Asymptotics of the partition function of the perturbed Gross-Witten-Wadia unitary matrix model

TL;DR

This paper analyzes the asymptotic behavior of the partition function of a perturbed Gross-Witten-Wadia matrix model, revealing phase transitions and expressing constants via special functions.

Contribution

It derives detailed asymptotic expansions of the partition function with an added logarithmic potential term, connecting to Painlevé equations and special functions.

Findings

Asymptotic expansions include constant terms with zeta and Barnes G-functions.

Identifies a third-order phase transition in the model.

Expresses the partition function as a Toeplitz determinant linked to Painlevé III'.

Abstract

We consider the asymptotics of the partition function of the extended Gross-Witten-Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a $\tau$-function sequence of the Painlev\'e III' equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta-function and the Barnes $G$-function. A third-order phase transition in the leading terms of the asymptotic expansions is also observed.