Loop-erased partitioning via parametric spanning trees: Monotonicities & 1D-scaling

We consider a parametric version of the UST (Uniform Spanning Tree) measure on arbitrary directed weighted finite graphs with tuning (killing) parameter q>0. This is obtained by considering the standard random weighted spanning tree on the extended graph built by adding a ghost state † and directed edges to it, of constant weights q, from any vertex of the original graph. The resulting measure corresponds to a random spanning rooted forest of the graph where the parameter q tunes the intensity of the number of trees as follows: partitions with many trees are favored for q>1, while as q→0, the standard UST of the graph is recovered. We are interested in the behavior of the induced random partition, referred to as Loop-erased partitioning, which gives a correlated cluster model, as the multiscale parameter q∈[0,∞) varies. Emergence of giant clusters in this correlated percolation model as a function of q has been recently explored on certain dense growing graphs Avena et al. (2022). Herein we derive two types of results. First, we explore monotonicity properties in q of this forest measure on general graphs showing in particular some counter-intuitive subtleties in non-reversible settings where the electrical-network interpretation of the UST observables gets partially lost. Second, by analyzing 2-points correlations on trees and various very sparse growing graph models, we characterize emerging macroscopic clusters, as q scales with the graph size, and derive related phase diagrams.