Brownian loops on non-smooth surfaces and the Polyakov-Alvarez formula (Q6581819)

Consider the \(2\)-dimensional Brownian loop \((L(s))_{s\in [0,t]}\) with length \(t\) and center \(t^{-1}\int_0^t L(s) ds\), constructed by rescaling the unit-length Brownian loop \((l(s))_{s\in [0,1]}\) such that \(l(0)=l(1)\) and \(\int_0^1 l(s) ds =0\). Given \(\rho\) a Lipschitz function on a compact domain \(D\) in \({\mathbb R}^2\), the \(\rho\)-length of \((L(s))_{s\in [0,t]}\) is defined as \(\int_0^t e^{\rho (L(s))} ds\). The main result of the paper states that the Brownian loop measure of the set of loops with center in \(D\) and \(\rho\)-length at least \(\delta >0\) is given by \[ \frac{1}{2\pi \delta} \int_D e^{\rho (x)} dx + \frac{1}{48 \pi} \int_D | \nabla \rho (x)|^2 dx +o(1) \] as \(\delta\) tends to zero. In addition, it is shown that the \(o(1)\) error term can be bounded uniformly over families of functions \(\rho\) that are precompact in \(W^{1,1}(D)\) and have uniformly bounded Lipschitz constants. The above result is also extended to compact smooth two-dimensional Riemannian manifolds.