I ran into this problem on the Bernoulli site percolation on $\mathbb{Z}^2$ coming from another area. I know there's a lot of theory on this and I'm hoping that mathoverflow might help point me in the right direction.

To fix notation consider $\mathbb{Z}^2$ as a graph in the standard way (each vertex has 4 neighbors) and let $P$ be the random subgraph obtained by keeping each vertex with probability $p$ and removing it with probability $1-p$ (all choices being independent).

By a closed path I mean a sequence of vertices $x_1,x_2,\ldots,x_n$ in $P$ where each vertex is a neighbor of the next and $x_n$ is a neighbor of $x_1$.

I say a closed path $x_1,\ldots,x_n$ in $P$ separates two vertices $x,y \in \mathbb{Z}^2$ if any path in $\mathbb{Z}^2$ joining $x$ to $y$ must contain some $x_i$.

My question is the following: Assuming $p$ is close to $1$. What's the number of disjoint closed paths in $P$ which separate $(0,0)$ from $(n,0)$?

To make this more precise, let $S_n$ be the maximal number of disjoint closed paths in $P$ one can take which separate $(0,0)$ from $(n,0)$. Since any such path must meet the line segment joining $(0,0)$ and $(n,0)$ one has $S_n \le n+1$.

Question: Assuming $p$ is close to $1$. Does $S_n/n$ have a positive limit (or liminf) when $n \to +\infty$ almost surely?

In general any information about $S_n$ might be useful to me. Thanks!


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