Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers [1,20] and sample covariance matrices [4]. We consider the case when the n × n external source matrix has two distinct real eigenvalues: a with multiplicity r and zero with multiplicity n − r. The source is small in the sense that r is finite or r = O(n γ ), for 0 < γ < 1. For a Gaussian potential, Péché [29] showed that for |a| sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for |a| sufficiently large (the supercritical regime) r eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of r × r Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.
In Hele-Shaw flows a boundary of a viscous fluid develops unstable fingering patterns. At vanishing surface tension, fingers evolve to cusp-like singularities preventing a smooth flow. We show that the Hele-Shaw problem admits a weak solution where a singularity triggers viscous shocks. Shocks form a growing, branching tree of a line distribution of vorticity where pressure has a finite discontinuity. A condition that the flow remains curl-free at a macroscale uniquely determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2,3)-cusp singularity -an elementary branching event of a branching shock graph.
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