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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:TALK133288@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. <br> <br> This g
 eneral result is applied to the hermitian random matrix model with externa
 l source and general polynomial potential\, when the source has two distin
 ct eigenvalues but is otherwise arbitrary. We prove that under some additi
 onal assumptions any limiting zero distribution for the average characteri
 stic 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 pow
 er of the cubic (but no higher order cubics can appear).<br> <br> This is 
 a joint work with Guilherme Silva (U. Michigan\, Ann Arbor).<br> <br> We a
 lso compare our findings with the most general results available in the li
 terature\, showing that once an additional symmetry is imposed\, our vecto
 r critical measure contains enough information to recover the solutions to
  the constrained equilibrium problem that was known to describe the limiti
 ng eigenvalue distribution in this symmetric situation.<br>
LOCATION:Seminar Room 1\, Newton Institute
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