Strong Szegő theorem on a Jordan curve

Abstract: We consider certain determinants with respect to a sufficiently regular Jordan curve $\gamma$ in the complex plane that generalize Toeplitz determinants which are obtained when the curve is the circle. This also corresponds to studying a planar Coulomb gas on the curve at inverse temperature $\beta=2$. Under suitable assumptions on the curve we prove a strong Szeg\H{o} type asymptotic formula as the size of the determinant grows. The resulting formula involves the Grunsky operator built from the Grunsky coefficients of the exterior mapping function for $\gamma$. As a consequence of our formula we obtain the asymptotics of the partition function for the Coulomb gas on the curve. This formula involves the Fredholm determinant of the absolute value squared of the Grunsky operator which equals, up to a multiplicative constant, the Loewner energy of the curve. Based on this we obtain a new characterization of curves with finite Loewner energy called Weil-Petersson quasicircles.