Random matrices: Law of the determinant

By Hoi H. Nguyen, Van H. Vu. 2011

73 citations recorded by the source corpus.

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*}

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cfcc5a38d365c4cc6a71bc7970db99ff641af17d

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, background): “cessary; thus, (8.2) |G(x1,...,xk)|≤1 17Alternatively, one can allow the moments to deviate from each other by, say, O(n−100), which one can verify will not aﬀect the argument. See [3], Chapter 2, or [36], Appendix A, for details. UNIVERSALITY FOR NON-HERMITIAN MATRICES 49 for all x1,...,xk ∈R. Fix ”