Quenched Free Energy and Large Deviations for Random Walks in Random Potentials

Abstract

We study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps. The potential can be unbounded and can depend on a few steps of the walk. Directed, undirected and stretched polymers, as well as random walk in random environment, are covered. The restriction needed is on the moment of the potential, in relation to the degree of mixing of the ergodic environment. We derive two variational formulas for the limiting quenched free energy and prove a process-level quenched large deviation principle for the empirical measure. As a corollary we obtain LDPs for types of random walk in random environment not covered by earlier results.

Variational Formula for the Time Constant of First‐Passage Percolation

“…They prove discrete variational formulas for the directed polymer model at zero and finite temperature, and for the closely related last-passage percolation model. Their ideas originate in the works of Rosenbluth [35], Rassoul-Agha and Seppäläinen [32] and Rassoul-Agha et al [33] for quenched large-deviation principles for random-walk in random environment.…”

“…Such a route has been taken to prove variational formulas for the large deviations of random walks in random environments by Rosenbluth [35]. This work was considerably generalized by Rassoul-Agha et al [33] and Rassoul-Agha and Seppäläinen [32]. Subsequently, Georgiou et al [16] extended these ideas to prove variational formulas for the directed random polymer, and for last-passage percolation.…”

Moderate deviations for diffusion in time dependent random media

“…This was proved in [7] for discrete models of random walks in random environment in dimension 1+1, and in [8] for a certain class of continuous diffusions which we discuss below in Section 3.1. In the large deviation regime (i.e along the ray x = vt with v = v 0 ), the fact that the probability distribution admits a large deviation principle for almost every environment was proved in [9] in a quite general setting. It was then shown in [1], using the exactly solvable Beta RWRE, that the second order corrections to the large deviation principle fluctuate sample to sample according to the Tracy-Widom distribution, as we have already mentioned.…”

Countable Alphabet Random Subhifts of Finite Type with Weakly Positive Transfer Operator

“…Such random walks are called inhomogenous random walks in random environments [23] Related to these examples are the random walks with random potentials considered in [16]. For appropriate choices of the potential V in [16] we are again in the scope of the present paper. To the following more general example all the results of the present paper do apply: consider a random walk in Z where the number of neighbors of a site i is some function η(i) ∈ N \ {0}.…”