Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights

TL;DR

This paper proves Tracy-Widom asymptotics for the partition function of a gamma-weighted random polymer model, connecting it to random matrix theory and Whittaker functions.

Contribution

It introduces a novel approach to analyze the asymptotics of the partition function using geometric RSK and Whittaker functions.

Findings

Establishes Tracy-Widom asymptotics for the model

Connects the partition function to random matrix theory

Provides a new analytical framework for gamma-distributed weights

Abstract

We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Sepp\"al\"ainen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.