Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

TL;DR

This paper provides exact formulas and asymptotic analysis for the point-to-point partition sum of the Beta polymer, revealing new fluctuation behaviors of a random walk in a time-dependent environment, including Gamma and Tracy-Widom distributions.

Contribution

It introduces exact solutions and asymptotic results for the Beta polymer model, connecting random walk fluctuations to Gamma and Tracy-Widom distributions, and explores crossover regimes.

Findings

Gamma distributed fluctuations near the optimal direction

Tracy-Widom GUE distribution in large deviations regime

Identification of a crossover regime at x ~ t^{3/4}

Abstract

We consider the Beta polymer, an exactly solvable model of directed polymer on the square lattice, introduced by Barraquand and Corwin. We study the statistical properties of its point to point partition sum. The problem is equivalent to a model of a random walk in a time-dependent (and in general biased) 1D random environment. In this formulation, we study the sample to sample fluctuations of the transition probability distribution function (PDF) of the random walk. Using the Bethe ansatz we obtain exact formulas for the integer moments, and Fredholm determinant formulas for the Laplace transform of the directed polymer partition sum/random walk transition probability. The asymptotic analysis of these formulas at large time $t$ is performed both (i) in a diffusive vicinity, $x \sim t^{1/2}$, of the optimal direction (in space-time) chosen by the random walk, where the fluctuations of the PDF are found to be Gamma distributed; (ii) in the large deviations regime, $x \sim t$, of the random walk, where the fluctuations of the logarithm of the PDF are found to grow with time as $t^{1/3}$ and to be distributed according to the Tracy-Widom GUE distribution. Our exact results complement those of BC for the cumulative distribution function of the random walk in regime (ii), and in regime (i) they unveil a novel fluctuation behavior. We also discuss the crossover regime between (i) and (ii), identified as $x \sim t^{3/4}$. Our results are confronted to extensive numerical simulations of the model.