### Spread-out percolation on transitive graphs of polynomial growth

Combinatorics Seminar

26th March 2024, 11:00 am – 12:00 pm

Fry Building, 2.04

Let G be a vertex-transitive graph of superlinear polynomial growth. Given r > 0, let G_r be the graph on the same vertex set as G, with two vertices joined by an edge if and only if they are at graph distance at most r apart in G. We show that as r goes to infinity the critical probability p_c(G_r) for Bernoulli bond percolation on G_r is asymptotically 1/deg(G_r); this should be compared to the fact that p_c(G) > 1/deg(G) for any infinite connected graph G. This significantly extends work of Penrose (1993) and Bollobás-Janson-Riordan (2007), who considered the case G=Z^d, and verifies a special case of a recent conjecture of Easo and Hutchcroft. Joint with Panagiotis Spanos.

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