Near-critical spanning forests and renormalization
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Date
2020
Publication Type
Journal Article
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yes
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Abstract
© Institute of Mathematical Statistics, 2020. 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 SLE2 (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the looperased random walk to the "natural parametrization" of the SLE2 (a recent result by Lawler and Viklund).
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published
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Journal / series
Volume
48 (4)
Pages / Article No.
1980 - 2013
Publisher
Institute of Mathematical Statistics
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Edition / version
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Date collected
Date created
Subject
Uniform spanning trees; Schramm–Loewner evolution; Renormalization
Organisational unit
09453 - Werner, Wendelin (ehemalig) / Werner, Wendelin (former)
Notes
Funding
155922 - Exploring two-dimensional continuous structures (SNF)
175505 - Loops, paths and fields (SNF)
175505 - Loops, paths and fields (SNF)
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