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 = q2m^x2m. More precisely, let P_n(dH) = C_ne^-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) = q2m^x2m with q2m = (Γ(m)Γ(1/2))/(Γ((2m+1)/2), and m ≥ 1. Let x₁ ≤…≤ 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) :=∫_-1^s ρV (x)dx, -1 ≤ s ≤ 1,
and set t := G^-1(k/n). We prove that, as n→∞,
(x_k-t)/(√logn/√2π²nρV (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.