Large deviation principle for some beta ensembles

Abstract

Let L L be a positive line bundle over a projective complex manifold X X , L p L^p its tensor power of order p p , H 0 ( X , L p ) H^0(X,L^p) the space of holomorphic sections of L p L^p , and N p N_p the complex dimension of H 0 ( X , L p ) H^0(X,L^p) . The determinant of a basis of H 0 ( X , L p ) H^0(X,L^p) , together with some given probability measure on a weighted compact set in X X , induces naturally a β \beta -ensemble, i.e., a random N p N_p -point process on the compact set. Physically, depending on X X and the value of β \beta , this general setting corresponds to a gas of free or interacting fermions on X X and may admit an interpretation in terms of some random matrix models. The empirical measures, associated with such β \beta -ensembles, converge almost surely to an equilibrium measure when p p goes to infinity. We establish a large deviation theorem (LDT) with an effective speed of convergence for these empirical measures. Our study covers a large class of β \beta -ensembles on a compact subset of the unit sphere S n ⊂ R n + 1 \mathbb {S}^n\subset \mathbb {R}^{n+1} or of the Euclidean space R n \mathbb {R}^n .