Critical interfaces and SLE curves: power law rate of convergence

Modern Mathematics Lecture

Speaker：

Ilia Binder (University of Toronto)

Time：

Thur., 16:00-17:00, May 28, 2026

Venue：

C548, Shuangqing Complex Building A

Online：

Zoom Meeting ID: 271 534 5558Passcode: YMSC

Title：

Critical interfaces and SLE curves: power law rate of convergence

Abstract:

Many two‑dimensional lattice models from statistical physics are believed to exhibit universal scaling limits at criticality, described by Schramm–Loewner Evolution (SLE). While convergence to SLE has been established in several celebrated cases, much less is known about how fast this convergence occurs. In this talk, I will discuss recent progress on quantitative convergence results, focusing on polynomial (power‑law) rates of convergence of discrete random interfaces to SLE. I will present a general framework that yields such rates in a unified way, applicable to a broad class of lattice models. As a central example, we consider the exploration process in critical percolation and show that, for any “reasonable’’ critical percolation model, convergence to SLE follows automatically, together with a polynomial rate. In particular, this result holds unconditionally for critical site percolation on the hexagonal lattice and several of its generalizations. I will also indicate how the same ideas extend to other models, including the Harmonic Explorer and the Ising model. This talk is based on joint work with L. Chayes, D. Chelkak, H. Lei, and L. Richards.