Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions

TL;DR

This paper derives explicit formulas for the distribution of the free energy of a 1+1 dimensional continuum directed random polymer, revealing a crossover from Gaussian to Tracy-Widom distributions over time.

Contribution

It provides the first rigorous derivation of crossover distributions for the free energy of the continuum directed polymer, connecting stochastic PDEs with integrable probability.

Findings

Distribution transitions from Gaussian to Tracy-Widom over time

Explicit formulas for crossover distributions are obtained

Analysis links asymmetric exclusion processes to continuum equations

Abstract

We consider the solution of the stochastic heat equation \partial_T \mathcal{Z} = 1/2 \partial_X^2 \mathcal{Z} - \mathcal{Z} \dot{\mathscr{W}} with delta function initial condition \mathcal{Z} (T=0)= \delta_0 whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We obtain explicit formulas for the one-dimensional marginal distributions -- the {\it crossover distributions} -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.