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

This paper enhances the result of the work [G. Kozma, B. Tóth, 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.