Open Access
January 2014 Random matrices: Law of the determinant
Hoi H. Nguyen, Van Vu
Ann. Probab. 42(1): 146-167 (January 2014). DOI: 10.1214/12-AOP791


Let $A_{n}$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_{n}|$ satisfies a central limit theorem. More precisely,

\begin{eqnarray*}&&\sup_{x\in{\mathbf {R}}}\biggl|{\mathbf{P} }\biggl(\frac{\log(|\det A_{n}|)-({1}/{2})\log(n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf{P} }\bigl(\mathbf{N} (0,1)\le x\bigr)\biggr|\\&&\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}


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Hoi H. Nguyen. Van Vu. "Random matrices: Law of the determinant." Ann. Probab. 42 (1) 146 - 167, January 2014.


Published: January 2014
First available in Project Euclid: 9 January 2014

zbMATH: 1299.60005
MathSciNet: MR3161483
Digital Object Identifier: 10.1214/12-AOP791

Primary: 60B20 , 60F05

Keywords: random determinant , random matrices

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 1 • January 2014
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