2023/02/05 by Minjae Park, Park, Minjae, Joshua Pfeffer +3
Mathematics · #58J52 (Secondary) #60D05 (Primary) 58J65 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2302.02358
openalex publication_date 2023/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ρ be compactly supported on D ⊂ \mathbb R2. Endow \mathbb R2 with the metric eρ(dx12 + dx22). As δ→ 0 the set of Brownian loops centered in D with length at least δ has measure (area(D))/(2πδ) + (1)/(48π)(ρ,ρ)∇+ o(1). 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. This fact can alternatively be expressed in terms of heat kernel traces, eigenvalue asymptotics, or zeta regularized determinants. 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( (1)/(48π)(ρ,ρ)∇). When the measure is a certain regularized Liouville quantum gravity measure, a companion work [APPS20] shows that this weighting has the effect of changing the so-called central charge of the surface.