Local limit of massive spanning forests on the complete graph

Abstract

We identify the local limit of massive spanning forests on the complete graph. This generalizes a well-known theorem of Grimmett on the local limit of uniform spanning trees on the complete graph.

Figures (2)

• Figure 1: The distribution of the local limit $\mathcal{T}_{\alpha}$.
• Figure 2: Samples of ${\rm \bf \lambda SF}(\mathbf{K}_{200})$ obtained via Wilson's algorithm with a uniform, positive killing rate $r=1-\mu=\frac{\lambda}{\lambda+n-1}$. From left to right: $r=0$ (i.e. a sample of ${\bf UST}(\mathbf{K}_{200})$), $r=\frac{1}{40}$ and $r=\frac{1}{3}$.

Theorems & Definitions (16)

• Theorem 1: Local limit of ${\rm \bf \lambda SF}(K_{n})$
• Remark 1.1
• Remark 1.2
• Theorem 2.1: Kirchhoff's or Matrix-Forest Theorem Kir
• Proposition 2.1
• proof
• Proposition 3.1: Resolvent and transfer current matrix
• proof
• Lemma 3.2: Inclusion probability
• proof