Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory

TL;DR

This paper introduces the mathematical framework of Schramm-Loewner evolution (SLE) and boundary conformal field theory to analyze the fractal geometry of critical curves in two-dimensional statistical mechanics, providing pedagogical explanations and alternative methods for calculating multifractal spectra.

Contribution

It offers a comprehensive pedagogical introduction to SLE and presents an alternative boundary CFT approach for deriving multifractal spectra of critical curves.

Findings

Derivation of fractal dimensions and crossing probabilities for critical curves.

Explanation of SLE properties and their relation to conformal maps.

Application of boundary CFT to compute multifractal spectra without quantum gravity.

Abstract

Conformally-invariant curves that appear at critical points in two-dimensional statistical mechanics systems, and their fractal geometry have received a lot of attention in recent years. On the one hand, Schramm has invented a new rigorous as well as practical calculational approach to critical curves, based on a beautiful unification of conformal maps and stochastic processes, and by now known as Schramm-Loewner evolution (SLE). On the other hand, Duplantier has applied boundary quantum gravity methods to calculate exact multifractal exponents associated with critical curves. In the first part of this paper I provide a pedagogical introduction to SLE. I present mathematical facts from the theory of conformal maps and stochastic processes related to SLE. Then I review basic properties of SLE and provide practical derivation of various interesting quantities related to critical curves, including fractal dimensions and crossing probabilities. The second part of the paper is devoted to a way of describing critical curves using boundary conformal field theory (CFT) in the so-called Coulomb gas formalism. This description provides an alternative (to quantum gravity) way of obtaining the multifractal spectrum of critical curves using only traditional methods of CFT based on free bosonic fields.