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Spectral curves, variational problems, and the hermitian matrix model with external source

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CATW02 - Complex analysis in mathematical physics and applications

We show that to any cubic equation from a special class (`a “spectral curve”) it corresponds a unique vector-valued measure with three components on the complex plane, characterized as a solution of a variational problem stated in terms of their logarithmic energy. We describe all possible geometries of the supports of these measures: the third component, if non-trivial, lives on a contour on the plane and separates the supports of the other two measures, both on the real line. This general result is applied to the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. We prove that under some additional assumptions any limiting zero distribution for the average characteristic polynomial can be written in terms of a solution of a spectral curve. Thus, any such limiting measure admits the above mentioned variational description. As a consequence of our analysis we obtain that the density of this limiting measure can have only a handful of local behaviors: Sine, Airy and their higher order type behavior, Pearcey or yet the fifth power of the cubic (but no higher order cubics can appear). We also compare our findings with the most general results available in the literature, showing that once an additional symmetry is imposed, our vector critical measure contains enough information to recover the solutions to the constrained equilibrium problem that was known to describe the limiting eigenvalue distribution in this symmetric situation.

This talk is part of the Isaac Newton Institute Seminar Series series.

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