Variational formulas and disorder regimes of random walks in random potentials

TL;DR

This paper develops variational formulas for the quenched and annealed free energies of random walks in random potentials, analyzing their minimizers and disorder regimes, including directed, undirected, and random walk in random environment cases.

Contribution

It introduces new variational formulas for free energies in RWRP, compares them to existing formulas, and characterizes minimizers across different disorder regimes.

Findings

(qVar0) always has a minimizer

(aVar2) has no minimizers unless RWRP is RWRE

(aVar1) has a minimizer iff in weak disorder regime

Abstract

We give two variational formulas (qVar1) and (qVar2) for the quenched free energy of a random walk in random potential (RWRP) when (i) the underlying walk is directed or undirected, (ii) the environment is stationary and ergodic, and (iii) the potential is allowed to depend on the next step of the walk which covers random walk in random environment (RWRE). In the directed i.i.d. case, we also give two variational formulas (aVar1) and (aVar2) for the annealed free energy of RWRP. These four formulas are the same except that they involve infima over different sets, and the first two are modified versions of a previously known variational formula (qVar0) for which we provide a short alternative proof. Then, we show that (qVar0) always has a minimizer, (aVar2) never has any minimizers unless the RWRP is an RWRE, and (aVar1) has a minimizer if and only if the RWRP is in the weak disorder regime. In the latter case, the minimizer of (aVar1) is unique and it is also the unique minimizer of (qVar1), but (qVar2) has no minimizers except for RWRE. In the case of strong disorder, we give a sufficient condition for the nonexistence of minimizers of (qVar1) and (qVar2) which is satisfied for the log-gamma directed polymer with a sufficiently small parameter. We end with a conjecture which implies that (qVar1) and (qVar2) have no minimizers under very strong disorder.