Asymptotic bootstrap for unitary matrix integrals at complex coupling

TL;DR

This work develops and applies an asymptotic bootstrap estimate to non-perturbative unitary matrix integrals, combining exact recursion relations with controlled large-mode asymptotics to achieve high numerical precision without positivity constraints. The method is validated by computing Wilson loop observables and matching exponentially small instanton contributions against analytical instanton calculations, including a Lefschetz-thimble perspective. It also enables systematic exploration of phase diagrams in complex coupling space, where Stokes and anti-Stokes structures proxy for phase boundaries and topology changes of the eigenvalue spectral density. Overall, the approach offers a practical, precision-focused toolkit for probing non-perturbative structure in unitary matrix models and related double-scaled theories.

Abstract

We apply an asymptotic bootstrap estimate method to the non-perturbative study of unitary matrix integrals. The method combines exact recursion relations with asymptotic control of large modes to achieve very high numerical precision without relying on positivity or semidefinite programming. We demonstrate its effectiveness in large-$N$ unitary matrix models by computing Wilson loop expectation values with sensitivity to exponentially small instanton effects and validating them against analytical instanton calculations. We further use the method to explore phase diagrams of unitary matrix models in complex 't Hooft coupling space, where positivity is absent, and observe that Stokes lines provide a useful proxy for additional phase boundaries. Our results show that asymptotic bootstrap estimates offer a practical and precise tool for probing the non-perturbative structure of unitary matrix integrals.

Figures (23)

• Figure 1: Representative root plots of the orthogonal polynomial $p_n$ for $n=N =125$, evaluated at five values of the complex coupling $g$ of the unitary matrix model with a quadratic single-trace potential $V(U)\propto g^{-1}(\text{Tr}(U)+\text{Tr}(U^{-1})+\text{Tr}(U^2)+\text{Tr}(U^{-2}))$ (see \ref{['eq:promotedpotential']}), with one sample taken from each colored region of the phase diagram. The central figure shows the phase diagram in the complex $g$-plane, while the surrounding panels display the corresponding root distributions in the eigenvalue $z$-plane. These colored regions correspond to phase approximations classified by the number of cuts in the associated eigenvalue spectral density. Each root plot is displayed alongside the phase diagram, with the corresponding sample indicated by a black dot and connected to its root plot. Within each panel, the polynomial roots are shown as purple dots, and the unit circle is indicated by a black dashed line. The specific sample values used here are listed in subsection \ref{['subsec:Complicatedmodel']}.
• Figure 2: Absolute error for the example \ref{['eq:example error']}, as a function of the truncation order $k$.
• Figure 3: Ratio of the numerical error divided by an estimated asymptotic error \ref{['eq:asym error']}, shown for both $a_1$ (in blue) and $a_2$ (in yellow). The errors appear highly correlated and exhibit a mild downward curvature.
• Figure 4: Plot of expectation values of (unnormalized) Wilson loops for the potential \ref{['eq:potential']} at increasing values of $N$. The vertical line indicates the expected transition to strong coupling. We see that at large enough $N$, $\widehat{W}_1\simeq 10,\widehat{W}_2\simeq 20, \widehat{W}_3\simeq 0$ as expected. The convergence to these values occurs very rapidly once $N$ surpasses the estimated transition value indicted in red.
• Figure 5: Plot of the residual between the numerical finite-$N$ result versus the expected large-$N$ result, for increasing values of $N$. The expected transition to strong coupling is indicated in red.