  We provide a concrete and systematic connection between the statistical
physics of the Ising ferromagnet on a Cayley tree, and the study of memory in
exponentially expanding spaces. Memory turns out to be a clear signal of the
`Bethe-Peierls' phase transition, and the average of memory divided by its
standard deviation provides a clear signal of the `spin-glass' transition
temperature. Numerical Monte Carlo simulations are used to make transparent the
existence of the two different transition temperatures. The quantities used to
spot the phase transitions with Monte Carlo could be useful when studying other
systems where analytical methods don't work.
