Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula

Abstract

Let ρ be compactly supported on D⊂R2. Endow R2 with the metric eρ|dx|2. As δ→0 the set of Brownian loops centered in D with length at least δ has measureVol(D)2πδ+148π(ρ,ρ)∇+o(1) where (ρ,ρ)∇=∫D|∇ρ(x)|2dx. When ρ is smooth, this follows from the classical Polyakov-Alvarez formula. We show that the above also holds if ρ is not smooth, e.g. if ρ is only Lipschitz. Variants of this statement apply to more general non-smooth manifolds on which one considers all loops (not only those centered in a domain D).We also show that the o(1) error is uniform for any family of ρ satisfying certain conditions. This implies that if we weight a measure ν on this family by the (δ-truncated) Brownian loop soup partition function, and take the vague δ→0 limit, we obtain a measure whose Radon-Nikodym derivative with respect to ν is exp⁡(148π(ρ,ρ)∇). When the measure is a certain regularized Liouville quantum gravity measure, a companion work [5] shows that this weighting has the effect of changing the so-called central charge of the surface.