Using memory to identify phase transitions on a Cayley Tree

TL;DR

This paper links memory in exponentially expanding spaces to phase transitions in the Ising model on a Cayley tree, using Monte Carlo simulations to identify Bethe-Peierls and spin-glass transitions.

Contribution

It establishes a systematic connection between memory signals and phase transitions in Cayley trees, providing practical methods for detecting these transitions numerically.

Findings

Memory signals indicate Bethe-Peierls transition.

Memory divided by standard deviation signals spin-glass transition.

Monte Carlo simulations confirm two distinct transition temperatures.

Abstract

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.