Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble.

Complex eigenvalues of random matrices J=GUE+iγdiag(1,0,…,0) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions N≫1 the eigenvalue density of undergoes an abrupt restructuring at γ=1, the critical threshold beyond which a single eigenvalue outlier ("broad resonance") appears. We provide a detailed description of this restructuring transition, including the scaling with of the width of the critical region about the outlier threshold γ=1 and the associated scaling for the real parts ("resonance positions") and imaginary parts ("resonance widths") of the eigenvalues which are farthest away from the real axis.

Exact Persistence Exponent for the 2D-Diffusion Equation and Related Kac Polynomials.

We compute the persistence for the 2D-diffusion equation with random initial condition, i.e., the probability p_{0}(t) that the diffusion field, at a given point x in the plane, has not changed sign up to time t.