Large deviations for partition functions of directed polymers in an IID field

Abstract

Consider the partition function of a directed polymer in dimension d ≥ 1 in an IID field. We assume that both tails of the negative and the positive part of the field are at least as light as exponential. It is a well-known fact that the free energy of the polymer is equal to a deterministic constant for almost every realization of the field and that the upper tail of the large deviations is exponential. The lower tail of the large deviations is typically lighter than exponential. In this paper we provide a method to obtain estimates on the rate of decay of the lower tail of the large deviations, which are sharp up to multiplicative constants. As a consequence, we show that the lower tail of the large deviations exhibits three regimes, determined according to the tail of the negative part of the field. Our method is simple to apply and can be used to cover other oriented and non-oriented models including first/last-passage percolation and the parabolic Anderson model.

Properties of the Parabolic Anderson Model and the Anderson Polymer Model

“…An analogous result was established in [72] for solutions of (20). Generalizations of these results appeared in [71].…”

“…The concentration results are thus closely related to large deviation results established in [39,[71][72][73]. For example, by means of a block argument, it was shown in [39] that for every > 0 there is a ( ) > 0 such that…”

Upper Tail Large Deviations in First Passage Percolation

“…//. In [7], results of [13] were generalized to establish the n 2 speed of the lower tail large deviations for log Z under certain tail conditions on the weight distribution. A precise upper tail large deviation rate function was established for the n-speed upper tail for the exactly solvable log-gamma polymer [18].…”

Concentrations for the simple random walk in un- bounded nonnegative potentials

“…In this model, due to superadditive arguments, it is known that the upper tail of the large deviations is exponential for the quenched partition function. Ben-Ari [1] thus studied the explicit order of the lower tail large deviations. Corollary 1 of [1] provides a necessary and sufficient condition for order of the lower tail large deviations to be comparable to the exponential tail of the potential.…”

“…Ben-Ari [1] thus studied the explicit order of the lower tail large deviations. Corollary 1 of [1] provides a necessary and sufficient condition for order of the lower tail large deviations to be comparable to the exponential tail of the potential. There are few results for the explicit rate function of this model.…”