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

Phase transition for loop representations of Quantum spin systems on trees