Random walk loop soups and conformal loop ensembles

Conformal radii for conformal loop ensembles

The conformal loop ensembles CLEκ, defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model. We also compu...

Loop-Erased Random Walk

Random Walk Loop Soup

The Brownian loop soup introduced in [3] is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlós, M...

Exploration trees and conformal loop ensembles

We construct and study the conformal loop ensembles CLE(κ), defined for 8/3 ≤ κ ≤ 8, using branching variants of SLE(κ) called exploration trees. The CLE(κ) are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical ph...

Scaling Limit of Loop-erased Random Walk

The loop-erased random walk (LERW) was first studied in 1980 by Lawler as an attempt to analyze self-avoiding walk (SAW) which provides a model for the growth of a linear polymer in a good solvent. The self-avoiding walk is simply a path on a lattice that does not visit the same site more than once. Proving things about the collection of all such paths is a formidable challenge to rigorous math...