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Random spanning trees in random environment

Luca Makowiec, Michele Salvi, Rongfeng Sun

Abstract

We introduce a new spanning tree model called the random spanning tree in random environment (RSTRE), which interpolates between the uniform spanning tree and the minimum spanning tree as the inverse temperature (disorder strength) $β$ varies. On the complete graph with $n$ vertices and i.i.d.\ uniform disorder variables on the edges, we identify: (1) a low disorder regime with $β\leq C n/\log n$, where the diameter of the random spanning tree is typically of order $n^{1/2}$, the same as for the uniform spanning tree; (2) a high disorder regime with $β\geq n^{4/3} \log n$, where the diameter is typically of order $n^{1/3}$, the same as for the minimum spanning tree. We conjecture that for $β=n^α$ with $α\in (1, 4/3)$, the diameter is of order $n^{γ+o(1)}$ for some $γ=γ(α)$ strictly between $1/2$ and $1/3$.

Random spanning trees in random environment

Abstract

We introduce a new spanning tree model called the random spanning tree in random environment (RSTRE), which interpolates between the uniform spanning tree and the minimum spanning tree as the inverse temperature (disorder strength) varies. On the complete graph with vertices and i.i.d.\ uniform disorder variables on the edges, we identify: (1) a low disorder regime with , where the diameter of the random spanning tree is typically of order , the same as for the uniform spanning tree; (2) a high disorder regime with , where the diameter is typically of order , the same as for the minimum spanning tree. We conjecture that for with , the diameter is of order for some strictly between and .

Paper Structure

This paper contains 25 sections, 27 theorems, 153 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Related literature
  4. Other results, conjectures and open questions
  5. Proof strategy and structure of the paper
  6. Preliminaries
  7. Electrical networks, random walks and UST
  8. Wilson's algorithm and spatial Markov property
  9. Bottleneck ratio and mixing time
  10. Sufficient conditions for bounding the diameter
  11. Erdős-Rényi random graphs
  12. Low disorder regime
  13. Concentration of stationary distribution and bottleneck ratio
  14. Proof of Theorem \ref{['T:GeneralLow']}
  15. Lack of concentration outside low disorder regime
  16. ...and 10 more sections

Key Result

Theorem 1.1

For $\beta_n\geq 0$, let ${\mathcal{T}} ^\omega_{n, \beta_n}$ be the RSTRE on the complete graph $K_n=(V_n, E_n)$ with $n$ vertices and i.i.d. disorder variables $(\omega_e)_{e\in E_n}$ uniformly distributed on $[0,1]$. There exists a constant $C>0$ such that for any $\delta > 0$, if $\beta_n \leq C where $C_1(\delta)>0$ is a constant depending only on $\delta$. On the other hand, if $\beta_n \geq

Figures (2)

  • Figure 1: Typical vertices $x$ and $y$ are likely to be connected in ${\mathcal{T}}$ via a short path from $x$ to ${\mathcal{T}} _{{\mathcal{C}} _{1}(p_m)}$, a path inside ${\mathcal{T}} _{{\mathcal{C}} _{1}(p_m)}$ and another short path from ${\mathcal{T}} _{{\mathcal{C}} _{1}(p_m)}$ to $y$.
  • Figure 2: The path $\gamma_{\mathcal{T}} (u,v)$ can be decomposed into a path $\Gamma_u$, an edge $e$ and another path $\Gamma_v$. After the contraction of $\Gamma_u$ and $\Gamma_v$, the edge $e$ becomes an edge between $u$ and $v$ in $(G', \textup{w})$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Proposition 1.2
  • Conjecture 1.3
  • Theorem 2.1: Kirchhoff's Formula
  • Lemma 2.2: Series Law
  • Theorem 2.3: Rayleigh’s Monotonicity Principle
  • Theorem 2.4
  • Lemma 2.5: Spatial Markov Property
  • Definition 2.6: Bottleneck Ratio
  • Theorem 2.7
  • ...and 38 more