1003.2964 (L. Brightmore et al.)

A matrix model with a singular weight and Painleve' III    [PDF]

L. Brightmore, F. Mezzadri, M. Y. Mo

We investigate the matrix model with weight exp(-z^2/2x^2 + t/x - x^2/2) and unitary symmetry. In particular we study the double scaling limit as N -> infinity and zN^(1/2) -> 0, where N is the matrix dimension. Using the Deift-Zhou steepest descent method we compute the asymptotics of the partition function when z and t are O(N^(-1/2)). In this regime we discover a phase transition in the (t,z)-plane characterised by the Painleve' III equation. This is the first time that Painleve' 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 Painleve' III equation. We derive explicitly the initial conditions in the limit zN^(1/2) -> 0 of this solution.

View original: http://arxiv.org/abs/1003.2964