Symmetric exclusion as a random environment: Invariance principle

Jara, Milton; Menezes, Otávio

doi:10.1214/20-aop1466

Cited by 5 publications

(3 citation statements)

References 27 publications

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“…The asymptotic behavior of a random walker in this setup is studied in [HKT20, HS15], and the hydrodynamic limit for the exclusion process as seen by this walker is studied in [AFJV15]. In a more general setup for the jump rates of the walker, an invariance principle about the random walk when the exclusion process starts from equilibrium is studied in [JM20].…”

Section: Model and Resultsmentioning

confidence: 99%

Mixing time for the asymmetric simple exclusion process in a random environment

Lacoin,

Yang

2021

Preprint

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We consider the simple exclusion process in the integer segment 1, N with k ≤ N/2 particles and spatially inhomogenous jumping rates. A particle at site x ∈ 1, N jumps to site x−1 (if x ≥ 2) at rate 1−ωx and to site x+1 (if x ≤ N −1) at rate ωx if the target site is not occupied. The sequence ω = (ωx) x∈Z is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume E[log ρ1] < 0 where ρ1 := (1 − ω1)/ω1, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of N . More precisely, for the exclusion process with N β+o(1) particles where β ∈ [0, 1), we have in the large N asymptoticthe equation has no positive root) and C is a constant which depends on the distribution of ω. We conjecture that our lower bound is sharp up to sub-polynomial correction.

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Section: Model and Resultsmentioning

confidence: 99%

Mixing time for the asymmetric simple exclusion process in a random environment

Lacoin,

Yang

2021

Preprint

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“…In [15] it is proven that, for λ 0 = λ 1 = 1 the law of large numbers holds for all ρ, with only two possible exceptions, and when the speed is not zero a Gaussian central limit theorem holds. Moreover, when p 0 = 1 − p 1 (as in [2] and [17]) and ρ = 1/2 it was shown in [15] that the speed is zero, but it is an interesting open problem to determine the scale of the fluctuations in this case and there are several competing conjectures: in [19] it is conjectured that under the scaling t 3/4 the limiting process is a fractional Brownian motion with Hurst index H = 3/4; in [12], it is conjectured (for a related continuous model) that the fluctuations are either of order t 1/2 (for a fast particle) or t 2/3 (for a slow particle); on the other hand in [16] and [18], it is conjectured that for either fast or slow particle dynamics the fluctuations are always of order t 1/2 for time t sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Variable speed symmetric random walk driven by the simple symmetric exclusion process

Menezes

Peterson

Xie

2022

Electron. J. Probab.

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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.

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“…In this case, only the law of large numbers is known ( [14]). It is conjectured in [17], where the central limit theorem for a weakly asymmetric version of the model was considered (see also [4] and [5]), that the fluctuations are of order t 3/4 . A related model where space is continuous was introduced in [16].…”

Section: Introductionmentioning

confidence: 99%