## View abstract

### Session I.3 - Graph Theory and Combinatorics

Monday, June 12, 15:00 ~ 15:30

## On percolation in locally dependent random graphs

### Victor Falgas-Ravry

Consider a random subgraph of the square integer lattice $\mathbb{Z}^2$ obtained by including each edge independently at random with probability $p$, and omitting it otherwise. The Harris--Kesten theorem states that if $p\leq 1/2$, then almost surely all connected components in this random graph model are finite, while if $p \gt 1/2$ then almost surely the model percolates and 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 nearby edges. To what extent can we exploit such local dependencies to delay the emergence of an infinite component?
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