  A Dyson hierarchical model for Anderson localization, containing non-random
hierarchical hoppings and random on-site energies, has been studied in the
mathematical literature since its introduction by Bovier [J. Stat. Phys. 59,
745 (1990)], with the conclusion that this model is always in the localized
phase. Here we show that if one introduces alternating signs in the hoppings
along the hierarchy (instead of choosing all hoppings of the same sign), it is
possible to reach an Anderson localization critical point presenting
multifractal eigenfunctions and intermediate spectral statistics. The advantage
of this model is that one can write exact renormalization equations for some
observables. In particular, we obtain that the renormalized on-site energies
have the Cauchy distributions for exact fixed points. Another output of this
renormalization analysis is that the typical exponent of critical
eigenfunctions is always $\alpha_{typ}=2$, independently of the disorder
strength. We present numerical results concerning the whole multifractal
spectrum $f(\alpha)$ and the compressibility $\chi$ of the level statistics,
both for the box and the Cauchy distributions of the random on-site energies.
We discuss the similarities and differences with the ensemble of ultrametric
random matrices introduced recently by Fyodorov, Ossipov and Rodriguez [J.
Stat. Mech. L12001 (2009)].
