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arXiv · 1512.09122

Active spanning trees and Schramm-Loewner evolution

Adrien Kassel·David B. Wilson
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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$_κ$ loop, where $y=1-2\cos(4π/κ)$, corresponding to $4<κ\leq 8$. We argue that the same should hold for $0\le y<1$, which corresponds to $8<κ\leq 12$.

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BibTeXRIS

Adrien Kassel, David B. Wilson. 2016-05-13. Active spanning trees and Schramm-Loewner evolution. https://doi.org/10.1103/physreve.93.062121

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