Generalized potlatch and smoothing processes

We consider simple generalizations of the potlatch and smoothing processes which were introduced in [8] and studied in [5]. These generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition. The original potlatch and smoothing processes do not exhibit phase transition. Our results show that for the generalized processes, phase transition does not usually occur in one or two dimensions, but usually does occur in higher dimensions. Upper and lower bounds for the relevant critical values are obtained. As one application of our results, we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d→∞, thus answering a question we raised in [3]. This is done by comparing the contact process with an appropriate generalized smoothing process.