On coupling and “vacant set level set” percolation

Electronic Communications in Probability

Alain-Sol Sznitman¹ 

关键词: Gaussian free field;    random interlacements;    percolation;    coupling;   

DOI  :  10.1214/19-ECP217

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

In this note we discuss “vacant set level set” percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpins a certain strict monotonicity property that has significant consequences for the percolation diagram. It is presently open whether a similar looking diagram holds in the case of ${\mathbb Z}^d$, $d \ge 3$.

【 授权许可】

CC BY