Variational formulas and disorder regimes of random walks in random potentials

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.

The endpoint distribution of directed polymers

“…Theorem F ([73], Theorem 3.1). Let F denote the distribution function of λ, and assume In recent work, Rassoul-Agha, Seppäläinen and Yilmaz [59,60] expressed the limit (1.16) in terms of several variational formulas. 4 A method for finding minimizers to these formulas was proposed in [29].…”

“…4 A method for finding minimizers to these formulas was proposed in [29]. It has been noted ( [60], Remark 2.12) that they may not admit minimizers when β > β c . These results suggest the possibility of a more general correspondence between the disorder regimes and the existence of variational minimizers (see [60], Conjecture 2.13).…”

Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation

“…There are numerous results concerning infinite geodesics. In particular, existence-uniqueness of onesided planar geodesics with fixed slope and certain geometric features of the joint behavior of different geodesics are known for several models, see [HN97], [HN99], [HN01], [Wüt02], [CP11], [CP12], [DH14], [Bak16], [GRAS16], [RSY16], [GRS15], and a survey [AHD15]. However, results about thermodynamic limits for directed polymers are relatively new.…”

Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment

“…The first formula (1.2) g(V ) = inf expresses g(V ) as an infimum over the L 1 (P) closure of gradients F (ω, z) = f (T z ω) − f (ω), which we called the space of cocycles. Since this formula is not the topic of the present paper, we refer to [24,38,39] for precise definitions. The second formula gives g(V ) as the dual of an entropy adapted to the point of view of the particle:…”