Communications in Mathematical Physics

, Volume 358, Issue 3, pp 1065–1115 | Cite as

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

  • Ewain Gwynne
  • Adrien Kassel
  • Jason Miller
  • David B. Wilson


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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ewain Gwynne
    • 1
  • Adrien Kassel
    • 2
  • Jason Miller
    • 3
  • David B. Wilson
    • 4
  1. 1.MITCambridgeUSA
  2. 2.ENS LyonLyonFrance
  3. 3.University of CambridgeCambridgeUK
  4. 4.Microsoft ResearchRedmondUSA

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