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A singular partition function, universality and Painleve' III
Fri 15 February 2013, 14:00
Francesco Mezzadri
University of Bristol
Organiser: Nina Snaith
ABSTRACT
Matrix models with singular weights have recently become of interest in various
areas of physics and mathematics, like chaotic quantum transport, quantum field theory as well as number theory.
The singularity in the weight usually means that studying the asymptotic behaviour of quantities like partition functions or
linear statistics of the eigenvalues presents major challenges.
We shall discuss the asymptotics as N -> infinity of the
partition function of the matrix model with weight
w(x) := exp(-z^2/(2x^2) + t/x - x^2/2)
We discover a phase transition in the $(z,N)$-plane characterised by the Painlev\'e III
equation. This is the first time that Painlev\'e III appears in studies of double scaling limits in Random Matrix Theory and is
associated to the emergence of the essential singularity in the weighting function. The asymptotics of the partition function are
expressed in terms of a particular solution of the Painlev\'e III equation.
This is work with Lorna Brightmore and Man Yue Mo.