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

We study the fluctuations of the largest eigenvalue lambda(max) of N x 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 deviation tails. While the central part characterizes the it fluctuations of lambda(max)-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, in non-intersecting Brownian motions, conductance fluctuations in mesoscopic physics and entanglement in a bipartite system.