学术报告（高一帆 2024.10.15）

摘要：Bernoulli percolation, introduced 70 years ago, is a central subject in modern probability theory. The two-dimensional case is well understood now thanks to its connection to Schramm-Loewner evolution (SLE). Much progress has also been made in long-range correlated models in recent years, such as excursion sets of GFF and loop-soup percolation. In fact, the GFF and the loop soup themselves are very important in the study of 2D random conformal geometry. Contrary to the excursion sets, less is known about the two-sided level sets of 2D GFF. We present some recent progress in the study of this two-sided percolation. Thanks to Dynkin’s isomorphism, the problem for GFF is nicely translated into the language of loop soups. In particular, we show that the percolation for the occupation field of the random walk loop soup with subcritical intensity exhibits a non-trivial phase transition, and we also obtain some results for TSLS of GFF with varying levels. Along the way, we develop a set of useful tools for the study of arm events in the loop soup, such as separation lemmas, quasi-multiplicativity, and arm exponents.