Free energy fluctuations for directed polymers in random media in 1+1 dimension

TL;DR

This paper proves the KPZ universality for two models of directed polymers in 1+1 dimensions by analyzing their free energy fluctuations, showing convergence to Tracy-Widom distribution and exploring boundary effects.

Contribution

It establishes the KPZ universality for the semi-discrete and continuum directed polymers using Fredholm determinant formulas and analyzes boundary perturbations' impact on free energy statistics.

Findings

Free energy fluctuations converge to GUE Tracy-Widom distribution for large time.

Boundary perturbations lead to Baik-Ben Arous-Peche distributions in certain regimes.

Laplace transform formulas enable derivation of distribution of solutions to related stochastic PDEs.

Abstract

We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semi-discrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time tau, the probability distributions for the free energy fluctuations, when rescaled by tau^{1/3}, converges to the GUE Tracy-Widom distribution. We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semi-discrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik-Ben Arous-Peche distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm -- the Kardar-Parisi-Zhang equation. The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data.