2017/03/23 by Serban T. Belinschi, Belinschi, Serban, Hari Bercovici +3
Mathematics · #Random Matrices and Applications #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1703.08102
Given a selfadjoint polynomial P(X,Y) in two noncommuting selfadjoint\nindeterminates, we investigate the asymptotic eigenvalue behavior of the random\nmatrix P(A\_N,B\_N), where A\_N and B\_N are independent Hermitian random\nmatrices and the distribution of B\_N is invariant under conjugation by\nunitary operators. We assume that the empirical eigenvalue distributions of\nA\_N and B\_N converge almost surely to deterministic probability measures\n\μ and \ν, respectively. In addition, the eigenvalues of A\_N and\nB\_N are assumed to converge uniformly almost surely to the support of \μ\nand \ν, respectively, except for a fixed finite number of fixed eigenvalues\n(spikes) of A\_N. It is known that almost surely the empirical distribution\nof the eigenvalues of P(A\_N,B\_N) converges to a certain deterministic\nprobability measure \η (sometimes denoted \η=P^ square(\μ,\ν)) and,\nwhen there are no spikes, the eigenvalues of P(A\_N,B\_N) converge uniformly\nalmost surely to the support of \η. When spikes are present, we show that\nthe eigenvalues of P(A\_N,B\_N) still converge uniformly to the support of\n\η, with the possible exception of certain isolated outliers whose location\ncan be determined in terms of \μ,\ν,P, and the spikes of A\_N. We\nestablish a similar result when B\_N is replaced by a Wigner matrix. The\nrelation between outliers and spikes is described using the operator-valued\nsubordination functions of free probability theory. These results extend known\nfacts from the special case in which P(X,Y)=X+Y.\n