Loop-erased partitioning of a graph: mean-field analysis

Abstract

Given a weighted finite graph G, we consider a random partition of its vertex set induced by a measure on spanning rooted forests on G. The latter is a generalized parametric version of the classical Uniform Spanning Tree measure which can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter $q>0$. The related random trees—identifying the blocks of the partition—tend to cluster nodes visited by the random walk on time scale $1∕q$. We explore the emerging macroscopic structure by analyzing two-point correlations, as a function of the tuning parameter q. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block. This interaction potential can be seen as an affinity measure for “densely connected nodes” and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structure. For such geometries we show phase-transitions in the behavior of the random partition as a function of the tuning parameter and the edge weights. Moreover, as a corollary of our main results, we infer the right scaling of the parameters that give rise to the emergence of “giant” blocks.