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.

We introduce a random walk in random environment associated to an underlying directed polymer model in $1+1$ dimensions. This walk is the positive temperature counterpart of the competition interface of percolation and arises as the limit of quenched polymer measures. We prove this limit for the exactly solvable log-gamma polymer, as a consequence of almost sure limits of ratios of partition functions. These limits of ratios give the Busemann functions of the log-gamma polymer, and furnish centered cocycles that solve a variational formula for the limiting free energy. Limits of ratios of point-to-point and point-to-line partition functions manifest a duality between tilt and velocity that comes from quenched large deviations under polymer measures. In the log-gamma case, we identify a family of ergodic invariant distributions for the random walk in random environment.Comment: Published at http://dx.doi.org/10.1214/14-AOP933 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. We take the point of view of a particle performing random walk with bounded jumps on Z d in a stationary and ergodic random environment. We prove the quenched large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function. We propose an Ansatz for the minimizer of this formula. When d = 1, we verify this Ansatz and generalize the nearest-neighbor result of Comets, Gantert and Zeitouni to walks with bounded jumps.

Abstract:We consider the quenched and the averaged (or annealed) large deviation rate functions I q and I a for space-time and (the usual) space-only RWRE on Z d . By Jensen's inequality, I a ≤ I q . In the space-time case, when d ≥ 3 + 1, I q and I a are known to be equal on an open set containing the typical velocity ξ o . When d = 1 + 1, we prove that I q and I a are equal only at ξ o . Similarly, when d = 2 + 1, we show that I a < I q on a punctured neighborhood of ξ o . In the space-only case, we provide a class of non-nestling walks on Z d with d = 2 or 3, and prove that I q and I a are not identically equal on any open set containing ξ o whenever the walk is in that class. This is very different from the known results for non-nestling walks on Z d with d ≥ 4.

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.

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on Z d with d ≥ 1, and gives a variational formula for the corresponding rate function Ia. Under Sznitman's transience condition (T), we show that Ia is strictly convex and analytic on a non-empty open set A, and that the true velocity of the particle is an element (resp. in the boundary) of A when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan's variational formula at any velocity in A. n→∞ 1 n log Q n (G) ≤ − inf x∈Ḡ I(x).

We prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.

We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk {X i } on the set of integers. The cost function is the expectation of exponential of the path sum of a random stationary and ergodic bounded potential plus θXn. The random walk policies are measurable with respect to the random potential, and are adapted, with their drifts uniformly bounded in magnitude by a parameter δ ∈ [0, 1]. Under natural conditions on the potential, we prove that the normalized logarithm of the optimal cost function converges. The proof is constructive in the sense that we identify asymptotically optimal policies given the value of the parameter δ, as well as the law of the potential. It relies on correctors from large deviation theory as opposed to arguments based on subadditivity which do not seem to work except when δ = 0.The Bellman equation associated to this control problem is a second-order Hamilton-Jacobi (HJ) stochastic partial difference equation with a separable random Hamiltonian which is nonconvex in θ unless δ = 0. We prove that this equation homogenizes under linear initial data to a first-order HJ deterministic partial differential equation. When δ = 0, the effective Hamiltonian is the tilted free energy of random walk in random potential and it is convex in θ. In contrast, when δ = 1, the effective Hamiltonian is piecewise linear and nonconvex in θ. Finally, when δ ∈ (0, 1), the effective Hamiltonian is expressed completely in terms of the tilted free energy for the δ = 0 case and its convexity/nonconvexity in θ is characterized by a simple inequality involving δ and the magnitude of the potential, thereby marking two qualitatively distinct control regimes.