Random curves, scaling limits and Loewner evolutions

TL;DR

This paper develops a framework linking probabilistic estimates to the convergence of 2D statistical mechanics models' interfaces to Schramm-Loewner Evolution (SLE) curves, advancing understanding of their scaling limits.

Contribution

It introduces a new estimate-based approach for proving the convergence of random curves to SLE, applicable to models like Ising and FK Ising, and extends to branching interface trees.

Findings

Weak annulus crossing probability estimates imply SLE scaling limits.

Existence of conformally-invariant observables suffices for convergence.

Framework supports convergence proofs for Ising and FK Ising interfaces.

Abstract

In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally-invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves, moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.