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Partial sampling of a random spanning tree

Yves Le Jan

TL;DR

This work analyzes how a fixed subset of vertices in a random spanning tree of the complete graph $K_n$ is connected in the large-$n$ limit. Using Wilson's loop-erased paths and a Green-function framework, it defines $L$-trees, their reductions, and a canonical $Q_L$ and $u_L$ encoding; it proves that asymptotically the subtree reduces to a binary shape with a uniform distribution over $\mathcal{B}_l$ and that the joint distribution of its geometry converges to a universal law, with leaf-to-root distances, when scaled by $\sqrt{n}$, following an explicit density. The limiting family is consistent in $l$ and connects to Aldous' Continuum Random Tree, providing a discrete-to-continuum bridge and a potential basis for CRT-inspired limit descriptions of subtrees in random spanning trees. The findings hold independently of the root-weight parameter $\kappa$ and illuminate the structure of large random trees on complete graphs with implications for related random-graph models.

Abstract

We investigate the distributions of subtrees connecting several vertices in the spanning trees of the complete graphs and their asymptotics.

Partial sampling of a random spanning tree

TL;DR

This work analyzes how a fixed subset of vertices in a random spanning tree of the complete graph is connected in the large- limit. Using Wilson's loop-erased paths and a Green-function framework, it defines -trees, their reductions, and a canonical and encoding; it proves that asymptotically the subtree reduces to a binary shape with a uniform distribution over and that the joint distribution of its geometry converges to a universal law, with leaf-to-root distances, when scaled by , following an explicit density. The limiting family is consistent in and connects to Aldous' Continuum Random Tree, providing a discrete-to-continuum bridge and a potential basis for CRT-inspired limit descriptions of subtrees in random spanning trees. The findings hold independently of the root-weight parameter and illuminate the structure of large random trees on complete graphs with implications for related random-graph models.

Abstract

We investigate the distributions of subtrees connecting several vertices in the spanning trees of the complete graphs and their asymptotics.
Paper Structure (4 sections, 2 theorems, 8 equations)

This paper contains 4 sections, 2 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Definitions, first properties and statement
  3. Proof
  4. Additional comments

Key Result

Lemma 2.1

$\mathbb{P}^{(n),\kappa}(Y=\Upsilon_L)=\frac{\kappa^{r-1}(a+\kappa)}{(n+\kappa)^d}$

Theorems & Definitions (2)

  • Lemma 2.1
  • Theorem 2.1