Random matrices: Law of the determinant

@inproceedings{Nguyen2014RandomML,
  title={Random matrices: Law of the determinant},
  author={Hoi H. Nguyen and Văn Hải Vũ},
  year={2014}
}
  • Hoi H. Nguyen, Văn Hải Vũ
  • Published 2014
  • Mathematics
  • 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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