Percolation Transition for Some Excursion Sets

Abstract

We consider a random field $(X_n)_{n\in\mathbb{Z}^d}$ and investigate when the set $A_h=\{k\in\mathbb{Z}^d; \vert X_k\vert \ge h\}$ has infinite clusters. The main problem is to decide whether the critical level $$h_c=\sup\{h\in R : P(A_h\text{ has an infinite cluster })>0\}$$ is neither $0$ nor $+\infty$. Thus, we say that a percolation transition occurs. In a first time, we show that weakly dependent Gaussian fields satisfy to a well-known criterion implying the percolation transition. Then, we introduce a concept of percolation along reasonable paths and therefore prove a phenomenon of percolation transition for reasonable paths even for strongly dependent Gaussian fields. This allows to obtain some results of percolation transition for oriented percolation. Finally, we study some Gibbs states associated to a perturbation of a ferromagnetic quadratic interaction. At first, we show that a transition percolation occurs for superstable potentials. Next, we go to the the critical case and show that a transition percolation occurs for directed percolation when $d\ge 4$. We also note that the assumption of ferromagnetism can be relaxed when we deal with Gaussian Gibbs measures, i.e., when there is no perturbation of the quadratic interaction.