Open Access
July 2020 Near-critical spanning forests and renormalization
Stéphane Benoist, Laure Dumaz, Wendelin Werner
Ann. Probab. 48(4): 1980-2013 (July 2020). DOI: 10.1214/19-AOP1413


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).


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Stéphane Benoist. Laure Dumaz. Wendelin Werner. "Near-critical spanning forests and renormalization." Ann. Probab. 48 (4) 1980 - 2013, July 2020.


Received: 1 July 2018; Revised: 1 September 2019; Published: July 2020
First available in Project Euclid: 20 July 2020

zbMATH: 07224966
MathSciNet: MR4124531
Digital Object Identifier: 10.1214/19-AOP1413

Primary: 60K35
Secondary: 60J67 , 82B20 , 82B26 , 82B28

Keywords: renormalization , Schramm–Loewner evolution , uniform spanning trees

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 4 • July 2020
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