Two-point generating function of the free energy for a directed polymer in a random medium

TL;DR

This paper extends the analysis of the free energy fluctuations of a 1+1 dimensional directed polymer in a random medium to two points, showing that long-term behavior aligns with the Airy process and KPZ universality.

Contribution

It provides a two-point generating function for the free energy of the directed polymer, connecting it to the Airy process and extending previous one-point results.

Findings

Long time free energy fluctuations follow the two-point Airy process distribution.

The results support the universality of KPZ class for directed polymers.

Extension of one-point to two-point free energy distribution analysis.

Abstract

We consider a 1+1 dimensional directed continuum polymer in a Gaussian delta-correlated space-time random potential. For this model the moments (= replica) of the partition function, Z(x,t), can be expressed in terms of the attractive delta-Bose gas on the line. Based on a recent study of the structure of the eigenfunctions, we compute the generating function for Z(x_1,t), Z(x_2,t) under a particular decoupling assumption and thereby extend recent results on the one-point generating function of the free energy to two points. It is established that in the long time limit the fluctuations of the free energy are governed by the two-point distribution of the Airy process, which further supports that the long time behavior of the KPZ equation is the same as derived previously for lattice growth models.