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Let G be an infinite connected graph with minimum degree δ and maximum degree Δ. Let Gp be a random induced subgraph of G obtained by selecting each vertex of G independently with probability p, , and let be the induced subgraph of Gp obtained by deleting all vertices of Gp with degree greater than k in Gp. We show that if and is not too large then almost surely has no...
Let be the set of points in a realization of a uniform Poisson process in ℝn. The set determines a Voronoĭ tiling of ℝn. Construct an infinite graph with vertex set and edges joining vertices when the corresponding Voronoĭ cells share a (n − 1)‐dimensional boundary face. We consider bond percolation models on obtained by declaring each edge xy of open...
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