Large Deviations Principle for Some Beta Ensembles

Let L be a positive line bundle over a projective complex manifold X , L its tensor power of order p, H(X,L) the space of holomorphic sections of L and Np the complex dimension of H(X,L). The determinant of a basis ofH(X,L), together with some given probability measure on a weighted compact set in X , induces naturally a βensemble, i.e., a random Np-point process on the compact set. Physically, this general setting corresponds to a gas of free fermions on X and may admit some random matrix models. The empirical measures, associated with such β-ensembles, converge almost surely to an equilibrium measure when p goes to infinity. We establish a large deviations principle (LDP) with an effective speed of convergence for these empirical measures. Our study covers the case of some β-ensembles on a compact subset of the unit sphere S ⊂ R or of the Euclidean space R. Classification AMS 2010: 32U15 (32L05, 60F10).