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.
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Zhang, D. Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I. Acta. Math. Sin.-English Ser. 31, 1487–1500 (2015). https://doi.org/10.1007/s10114-015-3685-y
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DOI: https://doi.org/10.1007/s10114-015-3685-y