On Natural Measures of SLE- and CLE-Related Random Fractals

TL;DR

This paper constructs and proves the uniqueness of natural measures on various SLE and CLE fractals, establishing their equivalence with discrete models and providing new proofs via CLE and LQG coupling.

Contribution

It introduces a new approach to defining and proving the uniqueness of natural measures on SLE and CLE fractals, and links these measures to discrete percolation models.

Findings

Natural measures on SLE cut points and boundary touching points are uniquely characterized.

CLE pivotal points and gasket measures are shown to be equivalent to discrete counting measures.

A new proof technique via CLE and LQG coupling is developed for natural measure existence and uniqueness.

Abstract

In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an application, we also show the equivalence between our natural measures defined in this paper (i.e. CLE pivotal and gasket measures) and their discrete analogs of counting measures in critical continuum planar Bernoulli percolation in [Garban-Pete-Schramm, J. Amer. Math. Soc.,2013]. Although the existence and uniqueness for the natural measure for CLE carpet/gasket have already been proved in [Miller-Schoug, arXiv:2201.01748], in this paper we provide with a different argument via the coupling of CLE and LQG.