$β$-Skewed Maximal Spanning Forests

Abstract: The Free $\mathbf{w}$-Maximal Spanning Forest (FMaxSF) is a weighted generalization of the classical Free Minimal Spanning Forest (FMSF) that is able to detect nonhyperfiniteness in percolation on nonunimodular graphs. We introduce a parameterized family of invariant random spanning forests that interpolates between these models. For every finite positive value of the parameter $β$, the construction retains many of the desired properties of FMaxSF while also admitting the finite-subtree forcing property of FMSF. We study local limits of these forests and, in particular, show that the small-$β$ limit of the wired variant coincides with FMSF if and only if $p_h=p_u$, where $p_h$ is the threshold for the existence of heavy clusters and $p_u$ is the uniqueness threshold for Bernoulli$(p)$ percolation. Finally, we show that the Free and the Wired $\mathbf{w}$-Maximal Spanning Forests may coincide even if $p_h<p_u$, providing a negative answer to a question of Terlov and Timár.