Escape of mass in zero-range processes with random rates

Abstract

Abstract:We consider zero-range processes in Z d with site dependent jump rates. The rate for a particle jump from site x to y in Z d is given by λxg(k)p(y− x), where p(·) is a probability in Z d , g(k) is a bounded nondecreasing function of the number k of particles in x and λ = {λx} is a collection of i.i.d. random variables with values in (c, 1], for some c > 0. For almost every realization of the environment λ the zero-range process has product invariant measures {ν λ,v : 0 ≤ v ≤ c} parametrized by v, the average total jump rate from any given site. The density of a measure, defined by the asymptotic average number of particles per site, is an increasing function of v. There exists a product invariant measure ν λ,c , with maximal density. Let µ be a probability measure concentrating mass on configurations whose number of particles at site x grows less than exponentially with x . Denoting by S λ (t) the semigroup of the process, we prove that all weak limits of {µS λ (t), t ≥ 0} as t → ∞ are dominated, in the natural partial order, by ν λ,c . In particular, if µ dominates ν λ,c , then µS λ (t) converges to ν λ,c . The result is particularly striking when the maximal density is finite and the initial measure has a density above the maximal.

Condensation in the Inclusion Process and Related Models

“…Our results for the asymmetric case also cover condensation phenomena in zero-range processes, which have attracted a lot of recent research interest [5,6]. For inhomogeneous systems, these have been studied before mainly in the context of a quenched disorder in the jump rates, which have to be non-decreasing functions of the number of particles [7,8,9,10]. For such systems, the use of coupling techniques allowed in special cases to also obtain results on the dynamics of condensation.…”

Condensation in Stochastic Particle Systems with Stationary Product Measures

“…In [16] coupling techniques are used to characterize convergence to the critical stationary measure for zero-range processes with random rates u x , initialzed at supercritical densities. This is extended to zero-range processes with general non-decreasing random rates u x (n) in [17]. Further related results on hydrodynamic limits in exclusion models with particle disorder can be found in [15].…”

Monotonicity and condensation in homogeneous stochastic particle systems

“…Monotone (attractive) particle systems preserve the partial order on the state space in time, which enables the use of powerful coupling techniques to derive rigorous results on large scale dynamic properties such as hydrodynamic limits (see [17] and references therein). These techniques have also been used to study the dynamics of condensation in attractive zero-range processes with spatially inhomogeneous rates [18][19][20][21][22], and more recently [23,24]. As we discuss in Appendix A, non-monotonicity in homogeneous systems with finite critical density can be related, on a heuristic level, to convexity properties of the canonical entropy.…”