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The wired minimal spanning forest on the Poisson-weighted infinite tree

Asaf Nachmias, Pengfei Tang

Abstract

We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\frac{3}{2}$ and its typical displacement exponent is $\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.

The wired minimal spanning forest on the Poisson-weighted infinite tree

Abstract

We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let be the tree containing the root in the WMSF on the PWIT and be a simple random walk on starting from the root. We show that almost surely has and with high probability. That is, the spectral dimension of is and its typical displacement exponent is , almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.
Paper Structure (27 sections, 29 theorems, 151 equations)

This paper contains 27 sections, 29 theorems, 151 equations.

Table of Contents

  1. Introduction and main results
  2. The WMSF on the PWIT
  3. Main results
  4. Outline of the paper
  5. Preliminaries
  6. The invasion percolation cluster $T$
  7. Poisson Galton--Watson aggregation process and the construction of $M$
  8. Poisson Galton--Watson aggregation process
  9. First moment of the size of $M_v$
  10. Several important functions
  11. Volume growth
  12. Upper bound on the volume growth
  13. Some estimates from Addario2013local_limit_MST_complete_graphs about the invasion percolation cluster $T$
  14. Proof of the upper bound \ref{['eq: improved upper bound on the volume']} from Theorem \ref{['thm: improved bounds on the volume']}
  15. Height of $M_v$
  16. ...and 12 more sections

Key Result

theorem 1

Almost surely $M$ has spectral dimension $3/2$. Moreover, there exist a constant $\beta>0$ and a random variable $C\in[1,\infty)$ that only depends on $M$, such that almost surely the transition density of $M$ satisfies the following: for all $n\geq 2$.

Theorems & Definitions (63)

  • theorem 1
  • theorem 2
  • remark 3
  • theorem 4
  • remark 5
  • definition 6
  • theorem 7
  • definition 8: One-step Poisson Galton--Watson aggregation
  • definition 9: Poisson Galton--Watson aggregation (PGWA) process
  • theorem 10
  • ...and 53 more