The effect of disorder on quenched and averaged large deviations for random walks in random environments: boundary behavior

Rodrigo Bazaes, Chiranjib Mukherjee, Alejandro Ramŕez, Santiago Saglietti

Published 2021-01-12Version 1

Consider a multidimensional random walk in a uniformly elliptic random environment (RWRE). Using very general arguments Varadhan [V03] showed that the rescaled location of the random walk satisfies both an almost sure (quenched) as well as an averaged (annealed) large deviation principle (LDP). Two fundamental questions remained open: First, can one characterize regions in the unit ball for which the quenched and the averaged rate functions are equal, and for which they are not; and next, if the two rate functions admit simpler expressions. The main results of the present article show that, for $d\geq 4$ and for any RWRE in uniformly elliptic and i.i.d. environments, the two rate functions agree on every compact subset of the boundary $\partial\mathbb D=\{x\in \mathbb Z^d: |x|_1=1\}$ of the $\ell^1$-unit ball (with the corners removed) for small enough disorder and also admit simple Cram\'er type variational formulas. It is also shown that there is a non-trivial critical disorder that determines a sharp phase-transition in the equality of rate functions, i.e. for a parametrized family of random environments, the equality of two rate functions continues to hold below and on the critical disorder and fails beyond it. These results do not assume any ballistic behavior of the RWRE and complement our recent result [BMRS19] where, under the same set up as the present article but using different methods, equality of the two rate functions was obtained also on every compact subset of the interior $\mathrm{int}\mathbb D=\{x\in \mathbb Z^d: |x|_1 <1\}$ with the origin removed.