BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Spectral curves\, variational problems\, and the hermitian matrix model with external source - Andrei Martinez-Finkelshtein (Baylor Universi ty\; University of Almeria) DTSTART:20191029T113000Z DTEND:20191029T123000Z UID:TALK133291@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:We show that to any cubic equation from a special class (`a "s pectral curve") it corresponds a unique vector-valued measure with three c omponents on the complex plane\, characterized as a solution of a variatio nal problem stated in terms of their logarithmic energy. We describe all possible geometries of the supports of these measures: the third compone nt\, if non-trivial\, lives on a contour on the plane and separates the su pports of the other two measures\, both on the real line. This general r esult is applied to the hermitian random matrix model with external source and general polynomial potential\, when the source has two distinct eigen values but is otherwise arbitrary. We prove that under some additional ass umptions any limiting zero distribution for the average characteristic pol ynomial can be written in terms of a solution of a spectral curve. Thus\, any such limiting measure admits the above mentioned variational descript ion. 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 a nd their higher order type behavior\, Pearcey or yet the fifth power of th e cubic (but no higher order cubics can appear). We also compare our find ings with the most general results available in the literature\, showing t hat once an additional symmetry is imposed\, our vector critical measure c ontains enough information to recover the solutions to the constrained equ ilibrium problem that was known to describe the limiting eigenvalue distri bution in this symmetric situation. LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR