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

We study the fluctuations of the largest eigenvalue λ max 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 λ max 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 λ max —of order —the large deviation tails are instead associated with extremely rare fluctuations—of order . 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.