Phase transition for loop representations of Quantum spin systems on trees

TL;DR

This paper investigates phase transitions in loop models on Galton-Watson trees, showing how certain offspring distributions lead to a shift from finite to infinite loops, which relate to quantum spin systems.

Contribution

It establishes conditions on offspring distributions that guarantee phase transitions in loop configurations on trees, connecting probabilistic models to quantum spin systems.

Findings

Phase transition from finite to infinite loops identified

Conditions on offspring distribution for phase transition derived

Models linked to quantum spin systems for various parameters

Abstract

We consider a model of random loops on Galton-Watson trees with an offspring distribution with high expectation. We give the configurations a weighting of $\theta^{\#\text{loops}}$. For many $\theta>1$ these models are equivalent to certain quantum spin systems for various choices of the system parameters. We find conditions on the offspring distribution that guarantee the occurrence of a phase transition from finite to infinite loops for the Galton-Watson tree.