Maxima of entries of Haar distributed matrices
Given an \(n\times n\) random matrix \(\Gamma_n=(\gamma_{ij})\) in \(O(n)\) with normalized Haar distribution, the behavior of the maximum of the absolute values of the entries, namely \(W_n=\max| \gamma_{ij}| \), is studied. The limit \(\lim_n \sqrt{n/\log n}\,W_n\) is shown to be 2 in probability, and \[ \lim_{n\rightarrow\infty}\,P(nW_n^2-4\log n+\log(\log n)\leq x)=\exp(-\sqrt{1/2\pi}\,e^{-x/2}). \] If independence of the sequence \(\{\Gamma_n\}\) of random matrices is required, then the sequence \(\{\sqrt{n/\log n}\,W_n\}\) is shown to be dense in \([2,\sqrt6]\) almost surely. Analog results are proven for matrices in \(\text{SO}(n)\), \(U(n)\), and \(\text{SU}(n)\).
scientific article; zbMATH DE number 2137181
scientific article