turkmath.org

Random matrix theory is roughly the study of the distribution of eigenvalues of large random square matrix in which the distribution of the entries is fixed. It features several mathematical structures : integrability, topological recursion, relation to intersection theory on the moduli space of curves, ... and conversely, the study of random matrices as a toy example can help understand better those structures, their interplay and their generalization. I will illustrate two of them, integrability and topological recursion, while describing the behavior of (real) eigenvalues in certain large random matrices. If $N\to \infty$ is the size of random matrix, the eigenvalues are globally distributed in a deterministic way. The deviations to this behavior are exponentially rare, and can be measured by concentration results. Then, the moments of $M$ can either have a $\frac{1}{N}$ asymptotic expansion, or feature an oscillatory behavior at all orders in $\frac{1}{N}$. In both cases, the coefficient of the expansion can be computed by the "topological recursion", which is a universal recursion involving algebraic geometry on a curve. In our case, the curve is determined only by the leading order of the mean and the covariance of eigenvalues. If time permits, I will explain how the topological recursion allows to predict the tails of the (universal) distribution of the fluctuations of the maximal eigenvalue near its mean in large random matrices in the beta ensemble, known as the beta Tracy-Widom function.