Random Matrix and Integrable Systems

The main topic of this proposal is the study of the Statistical Mechanics of Integrable systems, a particular class of dynamical systems for which the behaviour is fully predictable from the initial data. All relevant information about the dynamics is encoded in a particular matrix L, called Lax matrix.

We want to compute the maximum amplitude for the solution of the Ablowitz-Laddik lattice, and the correlation functions for the Volterra lattice, and the Exponential Toda one.

The first quantity is instrumental to study the phenomenon of rouge waves formations, and the second one to compute transport coefficients of specific lattices.

To compute these quantities, we need to obtain the distribution and the fluctuations of the eigenvalues of the Lax matrix when the initial data are sample according to a Generalized Gibbs Ensemble, thus the Lax matrix becomes a random matrix. To study these objects, we use Large Deviations principles.

Furthermore, we also considered the focusing Ablowitz--Laddik lattice, the focusing Schur flow, and the family of Itoh--Narita--Bogoyavleskii lattices.

The eigenvalues of the Lax matrices of these systems, when the initial data is sample according to a Generalized Gibbs Ensemble, lay on the complex plane.

We plan to compute the density of states, and the joint eigenvalues distribution of the random Lax matrices by using the Inverse Scattering Transform, that is a canonical transformation between the physical variables and the spectral variables of the Lax matrices, the Hermitization technique and the Brown measure characterization.

In the end, thanks to this analysis, we will be able to define some new random matrix ensembles on the complex plane, for which it is possible to compute the eigenvalues distribution, and the joint eigenvalues density explicitly. So, we will define some new beta-ensembles.