Uniqueness and full coverage in $k$-neighbor graphs

Abstract

We study the number of infinite clusters in the $2dp$-neighbor graph on $\mathbb Z^d$, $d\ge2$, in which every vertex independently selects $\lfloor 2dp\rfloor$ of its nearest neighbors uniformly at random, and one more with probability $2dp-\lfloor 2dp\rfloor$. The model has negatively correlated edges and, for $1\le2dp\le2d-1$, neither finite energy nor insertion or deletion tolerance. We prove that the undirected graph has at most one infinite cluster for every $p\in [0,1]$, and that in the planar case the same holds for forward and strong forward clusters of the directed graph when $p\in[1/2,1]$. We further exhibit a second phase transition: for $p\ge(d+1)/(2d)$ the open cluster of the origin is almost surely the whole lattice, and for $p>(d+1)/(2d)$ any two vertices of a box are connected inside the box of twice the size with probability exponentially close to one. In particular, the percolation function is continuous but not analytic on the supercritical interval $(p_c,1]$. Finally, we prove that the full-coverage phase transition at $p_f = (d+1)/2d$ is of order $2^d$.