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Percolation transition for some excursion sets. (English) Zbl 1065.60147
Summary: We consider a random field \((X_n)_{n\in\mathbb{Z}^d}\) and investigate when the set \(A_h= \{k\in \mathbb{Z}^d;|X_k|\geq h\}\) has infinite clusters. The main problem is to decide whether the critical level \[ h_c= \sup\{h\in\mathbb{R}; P\,(A_h\text{ has an infinite cluster)}> 0\} \] 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 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 critical case and show that a transition percolation occurs for directed percolation when \(d\geq 4\). 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G15 Gaussian processes
82B43 Percolation
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
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