We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas X2m ensemble with an external potential V -- q2m . More precisely, let Pn（dH） = Cne-nTrV（H）dH be the distribution of n &...We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas X2m ensemble with an external potential V -- q2m . More precisely, let Pn（dH） = Cne-nTrV（H）dH be the distribution of n × n Hermitian random matrices, py（x）dx the equilibrium measure, where Cn is a normalization constant, V（x）= q2mx2m with q2m , and m≥ 1. Let x1 ≤...≤xn 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 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.展开更多
文摘We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas X2m ensemble with an external potential V -- q2m . More precisely, let Pn（dH） = Cne-nTrV（H）dH be the distribution of n × n Hermitian random matrices, py（x）dx the equilibrium measure, where Cn is a normalization constant, V（x）= q2mx2m with q2m , and m≥ 1. Let x1 ≤...≤xn 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 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.
