Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model

TL;DR

This paper rigorously proves predictions about non-local observables in the 2D critical Ising model by connecting Schramm-Loewner evolution (SLE) with the model's scaling limit, and extends results to intersection probabilities.

Contribution

It provides a mathematical proof linking SLE(3) to non-local observables in the critical Ising model and computes intersection probabilities with Brownian excursions.

Findings

Confirmed theoretical predictions for non-local observables.

Derived intersection probabilities between SLE paths and Brownian excursions.

Established a rigorous connection between SLE and the Ising model's scaling limit.

Abstract

We present a mathematical proof of theoretical predictions made by Arguin and Saint-Aubin, as well as by Bauer, Bernard, and Kytola, about certain non-local observables for the two-dimensional Ising model at criticality by combining Smirnov's recent proof of the fact that the scaling limit of critical Ising interfaces can be described by chordal SLE(3) with Kozdron and Lawler's configurational measure on mutually avoiding chordal SLE paths. As an extension of this result, we also compute the probability that an SLE(k) path, k in (0,4], and a Brownian motion excursion do not intersect.