A note on the antisymmetry in the speed of a random walk in reversible dynamic random environment Oriane Blondel

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A note on the antisymmetry in the speed of a random walk in reversible

dynamic random environment

Oriane Blondel

January 13, 2023

Abstract

In this short note, we prove that v(−ε) = −v(ε). Here, v(ε)is the speed of a one-dimensional random walk

in a dynamic reversible random environment, that jumps to the right (resp. to the left) with probability 1/2 + ε

(resp. 1/2−ε) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where

v(ε), v(−ε)are well-deﬁned, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only

in the discrete setting.

1 Introduction

We consider the so-called “ε–random walk”: a random walk in one-dimensional dynamic random environment with

two values that jumps to the right (resp. to the left) with probability 1/2 +ε(resp. 1/2−ε) if it stands on an occupied

site, and vice-versa on an empty site (Figure 1).

2−

2+

Figure 1: Jump rates for the ε–random walk.

A lot of energy has been devoted to describing the behavior of this ε–RW in various random environments,

mainly to ﬁnd whether it satisﬁes the usual limit theorems (LLN, CLT...). This is generally a hard problem, since

the environment seen from the ε–RW is highly non-stationary (even when the environment is). Additionally, it

belongs to the class of nestling random walks, which do not have an a-priori preferred direction. It is possible to

ﬁnd settings in which the LLN does not hold. [BHT20, Section 9] proposes an example in which space-time traps

lead the random walk into longer and longer stretches of motion with drifts to the right or the left. Cases in which

the LLN has been shown include perturbative regimes (small ε) [ABF18], well-known environments like the exclusion

process [HS15, HKT20] or the contact process [MV15], uniform mixing hypotheses [CZ04, AdHR11, RV13], or fast

enough decay of correlations [BHT20].

When the random walk does satisfy a law of large numbers, let us call v(ε)its asymptotic speed (we will simply

say that v(ε)is well-deﬁned). We observed in [ABF16] that, if the environment is given by a reversible Markov process

with positive spectral gap, and |ε|is smaller than the spectral gap, v–in addition to being well-deﬁned– satisﬁes the

antisymmetry property

v(−ε) = −v(ε).(1)

This property also holds in higher dimensions for random walks with a certain symmetry property (Assumption 3

in [ABF16]). Simulations moreover suggest that this property extends out of the perturbative regime of [ABF16]

(where the law of large numbers was not yet even established; see Figure 3 in [ABF16]). The proof relies on an

expansion of the speed in ε, in which the terms of even degrees are shown to cancel due to reversibility.

It is worth pointing out that this antisymmetry does not seem to follow from obvious symmetry properties of the

system. Indeed, it is tempting at ﬁrst sight to think that the (−ε)–random walk reversed in time should have the same

distribution as the ε–random walk, from which (1) would immediately follow. But closer inspection reveals that this

is not true and those two processes have very diﬀerent trajectories in general (see for instance Figure 4 in [ABF16]).

Rather, (1) holds iﬀ the speed of the ε–random walk is the same in the reversible environment and in its image under

the mirror symmetry x7→ −x. While this property is obvious for reversible systems invariant under mirror symmetry

(e.g. simple symmetric exclusion process), there exist reversible, translation invariant processes which have no mirror-

invariance. One that has been extensively studied is the East model [JE91,FMRT13] , in which particles appear (resp.

disappear) at site xat rate p(resp. 1−p), but only if x+ 1 is empty. This dynamics is reversible w.r.t. the product

arXiv:2210.10427v2 [math.PR] 12 Jan 2023

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AnoteontheantisymmetryinthespeedofarandomwalkinreversibledynamicrandomenvironmentOrianeBlondelJanuary13,2023AbstractInthisshortnote,weprovethatv(")=v(").Here,v(")isthespeedofaone-dimensionalrandomwalkinadynamicreversiblerandomenvironment,thatjumpstotheright(resp.totheleft)withprobability1=2+"(resp.1...

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A note on the antisymmetry in the speed of a random walk in reversible dynamic random environment Oriane Blondel

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