Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I
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| 024 | 7 | 0 | |a 10.1007/s10114-015-3685-y |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10114-015-3685-y | ||
| 100 | 1 | |a Zhang |D Deng |u Department of Mathematics, Shanghai Jiao Tong University, 200240, Shanghai, P. R. China |4 aut | |
| 245 | 1 | 0 | |a Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I |h [Elektronische Daten] |c [Deng Zhang] |
| 520 | 3 | |a 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. | |
| 540 | |a Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg, 2015 | ||
| 690 | 7 | |a Bulk case |2 nationallicence | |
| 690 | 7 | |a central limit theorem |2 nationallicence | |
| 690 | 7 | |a the Costin-Lebowitz-Soshnikov theorem |2 nationallicence | |
| 690 | 7 | |a eigenvalues |2 nationallicence | |
| 690 | 7 | |a log-gas ensemble |2 nationallicence | |
| 773 | 0 | |t Acta Mathematica Sinica, English Series |d Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society |g 31/9(2015-09-01), 1487-1500 |x 1439-8516 |q 31:9<1487 |1 2015 |2 31 |o 10114 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10114-015-3685-y |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10114-015-3685-y |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Zhang |D Deng |u Department of Mathematics, Shanghai Jiao Tong University, 200240, Shanghai, P. R. China |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Acta Mathematica Sinica, English Series |d Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society |g 31/9(2015-09-01), 1487-1500 |x 1439-8516 |q 31:9<1487 |1 2015 |2 31 |o 10114 | ||