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Percolation Transition for Some Excursion Sets

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Percolation Transition for Some Excursion Sets
2.	Creator	Author's name, affiliation, country	Olivier Garet; Université d'Orléans, France
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2004-04-09
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/196
10.	Identifier	Digital Object Identifier	10.1214/EJP.v9-196
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 9
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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