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

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 $(Y_n{)}_{n\ge 0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3\mathrm{/}4+o(1)}$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(Y_0,Y_n)=n^{1\mathrm{/}4+o(1)}$ 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.