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Scaling limit of the Aldous-Broder chain on regular graphs: the transient regime

Osvaldo Angtuncio Hernández, Gabriel Berzunza Ojeda, Anita Winter

TL;DR

The paper proves that the rescaled Aldous–Broder chain on high-dimensional regular graphs converges, in the Gromov–Hausdorff sense, to the Root Growth with Re-grafting (RGRG) process, when started from the trivial rooted tree. The authors build a skeleton approximation that isolates non-ghost indices and long-loop behavior, and they couple this to a Poisson-driven RGRG, yielding weak convergence of the AB dynamics to RGRG in the Skorokhod topology on rooted metric trees. Key technical contributions include decomposition of random-walk segments into nearly independent pieces, detailed bounds on intersection and non-erased events, and a careful control of GH-distances via the skeleton and $c$-RGRG maps. The results extend CRT-type scaling limits to the AB chain on transient regimes, including regular graphs like high-dimensional tori, with explicit scaling parameters for time and edge lengths that ensure convergence to the RGRG limit. This provides a rigorous bridge between discrete spanning-tree dynamics and continuum random-tree-like limits in networked settings.

Abstract

The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder chain on a graph $G=(V,E)$ is a discrete-time stochastic process with values in the space of rooted trees whose vertex set is a subset of $V$ which is stationary under the uniform distribution on the space of rooted trees spanning $G$. In Evans, Pitman and Winter (2006) the so-called root growth with re-grafting process (RGRG) was constructed. Further it was shown that the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the continuum random tree is, with respect to the Gromov-weak topology, the scaling limit of the uniform spanning tree on $\mathbb{Z}_N^d$, $d\ge 5$. This result was recently strengthens in Archer, Nachmias and Shalev (2024) to convergence with respect to the Gromov-Hausdorff-weak topology, and therefore also with respect to the Gromov-Hausdorff topology. In the present paper we show that also the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology when initially started in the trivial rooted tree. We give conditions on the increasing graph sequence under which the result extends to regular graphs and give probabilistic expressions scales at which time has to be speed up and edge lengths have to be scaled down.

Scaling limit of the Aldous-Broder chain on regular graphs: the transient regime

TL;DR

The paper proves that the rescaled Aldous–Broder chain on high-dimensional regular graphs converges, in the Gromov–Hausdorff sense, to the Root Growth with Re-grafting (RGRG) process, when started from the trivial rooted tree. The authors build a skeleton approximation that isolates non-ghost indices and long-loop behavior, and they couple this to a Poisson-driven RGRG, yielding weak convergence of the AB dynamics to RGRG in the Skorokhod topology on rooted metric trees. Key technical contributions include decomposition of random-walk segments into nearly independent pieces, detailed bounds on intersection and non-erased events, and a careful control of GH-distances via the skeleton and -RGRG maps. The results extend CRT-type scaling limits to the AB chain on transient regimes, including regular graphs like high-dimensional tori, with explicit scaling parameters for time and edge lengths that ensure convergence to the RGRG limit. This provides a rigorous bridge between discrete spanning-tree dynamics and continuum random-tree-like limits in networked settings.

Abstract

The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with vertices. The Aldous-Broder chain on a graph is a discrete-time stochastic process with values in the space of rooted trees whose vertex set is a subset of which is stationary under the uniform distribution on the space of rooted trees spanning . In Evans, Pitman and Winter (2006) the so-called root growth with re-grafting process (RGRG) was constructed. Further it was shown that the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the continuum random tree is, with respect to the Gromov-weak topology, the scaling limit of the uniform spanning tree on , . This result was recently strengthens in Archer, Nachmias and Shalev (2024) to convergence with respect to the Gromov-Hausdorff-weak topology, and therefore also with respect to the Gromov-Hausdorff topology. In the present paper we show that also the suitable rescaled Aldous-Broder chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology when initially started in the trivial rooted tree. We give conditions on the increasing graph sequence under which the result extends to regular graphs and give probabilistic expressions scales at which time has to be speed up and edge lengths have to be scaled down.

Paper Structure

This paper contains 17 sections, 45 theorems, 382 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Estimates on RWs and locally erased RWs on regular graphs
  3. Decomposition into nearly independent random walk path segments
  4. Estimates on intersections probabilities of random walks
  5. Locally non-erased time indices on regular graphs
  6. Asymptotic estimates
  7. The RGRG driven by a Poisson point process
  8. Behavior of the RGRG map between re-grafting events
  9. The Aldous--Broder chain and the skeleton chain
  10. The sets of ghost indices
  11. The skeleton chain
  12. Distance between the Aldous-Broder chain and the skeleton chain
  13. Behavior of the skeleton chain between long loops
  14. Decomposable paths
  15. Bounding the GH-distance between the Aldous-Broder chain and the $c$-RGRG map
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\{Y^{\mathbb{Z}_N^d};\,N\in\mathbb{N}\}$ be a family of Aldous-Broder chains with values in rooted metric trees embedded in $\mathbb{Z}^d_N$, $d\ge 5$ which start in the rooted tree $Y^{\mathbb{Z}_N^d}(0)=(\{0\}, 0)$. Then with the constants $\beta(d)$ and $\alpha(d)$ from (e:059b) and (e:063), where $\Rightarrow$ stands for weak convergence in Skorokhod space on $\mathbb{T}_{\hbox{\tiny metr

Figures (4)

  • Figure 1: Illustrates an AB-move in the Aldous-Broder algorithm on a lattice graph in discrete time. The red cross indicates the current root. One can see that the new root was already contained in the tree, and thus it gets connected with the old root. At the same time, we erase the edge connecting the new root with the vertex visited one step after the last visit of the new root. Note that here we jump from a spanning tree of intrinsic height $13$ to one of intrinsic height $9$.
  • Figure 2: A path segment $\gamma=(\gamma(0),\gamma(1),...,\gamma(n))$, $n=1,...,8,9,...,14$ together with its loop erasure. Here ${\rm NE}^\gamma(1)=[0,1]$, ${\rm NE}^\gamma(8)=[0,8]$, ${\rm NE}^\gamma(9)=\{0,9\}$ and ${\rm NE}^\gamma(14)=\{0\}\cup[9,14]$.
  • Figure 3: We illustrate why we impose \ref{['eqnGhostIndexDefinitionNoLongLoops']} to later ensure that the subgraph restricted to the non-ghost indices is connected. A loop shorter than $s$ is closing at time index $n+1$. There exists $k\in[m_1,n]\setminus{\mathcal{G}}^{\gamma,s}(n)$ and $k_0\in[0,k-1]\setminus{\mathcal{G}}^{\gamma,s}(n)$ with $\gamma(k_0)=\gamma(k)$. If $k-k_0>s$, a loop longer than $s$ is closed at time index $k$. If we would erase the recently formed loop $[m_1,...,n]$, the graph would disconnect. We therefore need a condition that prevents that indices in $\{m_1,...,n\}$ are declared to be in ${\mathcal{G}}^{\gamma,s}(n+1)$.
  • Figure 4: Illustrates the dynamics of the skeleton chain. In the left-hand side, the skeleton up to a certain time is shown. In the first row, the last steps of the random walk are depicted in red. From left to right, there is root-growing, the Aldous-Broder algorithm is applied, or a small cycle is erased. In the bottom row the resulting skeleton is shown for each of the corresponding possibilities.

Theorems & Definitions (105)

  • Theorem 1: Convergence of the Aldous-Broder chain on $\mathbb{Z}^d_N$, $d\ge 5$
  • Theorem 2: Scaling the Aldous-Broder chain on regular graphs: transient regime
  • Lemma 2.1: Nearly independent after mixing
  • Remark 2.2
  • proof
  • Lemma 2.3: Distance to independence
  • proof
  • Proposition 2.4: Range self-intersection probabilities
  • proof
  • Corollary 2.5: Range self-intersection probabilities starting at zero
  • ...and 95 more