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<p>We prove a phase transition for the law of large numbers and fluctuations of F<sub><i>N</i></sub>, the maximum of the free energy of the log-gamma directed polymer with parameter <i>θ</i>, maximized over all possible starting and ending points in an <i>N</i> × <i>N</i> square. In particular, we find an explicit critical value <i>θ</i><sub><i>c</i></sub> = 2Ψ<sup>−1</sup>(0) &gt; 0 (Ψ is the digamma function) such that:<UnorderedList Mark="Bullet"> <ItemContent> <p>For <i>θ</i> &lt; <i>θ</i><sub><i>c</i></sub>, F<sub><i>N</i></sub>+ 2Ψ(<i>θ</i>/2)<i>N</i> has order <i>N</i><sup>1/3</sup> GUE Tracy–Widom fluctuations.</p> </ItemContent> <ItemContent> <p>For <i>θ</i> = <i>θ</i><sub><i>c</i></sub>, F<sub><i>N</i></sub> = Θ(<i>N</i><sup>1/3</sup>(log <i>N</i>)<sup>2/3</sup>).</p> </ItemContent> <ItemContent> <p>For <i>θ</i> &gt; <i>θ</i><sub><i>c</i></sub>, F<sub><i>N</i></sub> = Θ(log <i>N</i>).</p> </ItemContent> </UnorderedList></p><p>Using the same techniques for analyzing F<sub><i>N</i></sub>, we also show that an analogous phase transition occurs in a certain free start/end-point polymer measure with inverse gamma weights. By exploiting a connection between the log-gamma polymer and a certain random operator on the honeycomb lattice, recently found by Kotowski and Virág (Comm. Math. Phys. 370, 2019), we also deduce a similar phase transition for the asymptotic behavior of the smallest positive eigenvalue of the aforementioned random operator.</p>

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Maximal free energy of the log-gamma polymer

  • 期刊论文

  • 发表日期：

    2025年9月7日

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  • Guillaume Barraquand,
  • Ivan Corwin,
  • Evgeni Dimitrov

摘要

We prove a phase transition for the law of large numbers and fluctuations of FN, 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 × N square. In particular, we find an explicit critical value θc = 2Ψ−1(0) > 0 (Ψ is the digamma function) such that:

For θ < θc, FN+ 2Ψ(θ/2)N has order N1/3 GUE Tracy–Widom fluctuations.

For θ = θc, FN = Θ(N1/3(log N)2/3).

For θ > θc, FN = Θ(log N).

Using the same techniques for analyzing FN, we also show that an analogous phase transition occurs in a certain free start/end-point polymer measure with inverse gamma weights. By exploiting a connection between the log-gamma polymer and a certain random operator on the honeycomb lattice, recently found by Kotowski and Virág (Comm. Math. Phys. 370, 2019), we also deduce a similar phase transition for the asymptotic behavior of the smallest positive eigenvalue of the aforementioned random operator.