Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I

Abstract

We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas ensemble with an external potential V = q _2m x ^2m. More precisely, let P _n (dH) = C _n e ^-nTrV(H) dH be the distribution of n × n Hermitian random matrices, ρV (x)dx the equilibrium measure, where C _n is a normalization constant, V (x) = q _2m x ^2m with $q2m = \frac{{\Gamma \left( m \right)\Gamma \left( {\frac{1}{2}} \right)}}{{\Gamma \left( {\frac{{2m + 1}}{2}} \right)}}$, and m ≥ 1. Let x ₁ ≤... ≤ x _n be the eigenvalues of H. Let k:= k(n) be such that $\frac{{k\left( n \right)}}{n} \in \left[ {a,1 - a} \right]$ for n large enough, where a ∈ (0, 1/2). Define $G\left( s \right): = \int_{ - 1}^s {\rho v\left( x \right)dx, - 1 \leqslant s \leqslant 1} ,$ and set t:= G ⁻¹(k/n). We prove that, as n → ∞, $\frac{{xk - t}}{{\frac{{\left( {\sqrt {\log n} } \right)}}{{\sqrt {2{\pi ^2}} n\rho v\left( t \right)}}}} \to N\left( {0,1} \right)$ in distribution. Multi-dimensional central limit theorem is also proved. Our results can be viewed as natural extensions of the bulk central limit theorems for GUE ensemble established by J. Gustavsson in 2005.