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

Understand

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.