Recurrence and transience for infinite dimensional hypercube

The infinite-dimensional hypercube (IDH) is a novel example of a locally infinite graph and can play an important role in understanding Anderson localization and other related physical phenomena. In this paper, we investigate the IDH from the perspective of transience and recurrence. We examine the Green function associated with the heat semigroup on the IDH and establish several of its fundamental properties. By using the unilateral Green function, which we introduce, we provide conditions for the heat semigroup to be transient. Finally, we prove that the transience (or recurrence) of the heat semigroup is equivalent to that of a discrete-time Markov chain defined on the IDH.