Structure of spanning trees on the two-dimensional Sierpinski gasket

TL;DR

This paper rigorously derives the degree distribution probabilities for vertices in spanning trees on the two-dimensional Sierpinski gasket, providing exact rational values for the limiting distribution.

Contribution

It introduces a rigorous method to compute the degree distribution of vertices in spanning trees on the Sierpinski gasket, including exact limiting probabilities.

Findings

Exact rational probabilities for vertex degrees in spanning trees.

Limiting distribution of degrees as the gasket size tends to infinity.

Provides a detailed mathematical framework for analyzing spanning trees on fractals.

Abstract

Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage $n$ is a non-negative integer. For any given vertex $x$ of SG(n), we derive rigorously the probability distribution of the degree $j \in \{1,2,3,4\}$ at the vertex and its value in the infinite $n$ limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree $j$. The corresponding limiting distribution $\phi_j$ gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as $\phi_1=10957/40464$, $\phi_2=6626035/13636368$, $\phi_3=2943139/13636368$, $\phi_4=124895/4545456$.