Percolation Transition for Some Excursion Sets
Olivier Garet, Université d'Orléans, France
Abstract
We consider a random field (Xn)n ∈ Zd and investigate when the set
Ah={k ∈ Zd; |Xk| ≥ h}
has infinite clusters.
The main problem is to decide whether the critical level
hc=sup{h ∈ R ;P(Ah has an infinite cluster)>0}
is neither
0 nor +∞. 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.
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