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Active Spanning Trees with Bending Energy on Planar Maps and SLE-Decorated Liouville Quantum Gravity for \({\kappa > 8}\)

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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 Fortuin–Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield’s 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 \({{\rm SLE}_\kappa}\)-decorated γ-Liouville quantum cone with \({\kappa > 8}\) and \({\gamma = 4/ \sqrt\kappa \in (0,\sqrt 2)}\).

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Author information

Authors and Affiliations

  1. MIT, Cambridge, MA, USA

    Ewain Gwynne

  2. ENS Lyon, Lyon, France

    Adrien Kassel

  3. University of Cambridge, Cambridge, UK

    Jason Miller

  4. Microsoft Research, Redmond, WA, USA

    David B. Wilson

Authors
  1. Ewain Gwynne
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  2. Adrien Kassel
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  3. Jason Miller
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  4. David B. Wilson
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Correspondence to Ewain Gwynne.

Additional information

Communicated by H. Duminil-Copin

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Gwynne, E., Kassel, A., Miller, J. et al. Active Spanning Trees with Bending Energy on Planar Maps and SLE-Decorated Liouville Quantum Gravity for \({\kappa > 8}\). Commun. Math. Phys. 358, 1065–1115 (2018). https://doi.org/10.1007/s00220-018-3104-1

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