Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz

TL;DR

This paper extends the coordinate Bethe Ansatz solution of the log-gamma polymer to new boundary conditions, analyzes the large-time behavior, and reveals a crossover in the distribution of free energy from GUE to GOE Tracy--Widom, connecting to KPZ universality.

Contribution

It introduces a boundary condition extension for the log-gamma polymer and characterizes the crossover in free energy distribution using Fredholm determinants and scaling limits.

Findings

Identifies a crossover from GUE to GOE Tracy--Widom distributions.

Expresses the free energy distribution as a Fredholm determinant.

Reproduces the droplet to flat initial condition crossover in KPZ.

Abstract

The coordinate Bethe Ansatz solution of the log-gamma polymer is extended to boundary conditions with one fixed end and the other attached to one half of a one-dimensional lattice. The large-time limit is studied using a saddle-point approximation,and the cumulative distribution function of the rescaled free energy of a long polymer is expressed as a Fredholm determinant. Scaling limits of the kernel are identified, leading to a crossover from the GUE to the GOE Tracy--Widom distributions. The continuum limit reproduces the crossover from droplet to flat initial conditions of the Kardar--Parisi--Zhang equation.