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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Spectral curves\, variational problems\, and the h
ermitian matrix model with external source - Andre
i Martinez-Finkelshtein (Baylor University\; Unive
rsity of Almeria)
DTSTART;TZID=Europe/London:20191029T113000
DTEND;TZID=Europe/London:20191029T123000
UID:TALK133288AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/133288
DESCRIPTION:We show that to any cubic equation from a special
class (`a "spectral curve") it corresponds a uniqu
e vector-valued measure with three components on t
he complex plane\, characterized as a solution of
a variational problem stated in terms of their log
arithmic energy. We describe all possible geome
tries of the supports of these measures: the third
component\, if non-trivial\, lives on a contour o
n the plane and separates the supports of the othe
r two measures\, both on the real line.

This general result is applied to the hermitian ra
ndom matrix model with external source and general
polynomial potential\, when the source has two di
stinct eigenvalues but is otherwise arbitrary. We
prove that under some additional assumptions any l
imiting zero distribution for the average characte
ristic polynomial can be written in terms of a so
lution of a spectral curve. Thus\, any such limiti
ng measure admits the above mentioned variational
description. As a consequence of our analysis we
obtain that the density of this limiting measure c
an have only a handful of local behaviors: Sine\,
Airy and their higher order type behavior\, Pearce
y or yet the fifth power of the cubic (but no high
er order cubics can appear).

This is a jo
int work with Guilherme Silva (U. Michigan\, Ann A
rbor).

We also compare our findings with
the most general results available in the literatu
re\, showing that once an additional symmetry is i
mposed\, our vector critical measure contains enou
gh information to recover the solutions to the con
strained equilibrium problem that was known to des
cribe the limiting eigenvalue distribution in this
symmetric situation.

LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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