Directed random polymers via nested contour integrals

TL;DR

This paper derives exact formulas for the partition functions of continuum directed polymers in 1+1 dimensions, revealing Tracy-Widom fluctuations and providing a mathematical perspective on the replica method without relying on Bethe ansatz.

Contribution

It introduces nested contour integral formulas for moments of polymer partition functions and connects them to Tracy-Widom distributions, offering a rigorous perspective on the replica method.

Findings

Partition functions expressed as Fredholm determinants and Pfaffians.

Free energy fluctuations follow Tracy-Widom distributions with 1/3 scaling.

Provides a mathematical framework for the replica method without Bethe ansatz.

Abstract

We study the partition function of two versions of the continuum directed polymer in 1+1 dimension. In the full-space version, the polymer starts at the origin and is free to move transversally in the reals, and in the half-space version, the polymer starts at the origin but is reflected at the origin and stays in the negative reals. The partition functions solves the stochastic heat equation in full-space or half-space with mixed boundary condition at the origin; or equivalently the free energy satisfies the Kardar-Parisi-Zhang equation. We derive exact formulas for the Laplace transforms of the partition functions. In the full-space this is expressed as a Fredholm determinant while in the half-space this is expressed as a Fredholm Pfaffian. Taking long-time asymptotics we show that the limiting free energy fluctuations scale with exponent 1/3 and are given by the GUE and GSE Tracy-Widom distributions. These formulas come from summing divergent moment generating functions, hence are not mathematically justified. The primary purpose of this work is to present a mathematical perspective on the polymer replica method which is used to derive these results. In contrast to other replica method work, we do not appeal directly to the Bethe ansatz for the Lieb-Liniger model but rather utilize nested contour integral formulas for moments as well as their residue expansions.