Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs

TL;DR

This paper investigates phase transition behaviors of graphical representations of the Ising model on tree-like graphs, revealing non-uniqueness in certain models and conditions for uniqueness on regular trees.

Contribution

It constructs specific graphs demonstrating non-uniqueness of phase transitions and establishes conditions for phase transition uniqueness on regular trees.

Findings

Non-uniqueness of phase transitions for loop O(1) and random current models on certain graphs.

Existence of infinite graphs where percolation properties differ between subgraphs.

Uniqueness and coincidence of phase transitions on wired regular trees.

Abstract

We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\mathrm{O}(1)$ model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide.