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The largest eigenvalue of deformed Wigner random matrices

Mathematical Physics & Probability

Speaker: Sandrine Peche, UC Davis and Institut Fourier Grenoble
Location: 1147 MSB
Start time: Tue, Oct 3 2006, 2:10PM

We consider a finite rank perturbation of standard Hermitian random matrices: M_N=\frac{1}{\sqrt N} H +W_N, of size NxN, where the entries H_{ij}, i less than j, are i.i.d centered random variables with finite variance and W_N is a fixed rank deterministic matrix with largest eigenvalue \pi_1 independent of N. We then show that, for a large class of random matrix ensembles, the largest eigenvalue of M_N exhibits, in the large N-limit a phase transition according to the value of \pi_1. The limiting distribution of the properly scaled largest eigenvalue is also proved to be universal. The proof heavily relies on a preliminary universality result, due to A. Soshnikov, for the limiting fluctuations of the largest eigenvalues of standard Wigner random matrices.