Dimension jump at the uniqueness threshold for percolation in $\infty +d$ dimensions

Tom Hutchcroft, Minghao Pan

Consider percolation on $T\times {\mathbb{Z}}^d$, the product of a regular tree of degree $k\ge 3$ with the hypercubic lattice ${\mathbb{Z}}^d$. It is known that this graph has $0<p_c…