Random curves, Laplacians, and determinants

The loop-erased random walk (LERW), obtained from a random walk by chronologically erasing the loops created by self-intersection of the path, was introduced by Lawler in 1980 as an integrable model for a self-avoiding random path. In 2001 it was shown by Lawler, Schramm, and Werner to converge (in the scaling limit of small mesh size in Z²) to SLE₂, the Schramm-Loewner evolution of parameter κ=2, a conformally invariant random curve in the plane driven by a stochastic differential equation.

On a large finite graph, and in the long running-time limit, the statistics of loops erased by the LERW procedure are closely related to the statistics of the cycle of a uniformly sampled cycle-rooted spanning tree. I will give some precise estimates (obtained in joint work with Rick Kenyon and Wei Wu) on the shape (length and area) of this cycle in the infinite volume limit for some periodic planar graphs.

Wilson’s algorithm for sampling uniform spanning trees from LERW can be naturally modified to sample random cycle-rooted spanning forests (CRSF) (spanning graphs all of whose connected components is a cycle-rooted tree). The random collection of loops created by this procedure are ‘excursions’ of the LERW which ‘interact’ via the trees rooted on each of them. I will explain a construction (obtained in joint work with Rick Kenyon) of the scaling limit of naturally weighted CRSFs on graphs embedded on surfaces. This defines probability measures on the space Ω of multicurves of the surface which possess some properties which we hope characterize them in the space of probability measures on Ω.

As a conclusion, I will explain why the random CRSFs introduced above (seen as point processes on the set of edges of the graph) belong to a class of symmetric determinantal processes, slightly extending the usual class in the sense that their kernels take values in the skew field of quaternions.