Asymptotics of a singularly perturbed GUE partition function

Details Asymptotics of a singularly perturbed GUE partition function Asymptotics of a singularly perturbed GUE partition function

2010-03-15

arxiv.org/abs/1003.2964v1

Physics

Mathematical Physics

33 pages and 6 figures

Scientific paper

We study the double scaling asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble with weight exp(-z^2/2x^2 + t/x - x^2/2). We derive the asymptotics of the partition function when z and t are of O(N^(-1/2)). Our results are obtained using the Deift-Zhou steepest descent method and are expressed in terms of a solution of a fourth order nonlinear differential equation. We also compute the asymptotic limit of such a solution when zN^(1/2) -> 0. The behavior of this solution, together with fact that the partition function is an odd function in the variable t, allows us to reduce such a fourth order differential equation into a second order nonlinear ODE.

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