On the outlying eigenvalues of a polynomial in large independent random matrices
Résumé
Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu $ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\Pi_P$, and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\Pi_P$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\Pi_P$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu,\nu,P$ and the spikes of $A_N$. We establish a similar result when $B_N$ is a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extends known facts from the special case in which $P(X,Y)=X+Y$.
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