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 $q\to 0$, 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 $q\in [0,\infty )$ 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.