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We consider random partitions of the vertex set of a given finite graph thatcan be sampled by means of loop-erased random walks stopped at a randomexponential time of parameter $q>0$. The related random blocks tend to clusternodes visited by the random walk on time scale $1/q$. This random partitioningis induced by a measure of rooted spanning forest of the graph, whichgeneralizes the classical uniform spanning tree measure and which can beobtained as a zero-limit of FK-percolation with an external cemetery state.Some general properties of this rooted forest measure and related determinantalobservables, along with a number of applications in data analysis have beenrecently explored. We are here mainly interested in the structure the emergentpartitioning, referred to as loop-erased partitioning, as the scale parameter$q$ varies.  We first present two general results shedding light on subtle monoticityproperties in $q$ of these rooted forest and loop-erased partitioning measures.The first theorem characterizes monotone events in $q$ by deriving a Russo-likeformula. Our second general result concerns two-point correlations defined bythe probability that two vertices do not belong to the same block of thepartitioning. It states that, on undirected graphs, these correlation functionsare increasing in $q$. We then explore other types of results aiming atunderstanding the emerging asymptotic clusters on simple growing graph models,as $q$ scales with the graph size. Some first results in this direction havebeen investigated in the recent [arXiv:1906.03858] on dense geometries. Here welook at very sparse graphs. We offer a detailed analysis of the resultingpartitioning on line segments and we look at trees and other tree-likegeometries, without and with implanted modular structures. For the latter, wecharacterize the asymptotic detection of these implanted modules.

Loop-erased partitioning of a network: monotonicities & analysis of cycle-free graphs