Choosing a spanning tree for the integer lattice uniformly.

*(English)*Zbl 0758.60010Consider the nearest neighbour graph for the integer lattice \(\mathbb{Z}^ d\) in \(d\) dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly amongst all possible such subgraphs. It is shown that as the piece gets larger, this approaches a limiting measure on the spanning subgraphs of \(\mathbb{Z}^ d\), that this measure concentrates on spanning forests, which are trees 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. Extensive use is made of the connections between uniform spanning trees, random walks and electrical networks, and also with loop-erased random walks.

Reviewer: C.McDiarmid (Oxford)

##### MSC:

60C05 | Combinatorial probability |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |