# Independent bond percolation on upper density zero subgraphs of the square lattice can have a non-trivial critical point ?

Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where edge set $\mathbb{E}^2$ is given by the pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$. Let $V\subset\mathbb{Z}^2$ an infinite subset and suppose that $G[V]$, the induced subgraph, is connected. Let $\Lambda_n=([-n,n]\times[-n,n])\cap \mathbb{Z}^2$ be a sequence of squares on the two-dimensional lattice. Suppose additionally that $$\limsup_{n\to\infty} \frac{|V\cap \Lambda_n|}{|\Lambda_n|}=0.$$ Question: under the above conditions is it true that the independent bond percolation, with parameter $p$, on $G[V]$ is trivial, in the sense that for any $p\in [0,1)$ we do not have almost surely an infinite cluster ?

I suspect that the answer is affirmative and this is considered in the literature, but until now I only found trivial examples of such graphs $G[V]$, basically constructed from the one-dimensional lattice, where there is no percolation.

This is not true without additional assumptions on your subset. E.g. consider $$V = \{(x,y) \in \mathbb Z^2 : x>0, |y| < \sqrt x \}.$$ It clearly has zero asymptotic density, but on the other hand its critical point is equal to $1/2$ like that of $\mathbb Z^2$ itself. (You can prove that it percolates for every $p>1/2$ using exponential decay of dual clusters in the whole lattice, for instance.)