caa a22 4500 605461171 CHVBK 20210128100242.0 cr unu---uuuuu 210128e20150901xx s 000 0 eng 10.1007/s10114-015-3685-y doi (NATIONALLICENCE)springer-10.1007/s10114-015-3685-y Zhang Deng Department of Mathematics, Shanghai Jiao Tong University, 200240, Shanghai, P. R. China aut Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I [Elektronische Daten] [Deng Zhang] 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. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg, 2015 Bulk case nationallicence central limit theorem nationallicence the Costin-Lebowitz-Soshnikov theorem nationallicence eigenvalues nationallicence log-gas ensemble nationallicence Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/9(2015-09-01), 1487-1500 1439-8516 31:9<1487 2015 31 10114 https://doi.org/10.1007/s10114-015-3685-y text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10114-015-3685-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Zhang Deng Department of Mathematics, Shanghai Jiao Tong University, 200240, Shanghai, P. R. China aut NATIONALLICENCE 773 0- Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/9(2015-09-01), 1487-1500 1439-8516 31:9<1487 2015 31 10114