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.

Bio:

Professor Ilia Binder is a Professor of Mathematics at the University of Toronto and Chair of the Department of Mathematical and Computational Sciences at the University of Toronto Mississauga. He currently serves as the President of the Canadian Mathematical Society. He received his Ph.D. in Mathematics from the California Institute of Technology. He held positions at Harvard University, the Institute for Advanced Study Princeton, and the University of Illinois at Urbana-Champaign before joining the University of Toronto. Professor Binder’s research lies in geometric function theory, complex analysis, complex dynamics, Schramm–Loewner evolution, multifractal analysis, computability and complexity, and Schrödinger operators. He has made important contributions to the study of harmonic measures, conformally invariant random curves, Julia sets, and related topics in analysis, probability, and dynamical systems.