Strong Szego Theorem on a Jordan Curve

I will discuss certain determinants with respect to a sufficiently regular Jordan curve 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 Szego 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 the curve. As a consequence of our formula we obtain the asymptotics of the partition function on the curve. Interestingly, this formula involves the Fredholm determinant of the absolute value squared of the Grunsky operator which equals, up to a multiplicative constant, the so called Loewner energy of the curve.

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