Maximal free energy of the log-gamma polymer

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 ${\theta }_c=2{\mathrm{\Psi }}^{-1}(0)>0$ ($\mathrm{\Psi }$ is the digamma function) such that:

1. For $\theta <{\theta }_c$, ${\mathsf{F}}_N+2\mathrm{\Psi }(\theta \mathrm{/}2)N$ has order $N^{1\mathrm{/}3}$ GUE Tracy-Widom fluctuations.
2. For $\theta ={\theta }_c$, ${\mathsf{F}}_N=\mathrm{\Theta }(N^{1\mathrm{/}3}(\log N{)}^{2\mathrm{/}3})$.
3. For $\theta >{\theta }_c$, ${\mathsf{F}}_N=\mathrm{\Theta }(\log N)$. 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.