Annals of Probability
- Ann. Probab.
- Volume 48, Number 2 (2020), 817-871.
The endpoint distribution of directed polymers
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.
Ann. Probab., Volume 48, Number 2 (2020), 817-871.
Received: June 2018
Revised: February 2019
First available in Project Euclid: 22 April 2020
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60K37: Processes in random environments
Secondary: 82B26: Phase transitions (general) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82D60: Polymers
Bates, Erik; Chatterjee, Sourav. The endpoint distribution of directed polymers. Ann. Probab. 48 (2020), no. 2, 817--871. doi:10.1214/19-AOP1376. https://projecteuclid.org/euclid.aop/1587542681
- Appendix A: Remaining technical details. This appendix contains the proofs of Proposition 2.4, Theorem 2.8, Proposition 3.4, Lemma 6.1(a), measurability of the support number, Lemma 7.1, and the equivalence of two notions of asymptotic pure atomicity.
- Appendix B: Comparison to the Mukherjee–Varadhan topology. This appendix proves that the topology introduced by Mukherjee and Varadhan , when adapted to the discrete setting, is equivalent to the one constructed in this manuscript.