Resumen de Random curves on surfaces induced from the Laplacian determinant

Adrien Kassel, Richard W. Kenyon

• We define natural probability measures on finite multicurves (finite collections of pairwise disjoint simple closed curves) on curved surfaces. These measures arise as universal scaling limits of probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric, in the limit as the mesh size tends to zero. These in turn are defined from the Laplacian determinant and depend on the choice of a unitary connection on the surface.

  Wilson’s algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence.

  We set the framework for the study of these probability measures and their scaling limits and state some of their properties.