  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.
