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

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, .V (x)dx the equilibrium measure, where C n is a normalization constant, V (x) = q 2m x 2m with q2m=.(m).(12).(2m+12), and m . 1. Let x 1 .... . x n be the eigenvalues of H. Let k:= k(n) be such that k(n)n.[a,1.a] for n large enough, where a . (0, 1/2). Define G(s):=.s.1.v(x)dx,.1.s.1, and set t:= G .1(k/n). We prove that, as n . ., xk.t(logn.)2.2.n.v(t).N(0,1) 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.