The Brownian intersection exponents are computed by Lawler, Schramm and Werner (Acta Math, 2001a, 2001b, 2002) [conjectured by Duplantier and Kwon (Phys Rev Lett, 1988)]. These exponents are useful in determining the Hausdorff dimension of some exceptional sets of planar Brownian motion. In this talk, we discuss our recent development on such exponents. First, we show that a family of generalized exponents can be related to the fractal geometry of the Brownian loop soup (a Poissonian ensemble of Brownian loops in 2D). Second, we prove that the corresponding 3D exponents are analytic, which remains open to this day.

Speaker's Bio: Gao Yifan is currently a postdoctoral fellow at City University of Hong Kong, specializing in statistical physics and stochastic geometry (seepage, Gaussian free fields, SLE, etc.). He obtained his doctoral degree from Peking University in 2022 under the supervision of Professor Li Xinyi and Professor Zhang Fuxi. We have achieved many important results and have been accepted or published in journals such as Ann. Probab. and Comm. Math. Phys.