Dimension jump at the uniqueness threshold for percolation in $\infty+d$
dimensions
Dimension jump at the uniqueness threshold for percolation in $\infty+d$
dimensions
Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which percolation has $0$, $\infty$, and $1$ infinite clusters a.s., and it was proven by …