Interference percolation1. Issue 4 (5th March 2013) - Balister, Paul
Abstract 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, 0<p<1, and let Gp≤k be the induced subgraph of Gp obtained by deleting all vertices of Gp with degree greater than k in Gp. We show that if δ≥6 and Δ/δ is not too large then Gp≤3 almost surely has no infinite component. Moreover, this result is essentially best possible since there are examples where Gp≤k has an infinite component (a) when δ=Δ=3, 4, or 5, and k = 3; (b) when Δ≫δ for any δ and k = 3; and (c) when δ=Δ for any δ≥3 and k≥4. In addition, we show that if G is the d‐dimensional lattice ℤd then G1/d≤4 almost surely has an infinite component for sufficiently large d. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 399–418, 2014