Proof of the Path Localization Conjecture for Directed Polymers

Chatterjee, Sourav

doi:10.1007/s00220-019-03533-1

Cited by 11 publications

(8 citation statements)

References 19 publications

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“…Even less is known in d ě 2 due to the lack of integrable models within the same universality class. We mention that the strong disorder phase can alternatively be characterised by the fact that two polymer paths sampled independently in the same random environment have positive overlap, see [CH02,CSY03,V07] and the more recent results [Cha19,BC20a,BC20b,Ba21].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Critical 2d Stochastic Heat Flow

Caravenna¹,

Sun²,

Zygouras³

2021

Preprint

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We consider directed polymers in random environment in the critical dimension d " 2, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on R 2 with logarithmic correlations, which we call the Critical 2d Stochastic Heat Flow. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The Critical 2d Stochastic Heat Flow

Caravenna¹,

Sun²,

Zygouras³

2021

Preprint

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“…Replicas are important observables in statistical mechanics, as they can provide insights into the properties of the models. For polymer models, replicas can shed light on localization phenomena [4][5][6]21,22]. As a consequence of the connection with the persymmetric polymer, the present work leads to the computation of the Laplace transform of the replica partition function (1.6), which will be expressed as an integral of special functions called Whittaker functions (more in-depth explanations will be given later on and the detailed formulas can be found in Sect.…”

Section: Contributions Of This Workmentioning

confidence: 99%

The geometric Burge correspondence and the partition function of polymer replicas

Bisi

O’Connell

Zygouras

2021

Sel. Math. New Ser.

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We construct a geometric lifting of the Burge correspondence as a composition of local birational maps on generic Young-diagram-shaped arrays. We establish its fundamental relation to the geometric Robinson-Schensted-Knuth correspondence and to the geometric Schützenberger involution. We also show a number of properties of the geometric Burge correspondence, specializing them to the case of symmetric input arrays. In particular, our construction shows that such a mapping is volume preserving in log-log variables. As an application, we consider a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica. We prove that the distribution of the polymer replica partition function in a log-gamma random environment is a Whittaker measure, and deduce the corresponding Whittaker integral identity. For a certain choice of the parameters, we notice a distributional identity between our model and the symmetric log-gamma polymer studied by O’Connell, Seppäläinen, and Zygouras (2014).

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“…One should like said probability to be asymptotically equal to 1, meaning the specific realization of the disorder is irrelevant. Such was the advancement provided by Chatterjee [11], for sufficiently large β and a certain class of bounded random environments. In [7], Bates and Chatterjee proved an analogous (but less quantitative) statement in the Gaussian case, and then bootstrapped that result to the following, stronger one.…”

Section: Introductionmentioning

confidence: 99%

Full-path localization of directed polymers

Bates

2021

Electron. J. Probab.

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Certain polymer models are known to exhibit path localization in the sense that at low temperatures, the average fractional overlap of two independent samples from the Gibbs measure is bounded away from 0. Nevertheless, the question of where along the path this overlap takes place has remained unaddressed. In this article, we prove that on linear scales, overlap occurs along the entire length of the polymer. Namely, we consider time intervals of length εN , where ε > 0 is fixed but arbitrarily small.We then identify a constant number of distinguished trajectories such that the Gibbs measure is concentrated on paths having, with one of these distinguished paths, a fixed positive overlap simultaneously in every such interval. This result is obtained in all dimensions for a Gaussian random environment by using a recent non-local result as a key input.

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Proof of the Path Localization Conjecture for Directed Polymers

Cited by 11 publications

References 19 publications

The Critical 2d Stochastic Heat Flow

The Critical 2d Stochastic Heat Flow

The geometric Burge correspondence and the partition function of polymer replicas

Full-path localization of directed polymers

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