Maximal free energy of the log-gamma polymer

TL;DR

This paper establishes a phase transition in the maximum free energy of the log-gamma directed polymer, revealing different fluctuation regimes depending on the parameter , and connects this to eigenvalue behavior of a related random operator.

Contribution

It proves a phase transition in the free energy's law of large numbers and fluctuations for the log-gamma polymer, linking it to eigenvalue asymptotics of a random operator.

Findings

For <_c, fluctuations are of order N^{1/3} with Tracy-Widom distribution.

At =_c, free energy scales as N^{1/3} (\,log N)^{2/3}.

For >_c, free energy scales as (log N).

Abstract

We prove a phase transition for the law of large numbers and fluctuations of $\mathsf F_N$, the maximum of the free energy of the log-gamma directed polymer with parameter $\theta$, maximized over all possible starting and ending points in an $N\times N$ square. In particular, we find an explicit critical value $\theta_c=2\Psi^{-1}(0)>0$ ($\Psi$ is the digamma function) such that: 1. For $\theta<\theta_c$, $\mathsf F_N+2\Psi(\theta/2)N$ has order $N^{1/3}$ GUE Tracy-Widom fluctuations. 2. For $\theta=\theta_c$, $\mathsf F_N= \Theta(N^{1/3}(\log N)^{2/3})$. 3. For $\theta>\theta_c$, $\mathsf F_N=\Theta(\log N)$. Using a connection between the log-gamma polymer and a certain random operator on the honeycomb lattice, recently found by Kotowski and Vir\'ag (Commun. Math. Phys. 370, 2019), we deduce a similar phase transition for the asymptotic behavior of the smallest positive eigenvalue of the aforementioned random operator.