Choosing a Spanning Tree for the Integer Lattice Uniformly

Abstract

Consider the nearest neighbor graph for the integer lattice $\mathbf{Z}^d$ in $d$ dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for $\mathbf{Z}^d$. This is shown to be a tree if and only if $d \leq 4$. In this case, the tree has only one topological end, that is, there are no doubly infinite paths. When $d \geq 5$ the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.