The Type-problem on the Average for random walks on graphs

When averages over all starting points are considered, the Type Problem for the recurrence or transience of a simple random walk on an inhomogeneous network in general differs from the usual “local” Type Problem. This difference leads to a new classification of inhomogeneous discrete structures in terms of {\it recurrence} and {\it transience} {\it on the average}, describing their large scale topology from a “statistical” point of view. In this paper we analyze this classification and the properties connected to it, showing how the average behavior affects the thermodynamic properties of statistical models on graphs.