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 wellknown criterion implying...

Let Ξ be the set of points (we call the elements of Ξ centers) of Poisson process in Rd, d ≥ 2, with unit intensity. Consider the allocation of Rd to Ξ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume α ≤ 1. We prove that there is no percolation in the set of claimed sites if α is small enough, and that, for high dimensions, there ...

Let Ξ be the set of points (we call the elements of Ξ centers) of Poisson process in Rd, d ≥ 2, with unit intensity. Consider the allocation of Rd to Ξ which is stable in the sense of Gale-Shapley marriage problem and in which each center claims a region of volume α ≤ 1. We prove that there is no percolation in the set of claimed sites if α is small enough, and that, for high dimensions, there ...

The no-enclave percolation (NEP) model introduced recently by Sheinman et al. can be mapped to a problem of holes within a standard percolation backbone, and numerical measurements of such holes give the same size-distribution exponent τ=1.82(1) as found for the NEP model. An argument is given that τ=1+d_{B}/2≈1.822 for backbone holes, where d_{B} is the backbone dimension. On the other hand, a...