Central limit theorem for random walk in degenerate divergence-free random environment: $\mathcal H_{-1}$ reloaded with relaxed ellipticity

TL;DR

This paper proves a central limit theorem for random walks in divergence-free random environments under relaxed ellipticity conditions, broadening the applicability of previous results and simplifying the proof methodology.

Contribution

It extends the CLT for divergence-free RWRE by relaxing ellipticity assumptions and removing reliance on Nash's inequality, making the proof more elementary.

Findings

CLT holds under only integrability of reciprocal jump rates

Proof does not rely on Nash's inequality

Results generalize previous bounded ellipticity assumptions

Abstract

This paper enhances the result of the work [G. Kozma, B. T\'oth, Ann. Probab. vol. 45 (2017) 4307-4347] . We prove the central limit theorem (in probability w.r.t. the environment) for the displacement of a random walker in divergence-free (or, doubly stochastic) random environment, with substantially relaxed ellipticity assumptions. Integrability of the reciprocal of the symmetric part of the jump rates is only assumed (rather than their boundedness, as in previous works on this type of RWRE). Relaxing ellipticity involves substantial changes in the proof, making it conceptually elementary in the sense that it does not rely on Nash's inequality in any disguise.