Natural spanning trees of \({\mathbb{Z}}^ d\) are recurrent (Q1080429)

An old result of Polya states that the simple random walk on \({\mathbb{Z}}^ d\) is recurrent for \(d=1\) and 2, but transient for \(d\geq 3\). In the present paper it is shown that the natural spanning tree of \({\mathbb{Z}}^ d\) is recurrent for every \(d=1,2,3,... \). An asymptotic result on the probability of returning to the origin in n steps is proved by the method of Darboux.