Structure of spanning trees on the two-dimensional Sier- pinski gasket

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∈{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 ϕj 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 ϕ1=10957/40464, ϕ2=6626035/13636368, ϕ3=2943139/13636368, ϕ4=124895/4545456.