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

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 $${\gamma = 4/ \sqrt\kappa \in (0,\sqrt 2)}$$ .