Conformal restriction properties

Abstract

We give an introduction to some aspects of recent results concerning conformally invariant measures. We focus in this note on the conformal restriction properties of some measures on curves and loops in the plane, and see that these properties in fact almost characterize the measures and allow to classify them. For example, there basically exists a unique measure μ on the set of self-avoiding loops in the plane, such that for any two conformally equivalent domains D and D', the restrictions of μ to the set of loops remaining in D and D' are conformally equivalent.

This enables to show that a priori different discrete models define the same curves in the scaling limit and exhibit some surprising symmetries. It gives also a way to tie links between these concrete measures on curves and conformal field theory. Important roles in this theory are played by Brownian loops and by the Schramm–Loewner Evolutions (SLE).

Most of the results described in this paper were derived in joint work with Greg Lawler, and Oded Schramm.