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 M be the tree containing the root in the WMSF on the PWIT and ${({\mathit{Y}}_{\mathit{n}})}_{\mathit{n}\ge 0}$ be a simple random walk on M starting from the root. We show that almost surely M has $\mathbb{P}[{\mathit{Y}}_{2\mathit{n}}={\mathit{Y}}_0]={\mathit{n}}^{-3/4\mathbf{+}\mathit{o}(1)}$ and $dist({\mathit{Y}}_0,{\mathit{Y}}_{\mathit{n}})={\mathit{n}}^{1/4\mathbf{+}\mathit{o}(1)}$ with high probability. That is, the spectral dimension of M is $3/2$ and its typical displacement exponent is $1/4$, almost surely. These confirm Addario–Berry’s predictions (Addario-Berry (2013)).