic equations are affected by the parameters defining the model. The hydrodynamic equations are characterized by a class of super-diffusive operators that are given by the regional fractional Laplacian with some additional reaction terms and various boundary conditions. This work solves some questions left open in \cite{BJGO2} about the same model.
In this paper we focus on the open symmetric exclusion process with parameter $m$ (open SEP($m/2$)), which allows $m$ particles each site and has an open boundary. We generalize the result about hydrodynamic limit for the open SEP$(m/2)$ that was originally raised in Theorem 4.12 of arXiv:1908.02359. We prove that the hydrodynamic limit of the density profile for a $d-$dimensional open SEP$(m/2)$ solves the $(d+1)-$dimensional heat equation with certain initial condition and boundary condition.
Hydrodynamical behavior for the generalized symmetric exclusion with open boundary
本文研究了允许每个位置最多$α$个粒子的广义对称排斥过程，将其与强度由参数$θ$调节的随机水库接触，证明其流体动力学行为由热方程描述，且边界条件随$θ$变化，并在$α=1$时与已知结果一致。
We consider a one-dimensional simple symmetric exclusion process in equilibrium, constituting a dynamic random environment for a nearest-neighbor random walk that on occupied/vacant sites has two different local drifts to the right. We construct a renewal structure from which a LLN, a functional CLT and large deviation bounds for the random walk under the annealed measure follow. We further prove an Einstein relation under a suitable perturbation. A brief discussion on the topic of random walks in slowly mixing dynamic random environments is presented.
In this paper, we will study the long time behavior of the simple symmetric exclusion process in the ``channel'' $Λ_N=[1,N]\cap \mathbb{N}$ with reservoirs at the boundaries. These reservoirs are also systems of particles which can be exchanged with the particles in the channel. The size $M$ of each reservoir is much larger than the one of the channel, i.e. $M=N^{1+α}$ for a fixed number $α>0$. Based on the size of the channel and the holding time at each reservoir, we will investigate some types of rescaling time.
A quantitative central limit theorem for the simple symmetric exclusion process
本文通过比较 SSEP 密度涨落场与广义 Ornstein-Uhlenbeck 过程的生成元，并利用无穷维 Berry-Essen 界，证明了 d 维离散环面上简单对称排他过程的最优速率定量中心极限定理。
We consider a driven tagged particle in a symmetric exclusion process on Z with a removal rule. In this process, untagged particles are removed once they jump to the left of the tagged particle. We investigate the behavior of the displacement of the tagged particle and prove limit theorems of it: an (annealed) law of large numbers, a central limit theorem, and a large deviation principle. We also characterize a class of ergodic measures for the environment process. Our approach is based on analyzing two auxiliary processes with associated martingales and a regenerative structure.
Nonequilibrium Joint Fluctuations for Current and Occupation Time in The Symmetric Exclusion Process
本文研究一维非平衡 SSEP 中电流与占据时间的联合涨落，给出极限高斯过程协方差公式，并通过相关性估计、离散梯度估计和 Fourier 型 Kipnis–Varadhan 引理证明紧性。
We prove an upper bound for the $\varepsilon$-mixing time of the symmetric exclusion process on any graph G, with any feasible number of particles. Our estimate is proportional to $\mathsf{T}_{\mathsf{RW}(G)}\ln(|V|/\varepsilon)$, where |V| is the number of vertices in G, and $\mathsf{T}_{\mathsf{RW}(G)}$ is the 1/4-mixing time of the corresponding single-particle random walk. This bound implies new results for symmetric exclusion on expanders, percolation clusters, the giant component of the Erdos-Renyi random graph and Poisson point processes in $\mathbb{R}^d$. Our technical tools include a …
We consider a random walk (RW) driven by a simple symmetric exclusion process (SSE). Rescaling the RW and the SSE in such a way that a joint hydrodynamic limit theorem holds we prove a joint path large deviation principle. The corresponding large deviation rate function can be split into two components, the rate function of the SSE and the one of the RW given the path of the SSE. Such components have different structures (Gaussian and Poissoinian, respectively) and to overcome this difficulty we make use of the theory of Orlicz spaces. In particular, the component of the rate function correspo…
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{g…
We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process "seen from the tagged particle" has an ergodic invariant measure.
In this article we obtain the equilibrium fluctuations of a symmetric exclusion process in $\mathbb{Z}$ with long jumps. The transition probability of the jump from $x$ to $y$ is proportional to $|x-y|^{-γ-1}$. Here we restrict to the choice $γ\geq 2$ so that the system has a diffusive behavior. Moreover, when particles move between $\mathbb{Z}_{-}^{*}$ and $\mathbb N$, the jump rates are slowed down by a factor $αn^{-β}$, where $α>0$, $β\geq 0$ and $n$ is the scaling parameter. Depending on the values of $β$ and $γ$, we obtain several stochastic partial differential equations, corresponding t…
Correction to Annals of Probability 28 (2000) 277--302 [doi:10.1214/aop/1019160120].
We prove a Functional Central Limit Theorem for the position of a Tagged Particle in the one-dimensional Asymmetric Simple Exclusion Process in the hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the Tagged Particle at time $t$ depends on the initial configuration, by the number of empty sites in the interval $[0,(p-q)αt]$ divided by $α$ in the hyperbolic and in a longer time scale, namely $N^{4/3}$.
Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles
研究在泊松随机邻域图逼近带下界曲率的加权黎曼流形上对称排斥过程的水动力极限，推导福克-规划方程，并扩展到主丛，得到带有加权水平拉普拉斯算子的福克-规划方程。
We present the correct space of test functions for the Ornstein-Uhlenbeck processes defined in \cite{fgn2}. Under these new spaces, an invariance with respect to a second order operator is shown, granting the existence and uniqueness of those processes. Moreover, we detail how to prove some properties of the semi-groups, which are required in the proof of uniqueness.
In the boundary driven symmetric simple exclusion process consider an open set $\ms O$ of density profiles which does not contain the stationary density profile. We prove that the first time the empirical measure visits the set $\ms O$ converges to an exponential distribution.
In~\cite{bgs2013}, exclusion sensitivity and exclusion stability for symmetric exclusion processes on graphs were defined as a natural analogue of noise sensitivity and noise stability in this setting. As these concepts were defined for any sequence of connected graphs, it is natural to study the monotonicity properties of these definitions with respect to adding edges to the graphs, and in particular, whether some graphs are more stable or sensitive than others. The main purpose of this paper is to answer some such question from~\cite{bgs2013}. The main tool used is included results about the…
Diameters of symmetric and lifted simple exclusion models
本文确定了一维N粒子排他模型（Ssep和提升的Tasep）马尔可夫链的直径，为混合时间提供下界，并探讨了其对模型分析的意义。
We examine the entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes. In contrast with the Gibbs--Shannon entropy \cite{B, DLS2}, the entropy of nonequilibrium stationary states differs from the entropy of local equilibrium states.
In this article, we consider a one-dimensional symmetric exclusion process in weak contact with reservoirs at the boundary. In the diffusive time-scaling the empirical measure evolves according to the heat equation with Robin boundary conditions. We prove the associated dynamical large deviations principle.
We consider the symmetric exclusion process with jumps given by a symmetric, translation invariant, transition probability $p(\cdot)$. The process is put in contact with stochastic reservoirs whose strength is tuned by a parameter $θ\in\mathbb R$. Depending on the value of the parameter $θ$ and the range of the transition probability $p(\cdot)$ we obtain the hydrodynamical behavior of the system. The type of hydrodynamic equation depends on whether the underlying probability $p(\cdot)$ has finite or infinite variance and the type of boundary condition depends on the strength of the stochastic …
Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and normal distributions for various functionals of the process.
We consider the symmetric simple exclusion process in $\mathbb Z^d$ with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic behavior of the density field. While the hydrodynamic limit at fixed macroscopic times is obtained via a generalization to the time-inhomogeneous context of the strategy introduced in \cite{nagy_symmetric_2002}, in order to prove tightness for the sequence of em…
Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph $G=(V,E)$ dual to the biased voter model on $G$. Our main goal are tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as $|V|\rightarrow \infty$. Our results allow us to recover very directly and improve to $\ell^p$-mixing, $p\ge 2$, and to more general graphs, the mixing time r…
We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $ρ\in [0, 1]$ of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $ρ$ except for at most two values $ρ_-, ρ_+ \in [0, 1]$. The asymptotic speed we obtain in our LLN is a monotone function of…
Dormancy in random environment: Symmetric exclusion
本文构建随机环境下含休眠态的两类分支随机游走，借助 Feynman–Kac 公式与扩展的抛物 Anderson 模型分析期望种群规模的长时间渐近，量化休眠对增长与生存的影响。
We prove that the equilibrium fluctuations of the symmetric simple exclusion process in contact with slow boundaries is given by an Ornstein-Uhlenbeck process with Dirichlet, Robin or Neumann boundary conditions depending on the range of the parameter that rules the slowness of the boundaries.
We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it converges to the solution of the heat equation on the manifold.
Hydrodynamic limit of an exclusion process with vorticity
本文构造了具有伯努利乘积不变测度的不可逆排他过程，推导其扩散标度下的非对称扩散矩阵，揭示弱外场下迁移率矩阵与扩散矩阵对称部分的爱因斯坦关系，该现象在广义梯度模型中具有典型性。
The Quantum Symmetric Simple Exclusion Process (QSSEP) is a model of quantum particles hopping on a finite interval and satisfying the exclusion principle. Recently Bernard and Jin have studied the fluctuations of the invariant measure for this process, when the number of sites goes to infinity. These fluctuations are encoded into polynomials, for which they have given equations and proved that these equations determine the polynomials completely. In this paper, I give an explicit combinatorial formula for these polynomials, in terms of Schröder trees. I also show that, quite surprisingly, the…
Scaling Limits of Density Fluctuation Fields of the Symmetric Partial Exclusion Processes Approaching the Zero Range Process
本文研究一维对称部分排斥过程（SPEP）的密度涨落场标度极限，证明其在扩散标度下收敛到与对称零程过程（SZRP）一致的随机热方程解，为SPEP近似SZRP提供严格依据。
We give a new approach to the well-known convergence to the hydrodynamic limit for the symmetric simple exclusion process (SSEP). More precisely, we characterize any possible limit of its empirical density measures as solutions to the heat equation by passing to the limit in the gradient flow structure of the particle system.
Stationary fluctuations for the facilitated exclusion process
本文利用FEP与零程过程的映射，推导了一维FEP在对称、弱不对称和不对称情形下的平稳涨落，并在对称情形下通过系综等价性给出了直接证明。
We give bounds on the rate of convergence to equilibrium of the symmetric simple exclusion process in $\Z^d$. Our results include the existent results in the literature. We get better bounds and larger class of initial states via a unified approach. The method includes a comparison of the evolution of n interacting particles with n independent ones along the whole time trajectory.
Moderate deviations for the facilitated exclusion process in equilibrium
本文推导了一维促进排斥过程在平稳测度下对称与非对称情形的涨落场中度偏差原理，核心是通过基于对数Sobolev不等式证明Boltzmann-Gibbs原理的超指数版本。
Consider the symmetric exclusion process evolving on an interval and weakly interacting at the end-points with reservoirs. Denote by $I_{[0,T]} (\cdot)$ its dynamical large deviations functional and by $V(\cdot)$ the associated quasi-potential, defined as $V(γ) = \inf_{T>0} \inf_u I_{[0,T]} (u)$, where the infimum is carried over all trajectories $u$ such that $u(0) = \barρ$, $u(T) = γ$, and $\barρ$ is the stationary density profile. We derive the partial differential equation which describes the evolution of the optimal trajectory, and deduce from this result the formula obtained by Derrida, …
We provide a complete description of the equilibrium fluctuations for diffusive symmetric exclusion processes with long jumps in contact with infinitely extended reservoirs and prove that they behave as generalized Ornstein-Uhlenbeck processes with various boundary conditions, depending mainly on the strength of the reservoirs. On the way, we also give a general statement about uniqueness of the Ornstein-Uhlenbeck process originated by the microscopic dynamics of the underlying interacting particle systems and adapt it to our study.
Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1
本文研究 d≥2 维对称排斥过程在阶梯初始条件下越过边界的粒子数，证明其收敛到泊松分布，并给出极值位置的 Gumbel 极限分布。
Conservative stochastic PDE and fluctuations of the symmetric simple exclusion process
本文提出带非线性保守噪声的 SPDE 逼近序列，刻画对称简单排斥过程在流体动力学极限附近的涨落与稀有事件，并证明零噪声大偏差速率函数一致。
We consider the Symmetric Exclusion Process on a compact Riemannian manifold, as introduced in van Ginkel and Redig (2020). There it was shown that the hydrodynamic limit satisfies the heat equation. In this paper we study the equilibrium fluctuations around this hydrodynamic limit. We define the fluctuation fields as functionals acting on smooth functions on the manifold and we show that they converge in distribution in the path space to a generalized Ornstein-Uhlenbeck process. This is done by proving tightness and by showing that the limiting fluctuations satisfy the corresponding martingal…
Coupling hydrodynamics of several Facilitated Exclusion Processes with closed boundaries
本文通过构建映射将 Facilitated Exclusion Process 与 Simple Exclusion Process 耦合，证明了前者在对称、不对称及弱不对称机制下的流体动力学极限。
For the two-sided Bernoulli initial condition with density $ρ_-$ (resp. $ρ_+$) to the left (resp. to the right), we study the distribution of a tagged particle in the one dimensional symmetric simple exclusion process. We obtain a formula for the moment generating function of the associated current in terms of a Fredholm determinant. Our arguments are based on a combination of techniques from integrable probability which have been developed recently for studying the asymmetric exclusion process and a subsequent intricate symmetric limit. An expression for the large deviation function of the ta…
The symmetric exclusion process and the voter model are two interacting particle systems for which a dual finite particle system allows one to characterize its invariant measures. Adding spontaneous births and deaths to the two processes still allows one to use the dual to obtain information about the original process. We study the noisy voter-exclusion process which generalizes these processes by allowing for all of these interactions to take place. The dual process is used to characterize its invariant measures under various circumstances. Finally, an ergodic theorem for a related process is…
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