Asymptotics of spacing distributions 50 years later

Abstract

In 1962 Dyson used a physically based, macroscopic argument to deduce the first two terms of the large spacing asymptotic expansion of the gap probability for the bulk state of random matrix ensembles with symmetry parameter \beta. In the ensuing years, the question of asymptotic expansions of spacing distributions in random matrix theory has shown itself to have a rich mathematical content. As well as presenting the main known formulas, we give an account of the mathematical methods used for their proofs, and provide some new formulas. We also provide a high precision numerical computation of one of the spacing probabilities to illustrate the accuracy of the corresponding asymptotics.