The Scaling Limit of the Intrinsic Metric and Simple Random Walk on 2D Critical Percolation

Explore the mathematical foundations of conformal loop ensembles (CLE) and critical percolation through this 47-minute conference talk that demonstrates how CLE(κ) gaskets for κ values between 4 and 8 can be equipped with canonical intrinsic metrics and Brownian motion processes. Learn about the scaling limit behavior of critical percolation on triangular lattices, where shortest path distances and simple random walks on large clusters converge to continuum metrics and Brownian motion on CLE(6). Discover the joint convergence properties that connect discrete lattice models to their continuous counterparts, examining how loops interact with each other and domain boundaries in the critical regime. The presentation covers collaborative research findings on the geometric and probabilistic structures underlying two-dimensional critical systems, providing insights into the relationship between discrete random walk processes and their continuum limits through rigorous mathematical analysis of scaling behaviors and metric properties.