Fluctuations of the log-gamma polymer free energy with general parameters and slopes

TL;DR

This paper proves that the free energy fluctuations of the log-gamma polymer model converge to the GUE Tracy-Widom distribution under certain scaling conditions, extending results to non-i.i.d. weights and critical parameter regimes.

Contribution

It establishes convergence to Tracy-Widom distribution for general parameters and slopes, and introduces a generalized Baik-Ben Arous-Péché distribution for non-i.i.d. weights.

Findings

Convergence to GUE Tracy-Widom distribution under $M^{1/3}$ scaling.

Moderate deviation estimates for the upper tail of free energy.

Generalized Baik-Ben Arous-Péché distribution for critical non-i.i.d. weights.

Abstract

We prove that the free energy of the log-gamma polymer between lattice points $(1,1)$ and $(M,N)$ converges to the GUE Tracy-Widom distribution in the $M^{1/3}$ scaling, provided that $N/M$ remains bounded away from zero and infinity. We prove this result for the model with inverse gamma weights of any shape parameter $\theta>0$ and furthermore establish a moderate deviation estimate for the upper tail of the free energy in this case. Finally, we consider a non i.i.d. setting where the weights on finitely many rows and columns have different parameters, and we show that when these parameters are critically scaled the limiting free energy fluctuations are governed by a generalization of the Baik-Ben Arous-P\'ech\'e distribution from spiked random matrices with two sets of parameters.