[Paper Review] Uniform spanning trees on Sierpinski graphs

Masato Shinoda, Elmar Teufl+1 more|arXiv (Cornell University)|May 22, 2013

Stochastic processes and statistical mechanics14 references3 citations

TL;DR

This paper constructs a joint probability space for uniform spanning trees on finite Sierpiński graphs using a projective limit, showing convergence to a multi-type Galton-Watson tree. It establishes almost sure convergence of degree distributions and proves that loop-erased random walk on these graphs converges to a continuous limit process with Hausdorff dimension approximately 1.194, extending to other self-similar graphs with symmetry.

ABSTRACT

We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and show that this construction gives rise to a multi-type Galton-Watson tree. We derive a number of structural results, for instance on the degree distribution. The connection between uniform spanning trees and loop-erased random walk is then exploited to prove convergence of the latter to a continuous stochastic process. Some geometric properties of this limit process, such as the Hausdorff dimension, are investigated as well. The method is also applicable to other self-similar graphs with a sufficient degree of symmetry.

Motivation & Objective

• To construct a consistent joint probability space for uniform spanning trees across all finite Sierpiński graphs using a projective limit construction.
• To analyze the structural properties of uniform spanning trees on Sierpiński graphs, particularly the degree distribution and component sizes.
• To establish the convergence of loop-erased random walk on Sierpiński graphs to a continuous limit process and characterize its geometric properties.
• To extend the method to other self-similar graphs with sufficient symmetry, such as the modified Koch curve and higher-subdivision Sierpiński graphs.
• To investigate the metric structure induced by random spanning trees and the geometry of the interface between tree branches, including its Hausdorff dimension.

Proposed method

• Construct a projective limit of uniform spanning trees on finite Sierpiński graphs to define a limiting random tree on the infinite Sierpiński gasket.
• Use multi-type Galton-Watson processes to model the limiting tree structure, leveraging the recursive and symmetric nature of Sierpiński graphs.
• Apply the connection between uniform spanning trees and loop-erased random walk (via Wilson’s algorithm) to analyze the scaling limit of the walk.
• Prove almost sure convergence of the degree distribution in uniform spanning trees, showing that the proportion of vertices of degree i converges to a constant w(i).
• Use renormalization and scaling arguments to derive the limit distribution of loop-erased random walk lengths and compute its Hausdorff dimension.
• Analyze the metric induced by the random spanning tree, proving almost sure convergence to an R-tree and estimating the Hausdorff dimension of the interface set.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the degree of a typical vertex in a uniform spanning tree on a finite Sierpiński graph as the graph size grows?
• RQ2 How does loop-erased random walk on Sierpiński graphs behave asymptotically, and what is the nature of its scaling limit?
• RQ3 What is the Hausdorff dimension of the limit curve of loop-erased random walk on the Sierpiński gasket?
• RQ4 How does the interface—where different branches of the spanning tree touch—behave geometrically in the limit?
• RQ5 To what extent can the method used for Sierpiński graphs be generalized to other self-similar graphs with symmetry?

Key findings

• The proportion of vertices of degree i (i ∈ {1, 2, 3, 4}) in a uniform spanning tree of the n-th Sierpiński graph converges almost surely to a constant w(i) as n → ∞.
• The limit process of loop-erased random walk on the Sierpiński gasket is almost surely self-avoiding and has a Hausdorff dimension of log₂(4/3 + √205/15) ≈ 1.193995.
• The length of a loop-erased random walk from one corner to another in the n-th Sierpiński graph grows asymptotically like (4/3 + √205/15)^n, and the renormalized length converges in distribution.
• The interface of the spanning tree—the set where different branches touch—has Hausdorff dimension at most log(α̌)/log(2) ≈ 0.457029, where α̌ is a spectral radius related to the graph's resistance scaling.
• The method generalizes to other self-similar graphs with two or three boundary vertices, such as the modified Koch curve, yielding a limit process with Hausdorff dimension log(10/3)/log(3) ≈ 1.0959.
• For graphs with four or more boundary vertices, the method remains applicable asymptotically, though more complex forest types and non-exact counting formulas arise, leading to richer geometric phenomena like multiple traversals of subgraphs by loop-erased walks.