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

Yu Chen, Shuai-Xia Xu, Yu-Qiu Zhao

TL;DR

This work analyzes the large-$n$ asymptotics of the partition function for the extended Gross-Witten-Wadia unitary matrix model by recasting it as a Toeplitz determinant with $I$-Bessel entries and showing it forms a $\tau$-function sequence of Painlevé III$'$. Using a rigorous Deift-Zhou Riemann-Hilbert steepest-descent approach, the authors derive explicit asymptotic expansions for $\log D_{n,\nu}(n\tau)$ both for $0<\tau<1$ and for $\tau>1$, including constant terms expressed through $\log G(1-\nu)$ and $\zeta'(-1)$, and reveal a third-order phase transition at $\tau=1$ in the leading behavior. The analysis hinges on constructing a global parametrix and local parametrices (Airy and parabolic-cylinder types) around critical points, coupled with a differential identity in $\nu$ to access the constant terms. The results extend prior GWW asymptotics to general complex $\nu$, link the partition function to Painlevé dynamics, and provide exact constant-term expressions that involve Barnes $G$-function and zeta-function derivatives, highlighting the rich interplay between random matrix theory, special functions, and integrable systems.

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 $τ$-function sequence of the Painlevé 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.

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

TL;DR

This work analyzes the large- asymptotics of the partition function for the extended Gross-Witten-Wadia unitary matrix model by recasting it as a Toeplitz determinant with -Bessel entries and showing it forms a -function sequence of Painlevé III. Using a rigorous Deift-Zhou Riemann-Hilbert steepest-descent approach, the authors derive explicit asymptotic expansions for both for and for , including constant terms expressed through and , and reveal a third-order phase transition at in the leading behavior. The analysis hinges on constructing a global parametrix and local parametrices (Airy and parabolic-cylinder types) around critical points, coupled with a differential identity in to access the constant terms. The results extend prior GWW asymptotics to general complex , link the partition function to Painlevé dynamics, and provide exact constant-term expressions that involve Barnes -function and zeta-function derivatives, highlighting the rich interplay between random matrix theory, special functions, and integrable systems.

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 -function sequence of the Painlevé 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 -function. A third-order phase transition in the leading terms of the asymptotic expansions is also observed.

Paper Structure

This paper contains 20 sections, 4 theorems, 193 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Riemann-Hilbert problem for the orthogonal polynomials
  4. Asymptotics of $Y(z;n\tau)$: case $0<\tau<1$
  5. Normalization: $\widehat{Y} \to T$
  6. Deformation: $T\to S$
  7. Global parametrix: $N$
  8. Local parametrix: $P$
  9. Final transformation: $S\to R$
  10. Asymptotics of $Y(z;n\tau)$: case $\tau>1$
  11. Normalization: $\widehat{Y} \to T$
  12. Deformation: $T\to S$
  13. Global parametrix: $N$
  14. Local parametrices: $P^{({\pm} )}$
  15. Final transformation: $S\to R$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\nu\in \mathbb{C}$, $t=n\tau$ and $0<\tau<1$, we have the asymptotic approximation of the logarithm of the Toeplitz determinant associated with eq: weight where the error term is uniform for $\tau$ in any compact subsets of $(0,1)$. For $\nu=1,2,3\cdots,$ the asymptotic behavior of $\log D_{n,\nu}(t)$ can be obtained from Integral:logDn1 by using the symmetry $D_{n,\nu}(t)=D_{n,-\nu}(t)$ for

Figures (11)

  • Figure 1: The contour $\Gamma$
  • Figure 2: The regions for $\Omega_{k}$, $k=0,1,2$.
  • Figure 3: Contours of RH problem for $\widehat{Y}$
  • Figure 4: Contours and regions of the RH problem for $S(z)$
  • Figure 5: The contours and regions of the RH problem for $P(z)$ and their image under the map $\lambda(z)$
  • ...and 6 more figures

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1
  • proof
  • Proposition 2
  • proof