Department of Mathematics - Joint Probability Seminar - Multiple points on the boundaries of Brownian loop-soup clusters

The Brownian loop soup, introduced by Lawler and Werner in 2004, plays a crucial role in understanding two-dimensional models in statistical physics. In this talk, I will present the fractal property of the Brownian loop soup by the Hausdorff dimension of multiple points on the boundaries of clusters. The first part of the talk is to review the well-known results on the Hausdorff dimension of special points of the Brownian motion, which is intimately related to a family of exponents. In the second part, I will explain how to obtain our result via generalizing such exponents to the Brownian loop soup. The challenge is to prove a corresponding separation lemma which is unclear due to the interplay of Brownian motions and the Brownian loop soup. I will highlight this powerful lemma with a final remark on its use in our future work. Based on a joint work (arXiv:2205.11468) with Xinyi Li (Peking University) and Wei Qian (City University of Hong Kong).