Active Spanning Trees with Bending Energy on Planar Maps and SLE-Decorated Liouville Quantum Gravity for đ > 8
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Author(s)
Gwynne, Ewain
Kassel, Adrien
Miller, Jason
Wilson, David B.
Date Issued
February 2018
Journal
Communications in Mathematical Physics
Publisher
Springer Science and Business Media LLC
Citation
Gwynne, Ewain et al. "Active Spanning Trees with Bending Energy on Planar Maps and SLE-Decorated Liouville Quantum Gravity for đ
> 8." Communications in Mathematical Physics 358, 3 (February 2018): 1065â1115 © 2018 Springer-Verlag GmbH Germany, part of Springer Nature
Version
Author's final manuscript
Abstract
We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical FortuinâKasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffieldâs hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including the ones corresponding to maps decorated by an active spanning tree), the infinite-volume limit of spanning-tree-decorated planar maps sampled from our model converges in the peanosphere sense, upon rescaling, to an SLE Îș-decorated Îł-Liouville quantum cone with Îș > 8 and Îł = 4 / âÎș â (0, â2).
.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00220-018-3104-1
