Large deviation principle for some beta ensembles

Abstract:

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