Maximal free energy of the log-gamma polymer
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909737162899456 |
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| author | Barraquand, Guillaume Corwin, Ivan Dimitrov, Evgeni |
| author_facet | Barraquand, Guillaume Corwin, Ivan Dimitrov, Evgeni |
| contents | 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 $θ$, maximized over all possible starting and ending points in an $N\times N$ square. In particular, we find an explicit critical value $θ_c=2Ψ^{-1}(0)>0$ ($Ψ$ is the digamma function) such that:
1. For $θ<θ_c$, $\mathsf F_N+2Ψ(θ/2)N$ has order $N^{1/3}$ GUE Tracy-Widom fluctuations.
2. For $θ=θ_c$, $\mathsf F_N= Θ(N^{1/3}(\log N)^{2/3})$.
3. For $θ>θ_c$, $\mathsf F_N=Θ(\log N)$.
Using a connection between the log-gamma polymer and a certain random operator on the honeycomb lattice, recently found by Kotowski and Virág (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_05283 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Maximal free energy of the log-gamma polymer Barraquand, Guillaume Corwin, Ivan Dimitrov, Evgeni Probability Mathematical Physics 60K37, 82B23 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 $θ$, maximized over all possible starting and ending points in an $N\times N$ square. In particular, we find an explicit critical value $θ_c=2Ψ^{-1}(0)>0$ ($Ψ$ is the digamma function) such that: 1. For $θ<θ_c$, $\mathsf F_N+2Ψ(θ/2)N$ has order $N^{1/3}$ GUE Tracy-Widom fluctuations. 2. For $θ=θ_c$, $\mathsf F_N= Θ(N^{1/3}(\log N)^{2/3})$. 3. For $θ>θ_c$, $\mathsf F_N=Θ(\log N)$. Using a connection between the log-gamma polymer and a certain random operator on the honeycomb lattice, recently found by Kotowski and Virág (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. |
| title | Maximal free energy of the log-gamma polymer |
| topic | Probability Mathematical Physics 60K37, 82B23 |
| url | https://arxiv.org/abs/2105.05283 |