University of Birmingham > Talks@bham > Combinatorics and Probability seminar > Towards a 1-dependent version of the Harris--Kesten theorem

Towards a 1-dependent version of the Harris--Kesten theorem

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If you have a question about this talk, please contact Johannes Carmesin.

Consider a random subgraph of the square integer lattice Z2 obtained by including each edge independently at random with probability p, and leaving it out otherwise. The Harris—Kesten theorem states that if p is at most 1/2, then almost surely all connected components in this random subgraph are finite, while if p>1/2 then almost surely there exists a unique infinite connected component.

But now what if we introduced some local dependencies between the edges? More precisely, suppose each edge still has a probability p of being included in our random subgraph, but its state (present/absent) may depend on the states of edges it shares a vertex with. To what extent can we exploit such local dependencies to delay the appearance of an infinite component?

In this talk I will discuss this question, which first arose in work of Balister, Bollobás and Walters in 2005, and discuss some recent progress on it made in joint work with Nicholas Day, Robert Hancock and Vincent Pfenninger.

This talk is part of the Combinatorics and Probability seminar series.

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