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

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. (paper)