All-order asymptotics in beta ensembles in the multi-cut regime

Based on joint work with A. Guionnet (MIT). The beta ensemble is a particular model consisting of N strongly correlated real random variables. For specific values of beta, it is be realized by the eigenvalues of a random hermitian matrix whose distribution is invariant by conjugation, and in this case the model is exactly solvable in terms of orthogonal polynomials, and provide solutions to the Toda chain equations. I will present results on the large \(N\) asymptotics up to \(O(N^{-\infty})\) of the partition function, and the moments of the \(x_i\)'s, away from critical points. I will give an idea about the methods we use, based on the analysis of Schwinger-Dyson equations. Under fairly general assumptions, the \(x_i\) accumulate in the large N limit on a collection of \((g + 1)\) segments of the real line. The asymptotic behavior depends much on \(g\): if \(g = 0\), there is a \(1/N\) expansion, but if \(g > 0\), the model features an oscillatory behavior at all orders in \(1/N\). As an application, we obtain the all-order asymptotics of orthogonal polynomials away from their zero locus, and of solutions of the Toda chain in the continuum limit.