The endpoint distribution of directed polymers

TL;DR

This paper introduces a new framework for analyzing endpoint distributions of directed polymers in random environments without relying on integrability, using abstract limit objects and fixed point equations to describe phase transitions and localization phenomena.

Contribution

The paper develops a novel, non-integrable approach using partitioned subprobability measures and fixed point equations to study directed polymers, extending analysis beyond integrable models.

Findings

Endpoint distribution is purely atomic in low temperature phase with finite exponential moments.

In low temperature phase, endpoint distribution localizes in a stochastically bounded region.

Results hold in arbitrary dimensions without assuming integrability.

Abstract

Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called "partitioned subprobability measure", to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on large deviations of the occupation measure of Brownian motion, we define a suitable topology on the space of partitioned subprobability measures and prove that this topology is compact. Then using a variant of the cavity method from the theory of spin glasses, we show that any limit law of a sequence of endpoint distributions must satisfy a fixed point equation on this abstract space, and that the limiting free energy of the model can be expressed as the solution of a variational problem over the set of fixed points. As a first application of the theory, we prove that in an environment with finite exponential moment, the endpoint distribution is asymptotically purely atomic if and only if the system is in the low temperature phase. The analogous result for a heavy-tailed environment was proved by Vargas in 2007. As a second application, we prove a subsequential version of the longstanding conjecture that in the low temperature phase, the endpoint distribution is asymptotically localized in a region of stochastically bounded diameter. All our results hold in arbitrary dimensions, and make no use of integrability.