The processes described in the title always have reversible stationary distributions. In this paper, we give sufficient conditions for the existence of, and for the nonexistence of, nonreversible stationary distributions. In the case of an i.i.d. environment, these combine to give a necessary and sufficient condition for the existence of nonreversible stationary distributions.
We obtain the hydrodynamic limit of a simple exclusion process in an inhomogeneous environment of divergence form. Our main assumption is a suitable version of Gamma-convergence for the environment. In this way we obtain an unified approach to recent works on the field.
We prove that the diffusion coefficient for the asymmetric exclusion process diverges at least as fast as $t^{1/4}$ in dimension $d=1$ and $(\log t)^{1/2}$ in $d=2$. The method applies to nearest and non-nearest neighbor asymmetric exclusion processes.
Laws of large numbers, starting from certain nonequilibrium measures, have been shown for the integrated current across a bond, and a tagged particle in one-dimensional symmetric nearest-neighbor simple exclusion [Ann. Inst. Henri Poincare Probab. Stat. 42 (2006) 567-577]. In this article, we prove corresponding large deviation principles and evaluate the rate functions, showing different growth behaviors near and far from their zeroes which connect with results in [J. Stat. Phys. 136 (2009) 1-15].
Let $J(t)$ be the the integrated flux of particles in the symmetric simple exclusion process starting with the product invariant measure $ν_ρ$ with density $ρ$. We compute its rescaled asymptotic variance: \[ \lim_{t\to\infty} t^{-1/2} \V J(t) = \sqrt{2/π} (1-ρ)ρ\] Furthermore we show that $t^{-1/4}J(t)$ converges weakly to a centered normal random variable with this variance. From these results we compute the asymptotic variance of a tagged particle in the nearest neighbor case and show the corresponding central limit theorem, results previously proven by Arratia.
In \cite{J} M. Jara has presented a method, reducing the proof of the hydrodynamic limit of symmetric exclusion processes to an homogenization problem, as unified approach to recent works on the field as \cite{N}, \cite{F1}, \cite{F2} and \cite{FJL}. Although not stated in \cite{J}, the reduction of the hydrodynamic limit to an homogenization problem was already obtained (in a different way) in \cite{N}, \cite{F1}. This alternative and very simple relation between the two problems goes back to an idea of K.\ Nagy \cite{N}, is stated in \cite{F1}[Section B] for exclusion processes on $\bbZ^d$ a…
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.
The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Seppäläinen and Balázs recently proved this conjecture for the nearest neighbor case. We extend their results to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.
We study aspects of the hydrodynamics of one-dimensional totally asymmetric K-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We prove that the weak solution chosen by the particle system is the unique one with maximal current past any fixed location. A uniqueness result is needed because we can prove neither differentiability nor strict concavity of the flux function, so we cannot use the Lax-Oleinik formula or jump conditions to define entropy solutions. Next we prove laws of large numbers for a second class particle in K-exclusion. The macroscopic trajectories of second…
We consider a one-dimensional nearest-neighbor interacting particle system, which is a mixture of the simple exclusion process and the voter model. The state space is taken to be the countable set of the configurations that have a finite number of particles to the right of the origin and a finite number of empty sites to the left of it. We obtain criteria for the ergodicity and some other properties of this system using the method of Lyapunov functions.
For a sequence of i.i.d. random variables $\{ξ_x : x\in \bb Z\}$ bounded above and below by strictly positive finite constants, consider the nearest-neighbor one-dimensional simple exclusion process in which a particle at $x$ (resp. $x+1$) jumps to $x+1$ (resp. $x$) at rate $ξ_x$. We examine a quenched nonequilibrium central limit theorem for the position of a tagged particle in the exclusion process with bond disorder $\{ξ_x : x\in \bb Z\}$. We prove that the position of the tagged particle converges under diffusive scaling to a Gaussian process if the other particles are initially distribute…
We show that if $μ$ is an invariant measure for the long range exclusion process putting no mass on the full configuration, $L$ is the formal generator of that process and $f$ is a cylinder function, then $Lf\in\mathbf{L}^1(dμ)$ and $\int Lf dμ=0$. This result is then applied to determine (i) the set of invariant and translation-invariant measures of the long range exclusion process on $\mathbb{Z}^d$ when the underlying random walk is irreducible; (ii) the set of invariant measures of the long range exclusion process on $\mathbb{Z}$ when the underlying random walk is irreducible and either has…
We consider finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift. The diffusivity $D(t)$ is expected to be of ${\mathcal O}(t^{1/3})$. We prove that $D(t)\ge Ct^{1/3}$ in the weak (Tauberian) sense that $\int_0^\infty e^{-λt}tD(t)dt \ge Cλ^{-7/3}$ as $λ\to 0$. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that $tD(t)$ is mono…
We give a characterization of the invariant measures for the exclusion process on the integers with certain reversible transition kernels. Some examples include all nearest-neighbor kernels with asymptotic mean zero. One tool used is a necessary and sufficient condition for reversible measures to be extremal in the set of all invariant measures which is an interesting result in its own right.
We consider the PushTASEP (pushing totally asymmetric simple exclusion process, also sometimes called long-range TASEP) with the step initial configuration evolving in an inhomogeneous space. That is, the rate of each particle's jump depends on the location of this particle. We match the distribution of the height function of this PushTASEP with Schur processes. Using this matching and determinantal structure of Schur processes, we obtain limit shape and fluctuation results which are typical for stochastic particle systems in the Kardar-Parisi-Zhang universality class. PushTASEP is a close rel…
We consider the asymmetric exclusion process with a driven tagged particle on Z which has different jump rates from other particles and show that the tagged particle can have a ballistic behavior when the non-tagged particles have non-nearest-neighbor jump rates. We show the existence of some non-trivial invariant measures for the environment process viewed from the tagged particle. Our arguments are based on coupling, the martingale approach, and analyzing currents through fixed bonds.
We introduce a new method, which we call stochastic fusion, which takes an exclusion process and constructs an interacting particle systems in which more than one particle may occupy a lattice site. The construction only requires the existence of stationary measures of the original exclusion process on a finite lattice. If the original exclusion process satisfies Markov duality on a finite lattice, then the construction produces Markov duality functions (for some initial conditions) for the fused exclusion process. The stochastic fusion construction is based off of the Rogers-Pitman intertwini…
Consider a system of particles performing nearest neighbor random walks on the lattice $\ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $\a$--stable law, $0<\a<1$. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function $L$, under the super-diffusive time scaling $N^{1 + 1/α}L(N)$, the density profile evolves as the solut…
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.
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…
Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. In previous work we proved a law of large numbers for dynamic random environments satisfying a space-time mixing property called cone-mixing. If an attractive spin-flip system has a finite average coupling time at the origin for two copies starting from the all-occupied and the all-vacant configuration, respectively, t…
We consider a large class of nearest neighbor attractive stochastic interacting systems that includes the asymmetric simple exclusion, zero range, bricklayers' and the symmetric K-exclusion processes. We provide exact formulas that connect particle flux (or surface growth) fluctuations to the two-point function of the process and to the motion of the second class particle. Such connections have only been available for simple exclusion where they were of great use in particle current fluctuation investigations.
Let $Λ$ be a connected closed region with smooth boundary contained in the $d$-dimensional continuous torus $\bb T^d$. In the discrete torus $N^{-1} \bb T^d_N$, we consider a nearest neighbor symmetric exclusion process where occupancies of neighboring sites are exchanged at rates depending on $Λ$ in the following way: if both sites are in $Λ$ or $Λ^\complement$, the exchange rate is one; If one site is in $Λ$ and the other one is in $Λ^\complement$ and the direction of the bond connecting the sites is $e_j$, then the exchange rate is defined as $N^{-1}$ times the absolute value of the inner p…
We study a symmetric randomly moving line interacting by exclusion with a wall. We show that the expectation of the position of the line at the origin when it starts attached to the wall satisfies the following bounds: c_1t^{1/4} \le\Eξ_t(0) \le c_2 t^{1/4}\log t The result is obtained by comparison with a ``free'' process, a random line that has the same behavior but does not see the wall. The free process is isomorphic to the symmetric nearest neigbor one-dimensional simple exclusion process. The height at the origin in the interface model corresponds to the integrated flux of particles thro…
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…
We study the hydrodynamic limits of the simple exclusion processes and the zero range processes on crystal lattices. For a periodic realization of crystal lattice, we derive the hydrodynamic limit for the exclusion processes and the zero range processes, which depends on both the structure of crystal lattice and the periodic realization. Even through the crystal lattices have inhomogeneous local structure, for all periodic realizations, we apply the entropy method to derive the hydrodynamic limits. Also, we discuss how the limit equation depends on the choices of the realizations.
We consider the symmetric simple exclusion processes with a slow site in the discrete torus with $n$ sites. In this model, particles perform nearest-neighbor symmetric random walks with jump rates everywhere equal to one, except at one particular site, \textit{the slow site}, where the jump rate of entering that site is equal to one, but the jump rate of leaving that site is given by a parameter $g(n)$. Two cases are treated, namely $g(n)=1+o(1)$, and $g(n)=αn^{-β}$ with $β>1$, $α>0$. In the former, both the hydrodynamic behavior and equilibrium fluctuations are driven by the heat equation (wi…
We compute the growth fluctuations in equilibrium of a wide class of deposition models. These models also serve as general frame to several nearest-neighbor particle jump processes, e.g. the simple exclusion or the zero range process, where our result turns to current fluctuations of the particles. We use martingale technique and coupling methods to show that, rescaled by time, the variance of the growth as seen by a deterministic moving observer has the form |V-C|*D, where V and C is the speed of the observer and the second class particle, respectively, and D is a constant connected to the eq…
We investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diffusive scaling the shape of the interfaces converges in probability to the solution of a non-linear parabolic equation. This Law of Large Numbers is obtained from the Hydrodynamic limit of a coupling between an exclusion process and an inhomogeneous one dimensional zero range process with asymmetry at the origin.
