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 $4<\kappa\leq 8$. We argue that the same should hold for $0\le y<1$, which corresponds to $8<\kappa\leq 12$.