Active spanning trees and Schramm-Loewner evolution

Abstract

We consider the Peano curve separating a spanning tree from its dual spanning tree on an embedded planar graph, where the tree and dual tree are weighted by $y$ to the number of active edges, and "active" is in the sense of the Tutte polynomial. When the graph is a portion of the square grid approximating a simply connected domain, it is known ($y=1$ and $y=1+\sqrt{2}$) or believed ($1<y<3$) that the Peano curve converges to a space-filling SLE${}_{\kappa }$ loop, where $y=1-2\cos (4\pi /\kappa )$, corresponding to $4<\kappa \le 8$. We argue that the same should hold for $0\le y<1$, which corresponds to $8<\kappa \le 12$.

6 pages, 7 figures

References in corpus (6)

• A Guide to Stochastic Loewner Evolution and its Applications
• Duality of Chordal SLE
• Winding angle variance of Fortuin-Kasteleyn contours
• Watersheds are Schramm-Loewner Evolution curves
• Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case
• Random cluster dynamics for the Ising model is rapidly mixing

Cited by in corpus (3)

• Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa >8$
• The six-vertex model and Schramm-Loewner evolution
• Critical Behaviour of Spanning Forests on Random Planar Graphs