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

TL;DR

This paper investigates the spectral and diffusive properties of the wired minimal spanning forest on the Poisson-weighted infinite tree, confirming theoretical predictions about its spectral dimension and displacement behavior.

Contribution

It provides rigorous analysis of the spectral dimension and displacement exponents of the WMSF on the PWIT, confirming prior conjectures.

Findings

Spectral dimension of the WMSF is 3/2.

Typical displacement exponent is 1/4.

Results confirm Addario-Berry's predictions.

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.