Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales

TL;DR

This paper explores how multiple Schramm-Loewner Evolutions (SLEs) can model interfaces in 2D critical statistical mechanics, linking partition functions, martingales, and conformal blocks to physical phenomena.

Contribution

It introduces a framework for multiple SLEs satisfying a key statistical mechanics condition, connecting conformal blocks to interface configurations and physical interpretations.

Findings

Examples of multiple SLEs consistent with critical models

Relation between conformal blocks and interface geometry

Applications to percolation and Ising model

Abstract

A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs to satisfy this condition leads to some natural processes, which we study in this note. We give examples of such multiple SLEs and discuss how a choice of conformal block is related to geometric configuration of the interfaces and what is the physical meaning of mixed conformal blocks. We illustrate the general ideas on concrete computations, with applications to percolation and the Ising model.