Partition functions of the Ising model on some self-similar Schreier graphs

TL;DR

This paper investigates the partition functions and thermodynamic limits of the Ising model on three self-similar graph families derived from automorphism groups of rooted trees, linking group theory with statistical physics.

Contribution

It introduces a novel analysis of the Ising model on self-similar graphs associated with specific automorphism groups, expanding understanding of phase transitions on fractal-like structures.

Findings

Partition functions computed for each graph family.

Thermodynamic limits established for the models.

Connections made between group properties and statistical mechanics.

Abstract

We study partition functions and thermodynamic limits for the Ising model on three families of finite graphs converging to infinite self-similar graphs. They are provided by three well-known groups realized as automorphism groups of regular rooted trees: the first Grigorchuk's group of intermediate growth; the iterated monodromy group of the complex polynomial $z^2-1$ known as the Basilica group; and the Hanoi Towers group $H^{(3)}$ closely related to the Sierpinsky gasket.