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Random choice spanning trees

Eleanor Archer, Matan Shalev

Abstract

In this paper we introduce a new model of random spanning trees that we call choice spanning trees, constructed from so-called choice random walks. These are random walks for which each step is chosen from a subset of random options, according to some pre-defined rule. The choice spanning trees are constructed by running a choice modified version of Wilson's algorithm or the Aldous-Broder algorithm on the complete graph. We show that the scaling limits of these choice spanning trees are slight variants of random aggregation trees previously considered by Curien and Haas (2017). Moreover, we show that the loop-erasure of a choice random walk run on the complete graph converges after rescaling to a generalized Rayleigh process, extending a result of Evans, Pitman and Winter (2006). These are all natural extensions of similar results for uniform spanning trees.

Random choice spanning trees

Abstract

In this paper we introduce a new model of random spanning trees that we call choice spanning trees, constructed from so-called choice random walks. These are random walks for which each step is chosen from a subset of random options, according to some pre-defined rule. The choice spanning trees are constructed by running a choice modified version of Wilson's algorithm or the Aldous-Broder algorithm on the complete graph. We show that the scaling limits of these choice spanning trees are slight variants of random aggregation trees previously considered by Curien and Haas (2017). Moreover, we show that the loop-erasure of a choice random walk run on the complete graph converges after rescaling to a generalized Rayleigh process, extending a result of Evans, Pitman and Winter (2006). These are all natural extensions of similar results for uniform spanning trees.
Paper Structure (33 sections, 26 theorems, 126 equations, 4 figures)

This paper contains 33 sections, 26 theorems, 126 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and constructions
  3. Gromov-Hausdorff-Prokhorov topology
  4. Skorokhod-$J_1$ topology
  5. Constructions of uniform spanning trees
  6. Loop-erased random walk
  7. Rayleigh process
  8. Wilson's algorithm
  9. Laplacian random walk
  10. Aldous--Broder algorithm
  11. Constructions of choice spanning trees
  12. Choice random walk
  13. Generalized Rayleigh process
  14. Wilson's algorithm with choice
  15. Aldous--Broder algorithm with choice
  16. ...and 18 more sections

Key Result

Theorem 1.1

For all $k \geq 1$, with respect to the Gromov-Hausdorff-Prokhorov topology as $n \to \infty$. In addition, the Hausdorff dimension of $\mathcal{T}_{k, k-1}$ and $\mathcal{T}_{k, 0}$ is equal to $\frac{k+1}{k}$, almost surely.

Figures (4)

  • Figure 1: A $3$-choice spanning tree
  • Figure 2: Rayleigh process
  • Figure 3: $2$-Rayleigh process
  • Figure 5: Some choice trees

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Remark 2.7
  • ...and 38 more