Virasoro symmetries for the Ablowitz-Ladik hierarchy and non-intersecting Brownian motion models

(eng) Random matrix theory studies the distribution of the spectrum of matrices chosen randomly in various matrix ensembles. The link between random matrix theory and integrable systems has been exhibited in the 1980’s. This thesis is divided into two parts, both related to random matrix theory and integrable systems. The recurrent theme is the Virasoro algebra. The first part deals with the Ablowitz-Ladik (AL) hierarchy and the Circular Unitary Ensemble (CUE). The AL hierarchy is a hierarchy of compatible equations, of which the first one is the Ablowitz-Ladik equation, a space-discretization of the nonlinear cubic Schrödinger equation. The AL hierarchy comes up naturally when studying CUE, as the deformed gap probabilities of this ensemble are tau-functions for the AL hierarchy. We prove that the AL hierarchy admits a centerless Virasoro algebra of master symmetries. An explicit expression for these symmetries is given in terms of a generalisation of the Cantero-Moral-Velázquez matrices in the context of biorthogonal Laurent polynomials. Their action on the tau-functions of the hierarchy is described. The second part of this thesis deals with one-dimensional non-intersecting Brownian motions, starting from and going to several points. Of particular interest are the universal asymptotic processes which appear when the number of Brownian particles gets large. For one starting or ending point, the asymptotic processes are well understood. For several starting and ending points, some open problems remain. We study a finite number of non-intersecting Brownian motions starting from and going to an arbitrary number of points. The quantity of interest is the probability that at a given time all the Brownian particles are in a given set E, taken to be a union of intervals. We prove the existence of a partial differential equation satisfied by the log of this probability. The variables are the coordinates of the starting and ending points and the boundary points of the set E.