Mathematics

This presentation delves into the fascinating relationship between integrable systems theory and random matrix theory. In particular, a method to calculate the eigenvalue density of the Lax matrix   of the Ablowitz-Ladik lattice   sampled from the  Generalized Gibbs ensemble is derived.  This characterization is achieved by employing a large deviation principle (LDP). Additionally,  a connection between this LDP and the one of the Circular β ensemble in the high temperature regime is established. This permits to  explicitly compute the eigenvalue density of the Ablowitz-Ladik lattice random Lax matrix.