Active spanning trees and Schramm-Loewner evolution

TL;DR

This paper explores the conjectured convergence of Peano curves derived from active spanning trees on planar graphs to Schramm-Loewner evolution (SLE) loops, extending known results to a new parameter range.

Contribution

It extends the conjecture that active spanning tree Peano curves converge to SLE loops to the case where y is between 0 and 1, corresponding to 8<κ≤12.

Findings

Known convergence for y=1 and y=1+√2

Believed convergence for 1<y<3

Proposed extension for 0≤y<1, corresponding to 8<κ≤12

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 $8<\kappa\leq 12$.