Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

Mukherjee, Chiranjib

doi:10.1002/cpa.21700

Cited by 4 publications

(2 citation statements)

References 30 publications

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“…Below we mention some remaining questions concerning the methods of this manuscript. Given the subsequent applications [44,10] of Mukherjee and Varadhan's construction in [53] and of a variant construction in [52], as well as the success of the concentration compactness phenomenon in general, we are optimistic that the ideas presented in this study can fuel future work. For a broader discussion of open problems on directed polymers, we refer the reader to Section 12.9 of [27].…”

Section: Open Problemsmentioning

confidence: 97%

The endpoint distribution of directed polymers

Bates¹,

Chatterjee²

2020

Ann. Probab.

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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. Contents2010 Mathematics Subject Classification. 60K37, 82B26, 82B44, 82D60.

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Section: Open Problemsmentioning

confidence: 97%

The endpoint distribution of directed polymers

Bates¹,

Chatterjee²

2020

Ann. Probab.

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“…We collect here some crucial results from [11] and [8] (see also [10]) that will be used in the sequel.…”

Section: Earlier Results: Tube Properties Under P Tmentioning

confidence: 99%

Mean‐Field Interaction of Brownian Occupation Measures II: A Rigorous Construction of the Pekar Process

Bolthausen

König

Mukherjee

2017

Comm Pure Appl Math

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We consider mean‐field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self‐attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan in terms of the Pekar variational formula, which coincides with the behavior of the partition function of the polaron problem under strong coupling. Based on this, in 1986 Spohn made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the Pekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean‐field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean‐field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence is a contribution to the understanding of the “mean‐field approximation” of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed by Mukherjee and Varadhan in 2016; its extension to the uniform strong metric for the singular Coulomb interaction was carried out by König and Mukherjee in 2015, as well as an idea inspired by a partial path exchange argument appearing in 1997 in work by Bolthausen and Schmock.© 2017 Wiley Periodicals, Inc.

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Large deviation principle for the intersection measure of Brownian motions on unbounded domains

Mori

2023

Ann. Inst. H. Poincaré Probab. Statist.

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Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

Cited by 4 publications

References 30 publications

The endpoint distribution of directed polymers

The endpoint distribution of directed polymers

Mean‐Field Interaction of Brownian Occupation Measures II: A Rigorous Construction of the Pekar Process

Large deviation principle for the intersection measure of Brownian motions on unbounded domains

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