Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa > 8$

TL;DR

This paper introduces a two-parameter family of models for spanning trees on planar maps, linking them to Liouville quantum gravity and SLE processes, and proves convergence of these models to SLE-decorated quantum cones.

Contribution

It develops a new probabilistic model for spanning trees with bending energy on planar maps and establishes their convergence to SLE-decorated Liouville quantum gravity surfaces.

Findings

The model generalizes active spanning trees with bending energy.

Infinite-volume limits converge to SLE$_ppa$-decorated Liouville quantum cones.

The encoding via a generalized Sheffield bijection facilitates the convergence proof.

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 SLE$_\kappa$-decorated $\gamma$-Liouville quantum cone with $\kappa > 8$ and $\gamma = 4/\sqrt\kappa \in (0,\sqrt 2)$.