Top eigenvalue of a random matrix: large deviations and third order phase transition

Abstract

We study the fluctuations of the largest eigenvalue ${\lambda }_{\max }$ of $N\times N$ random matrices in the limit of large $N$. The main focus is on Gaussian $\beta$-ensembles, including in particular the Gaussian orthogonal ($\beta =1$), unitary ($\beta =2$) and symplectic ($\beta =4$) ensembles. The probability density function (PDF) of ${\lambda }_{\max }$ consists, for large $N$, of a central part described by Tracy-Widom distributions flanked, on both sides, by two large deviations tails. While the central part characterizes the typical fluctuations of ${\lambda }_{\max }$ -- of order $\mathcal{O}(N^{-2/3})$ --, the large deviations tails are instead associated to extremely rare fluctuations -- of order $\mathcal{O}(1)$. 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.

32 pages, 8 figures, contribution to Statphys25 (Seoul, 2013) proceedings. Revised version where references have been added and typos corrected

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