From Instantons to Large Deviations in Hermitian Random Matrices
Max Atkin (University of Bielefeld)
The typical fluctuations of the largest eigenvalue of a Hermitian random matrix about its mean has been known for a long time to be given by the Tracy-Widom distribution. More recently interest has focused on the ‘atypical’ fluctuations in which an eigenvalue appears very far from the bulk spectrum. This problem has been attacked using saddle point methods, loop equations and, to a much lesser extent, orthogonal polynomials. In this talk we review recent progress in the orthogonal polynomial approach which makes contact with instanton effects in the string theory literature. We use this framework to derive the distribution for large deviations in the case of multi-critical potentials.