  A 1-independent bond percolation model on a graph $G$ is a probability
distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint
sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent
of the states of the edges in $S_2$. Such a model is said to percolate if the
random subgraph has an infinite component with positive probability. In 2012
the first author and Bollob\'as defined $p_{\max}(G)$ to be the supremum of
those $p$ for which there exists a 1-independent bond percolation model on $G$
in which each edge is present in the random subgraph with probability at least
$p$ but which does not percolate.
  A fundamental and challenging problem in this area is to determine the value
of $p_{\max}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since
$p_{\max}(\mathbb{Z}^n)\leq p_{\max}(\mathbb{Z}^{n-1})$, it is also of interest
to establish the value of $\lim_{n\to\infty} p_{\max}(\mathbb{Z}^n)$. In this
paper we significantly improve the best known upper bound on this limit and
obtain better upper and lower bounds on $p_{\max}(\mathbb{Z}^2)$. In proving
these results, we also give an upper bound on the critical probability for a
1-independent model on the hypercube graph to contain a giant component
asymptotically almost surely.
