Universality for Random Matrices

Abstract

Traces of large powers of real-valued Wigner matrices are known to have Gaussian fluctuations: for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n}\in \mathbb{R}^{n \times n}, A=A^T$ with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d., symmetric, subgaussian, $\mathbb{E}[a^{2}_{11}]=1,$ and $p=o(n^{2/3}),$ as $n,p \to \infty,$ $\frac{\sqrt{\pi}}{2^{p}}(tr(A^p)-\mathbb{E}[tr(A^p)]) \Rightarrow N(0,1).$ This work shows the entries of $A^{2p},$ properly scaled, also have asymptotically normal laws when $n \to \infty, p=n^{o(1)}:$ the normalizations of the diagonal entries depend on $\mathbb{E}[a_{11}^4],$ contributions that become negligible as $p \to \infty,$ whereas their counterparts in $A^{2p+1}$ depend on all the moments of \(a_{11}\) when \(p\) is bounded or the moments grow fast relatively to $p.$ This result demonstrates large powers of Wigner matrices are roughly Wigner matrices with normal entries when $a_{11} \overset{d}{=} -a_{11},\mathbb{E}[a^{2}_{11}]=1, \mathbb{E}[|a_{11}|^{8+\epsilon_0}] \leq C(\epsilon_0),$ providing another perspective on eigenvector universality, which until now has been justified primarily via local laws. The last part of this paper finds the first-order terms of traces of Wishart matrices in the random matrix theory regime, rendering yet another connection between Wigner and Wishart ensembles, as well as an avenue to extend the results herein for the former to the latter. The primary tools employed %behind the entry CLTs are the method of moments and a simple identity the Catalan numbers satisfy.