Near-critical spanning forests and renormalization

Abstract

We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state space of abstract graphs with real-valued edge weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in order to relate the law of the scaling limit of a critical model to a stationary distribution of such a renormalization/Markov process. When starting from any two-dimensional lattice with constant edge weights, the Markov process does indeed converge in law to this stationary distribution that corresponds to a scaling limit of UST with Poissonian deletions. The results of this paper heavily build on the convergence in distribution of branches of the UST to $\mathrm{SLE}_{2}$ (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the loop-erased random walk to the “natural parametrization” of the $\mathrm{SLE}_{2}$ (a recent result by Lawler and Viklund).