Phase transition for loop representations of Quantum spin systems on trees

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.