$1$-independent percolation on $\mathbb{Z}^2 \times K_n$

Abstract: A random graph model on a host graph H is said to be 1-independent if for every pair of vertex-disjoint subsets A,B of E(H), the state of edges (absent or present) in A is independent of the state of edges in B. For an infinite connected graph H, the 1-independent critical percolation probability $p_{1,c}(H)$ is the infimum of the p in [0,1] such that every 1-independent random graph model on H in which each edge is present with probability at least p almost surely contains an infinite connected component. Balister and Bollob\'as observed in 2012 that $p_{1,c}(\mathbb{Z}^d)$ is nonincreasing and tends to a limit in [1/2, 1] as d tends to infinity. They asked for the value of this limit. We make progress towards this question by showing that [\lim_{n\rightarrow \infty}p_{1,c}(\mathbb{Z}^2\times K_n)=4-2\sqrt{3}=0.5358\ldots \ .] In fact, we show that the equality above remains true if the sequence of complete graphs $K_n$ is replaced by a sequence of weakly pseudorandom graphs on n vertices with average degree $\omega(\log n)$. We conjecture that the equality also remains true if $K_n$ is replaced instead by the n-dimensional hypercube $Q_n$. This latter conjecture would imply the answer to Balister and Bollob\'as's question is $4-2\sqrt{3}$. Using our results, we are also able to resolve a problem of Day, Hancock and the first author on the emergence of long paths in 1-independent random graph models on $\mathbb{Z}\times K_n$. Finally, we prove some results on component evolution in 1-independent random graphs, and discuss a number of open problems arising from our work that may pave the way for further progress on the question of Balister and Bollob\'as.