  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.
