Phase transitions in long-range Ising models and an optimal condition for factors of $g$-measures

TL;DR

This paper explores phase transitions in long-range Ising models and establishes an optimal condition for the existence of $g$-measures in factor maps, demonstrating the sharpness of the Berbee's condition.

Contribution

It weakens the known assumptions for $g$-measures in symbolic dynamics and provides a counterexample showing the optimality of Berbee's condition, along with results on inverse critical temperatures.

Findings

Berbee's condition is optimal for $g$-measures in this context.

Existence of an inverse critical temperature at most 8 times the standard critical temperature.

Counterexample demonstrating the sharpness of the condition.

Abstract

We weaken the assumption of summable variations in a paper by Verbitskiy \cite{verb} to a weaker condition, Berbee's condition, in order for a 1-block factor (a single site renormalisation) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample, that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is an inverse critical temperature in a one-sided long-range Ising model which is at most 8 times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.