Some recent aspects of random conformally invariant systems

TL;DR

This paper reviews recent developments in two-dimensional conformally invariant systems, focusing on conformally invariant measures on loops and conformal loop-ensembles, highlighting progress in understanding their properties and structures.

Contribution

It introduces recent results on conformally invariant measures and conformal loop-ensembles, advancing the theoretical understanding of these systems.

Findings

Analysis of conformally invariant measures on loops

Development of conformal loop-ensembles (CLE) theory

Insights into the structure of 2D conformally invariant systems

Abstract

These are the lecture notes from a course given in July 2005 at the summer school in Les Houches. We describe some recent results concerning two-dimensional conformally invariant systems. In particular, we discuss conformally invariant measures on loops and conformal loop-ensembles (CLE).