Universality and arcsine laws for random matrices \(A+U^{m}B(U^{*})^{m}\) (Q2755164)

This paper deals with an application of the REFORM (resolvent formulas and martingale) method for describing the limit normalized spectral functions for several classes of random unitary matrices. The main result of the article is the following.NEWLINENEWLINENEWLINEAssume that the entries \(\xi_{ij}^{(n)}, i,j=1,\ldots,n\) of the random matrix \(\Xi_{n}\) are independent for every \(n\), \({\mathbf E}\xi_{ij}^{(n)}=0\), \({\text{Var}}\xi_{ij}^{(n)}=n^{-1}, i,j=1,\ldots,n\), and for certain \(\delta>0\) the Lyapunov condition is fulfilled \(\sup\limits_{n}\max\limits_{1\leq i,j\leq n}{\mathbf E}|\xi_{ij}^{(n)}\sqrt{n}|^{4+\delta}<\infty\), \(A_{n}=(\delta_{ij}\alpha_{i}), B_{n}=(\delta_{ij}\beta_{i})\) and \(\sup\limits_{n}\max\limits_{1\leq i,j\leq n}[|\alpha_{i}|+|\beta_{i}|]\leq\infty\). Then for almost all \(x\): \(P\lim\limits_{n\to\infty}[\mu(x, A_{n}+U_{n}B_{n}U_{n}^{T})-F_{n}(x)]=0\), where \(U_{n}=\lim\limits_{\epsilon\downarrow 0}\Xi_{n}(\Xi_{n}^{T}\Xi_{n}+I\epsilon)^{-1/2}\); \(\mu(x, A_{n}+U_{n}B_{n}U_{n}^{T})\) is the normalized spectral function of the eigenvalues of the matrix \(A_{n}+U_{n}B_{n}U_{n}^{T}\); \(F_{n}(x)\) is the nonrandom distribution function whose Stieltjes transform \(m_{n}(z)=\int_{-\infty}^{\infty}(x-z) dF_{n}(x)\) satisfies the system of canonical equations NEWLINE\[NEWLINEm_{n}(z)=\int\limits_{-\infty}^{\infty}{d\mu_{n}(x,A_{n})\over x-z+f_{n}(z)}, m_{n}(z)=\int\limits_{-\infty}^{\infty}{d\mu_{n}(x,B_{n})\over x-z+g_{n}(z)}, 1=f_{n}(z)m_{n}(z)-(z-g_{n}(z))m_{n}(z).NEWLINE\]NEWLINE There exists a unique solution \(\{m_{n}(z),f_{n}(z),g_{n}(z)\}, z=t+ is\) of the system of canonical equations in the class of analytic functions \(L=\{[m_{n}(z),f_{n}(z),g_{n}(z)]:{\text{Im}} m_{n}(z)>0,{\text{Im}} f_{n}(z)>0,{\text{Im}} g_{n}(z)>0, z=t+is, {\text{Im}} z>0, \lim\limits_{s\to\infty}\sup\limits_{|t|}{1\over s}[|m_{n}(z)|+|f_{n}(z)|+|g_{n}(z)|]=0\}\).