Near-critical spanning forests and renormalization

TL;DR

This paper investigates the scaling limits of two-dimensional spanning forests, relating them to a renormalization flow modeled by a Markov process, and proves convergence to a stationary distribution connected to SLE$_2$.

Contribution

It establishes a rigorous connection between the scaling limit of critical spanning forests and a stationary distribution of a renormalization Markov process, extending prior SLE convergence results.

Findings

Convergence of the Markov process to a stationary distribution representing the scaling limit.

Relation of the scaling limit to a stationary distribution of a coalescent-type Markov process.

Rigorous implementation of a formal renormalization framework for critical models.

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 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 SLE$_2$ (a recent result by Lawler and Viklund).