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March 2020 The endpoint distribution of directed polymers
Erik Bates, Sourav Chatterjee
Ann. Probab. 48(2): 817-871 (March 2020). DOI: 10.1214/19-AOP1376

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.

Citation

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Erik Bates. Sourav Chatterjee. "The endpoint distribution of directed polymers." Ann. Probab. 48 (2) 817 - 871, March 2020. https://doi.org/10.1214/19-AOP1376

Information

Received: 1 June 2018; Revised: 1 February 2019; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199863
MathSciNet: MR4089496
Digital Object Identifier: 10.1214/19-AOP1376

Subjects:
Primary: 60K37
Secondary: 82B26, 82B44, 82D60

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • March 2020
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