Asymptotic bootstrap for unitary matrix integrals at complex coupling

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.