Top eigenvalue of a random matrix: large deviations and third order phase transition
Top eigenvalue of a random matrix: large deviations and third order phase transition
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DOI：
10.1088/1742-5468/2014/01/p01012
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发表时间：
2013-11
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影响因子：
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通讯作者：
S. Majumdar;G. Schehr
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文献类型：
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作者：
S. Majumdar;G. Schehr
研究了N × N随机矩阵最大特征值λmax在大N极限下的涨落.主要关注的是高斯β系综，特别是包括高斯正交（β = 1），酉（β = 2）和辛（β = 4）系综。当N较大时，λmax的概率密度函数（PDF）由Tracy-Widom分布描述的中心部分组成，两侧是两个大偏差尾部。而中心部分的特点是典型的波动λ max-的顺序O（N−2/3）？> - 大偏差尾部反而与O（1）阶的极其罕见的波动有关。在这里，我们回顾了一些最近的发展，这些极其罕见的事件的理论，使用库仑气体的方法。我们讨论特别是第三阶相变分离的左尾从右尾，过渡类似于所谓的Gross-Witten-Wadia相变中发现的2-D晶格量子色动力学。我们还讨论了在各种物理问题中类似的三阶跃迁的发生，包括不相交的布朗运动，介观物理中的电导涨落和二分系统中的纠缠。
We study the fluctuations of the largest eigenvalue λmax of N × N random matrices in the limit of large N. The main focus is on Gaussian β ensembles, including in particular the Gaussian orthogonal (β = 1), unitary (β = 2) and symplectic (β = 4) ensembles. The probability density function (PDF) of λmax consists, for large N, of a central part described by Tracy–Widom distributions flanked, on both sides, by two large deviation tails. While the central part characterizes the typical fluctuations of λmax—of order O(N−2/3)?> —the large deviation tails are instead associated with extremely rare fluctuations—of order O(1)?>. Here we review some recent developments in the theory of these extremely rare events using a Coulomb gas approach. We discuss in particular the third order phase transition which separates the left tail from the right tail, a transition akin to the so-called Gross–Witten–Wadia phase transition found in 2-d lattice quantum chromodynamics. We also discuss the occurrence of similar third order transitions in various physical problems, including non-intersecting Brownian motions, conductance fluctuations in mesoscopic physics and entanglement in a bipartite system.