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Pré-Publication, Document De Travail Année : 2021

Maximal free energy of the log-gamma polymer

Résumé

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.

Dates et versions

hal-03357593 , version 1 (28-09-2021)

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Guillaume Barraquand, Ivan Corwin, Evgeni Dimitrov. Maximal free energy of the log-gamma polymer. 2021. ⟨hal-03357593⟩
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