The OSSS Method in Percolation Theory

Sharpness of the Percolation Phase Transition for the Contact Process on $\mathbb{Z}^d$

We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raou…

平面Fortuin-Kasteleyn渗流的混合速率指数

Mixing rate exponent of planar Fortuin-Kasteleyn percolation

该研究定义了共形环系综（CLE）对应的混合速率指数，在FK渗流收敛于CLE的假设下证明二者指数一致，并借助刘维尔量子引力与CLE的耦合推导出CLE$_κ$的混合速率指数公式，解答了前人提出的相关问题。

Three theorems in discrete random geometry

These notes are focused on three recent results in discrete random geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is \sqrt{2+\sqrt 2}; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on Z^2 is \sqrt q/(1+\sqrt q). Background information on the relevant random processes is presented on route to these theorems. The emphasis is upon the communication …