De Luca Andrea, Le Doussal Pierre
Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626), Université Paris-Sud, Orsay, France.
Laboratoire de Physique Théorique de l'ENS, CNRS & Ecole Normale Supérieure de Paris, Paris, France.
Phys Rev E. 2017 Mar;95(3-1):030103. doi: 10.1103/PhysRevE.95.030103. Epub 2017 Mar 21.
Recently, it was shown that the probability distribution function (PDF) of the free energy of a single continuum directed polymer (DP) in a random potential, equivalently to the height of a growing interface described by the Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom distribution. The latter describes the fluctuations of the largest eigenvalue of a random matrix, drawn from the Gaussian unitary ensemble (GUE), and the result holds for a DP with fixed end points, i.e., for the KPZ equation with droplet initial conditions. A more general conjecture can be put forward, relating the free energies of N>1 noncrossing continuum DP in a random potential, to the sum of the Nth largest eigenvalues of the GUE. Here, using replica methods, we provide an important test of this conjecture by calculating exactly the right tails of both PDFs and showing that they coincide for arbitrary N.
最近有研究表明，单个连续定向聚合物（DP）在随机势场中的自由能概率分布函数（PDF），等同于由 Kardar-Parisi-Zhang（KPZ）方程描述的生长界面高度，在大尺度下收敛于 Tracy-Widom 分布。后者描述了从高斯酉系综（GUE）中抽取的随机矩阵最大特征值的涨落，该结果适用于具有固定端点的 DP，即具有液滴初始条件的 KPZ 方程。可以提出一个更一般的猜想，将 N>1 个非交叉连续 DP 在随机势场中的自由能与 GUE 的第 N 大特征值之和联系起来。在此，我们使用复制方法，通过精确计算两个 PDF 的右尾并表明它们对于任意 N 都一致，为该猜想提供了一个重要检验。
