Johannes Ehlert

Phase transition for random loops on trees

We consider the model of random loops on graphs introduced by Tóth, Aizenman, Nachtergaele and Ueltschi that generalizes the Random Interchange Model and which has connections to the Quantum Heisenberg Ferro- and Antiferro-magnet as well as to the Quantum XY-Model.
An interesting question is, whether or not there is an infinite loop with positive probability for given model parameters. Since this is still wide open for general graphs and in particular for Zd, we restrict ourselves to the d-ary tree. In this setting, certain edges exhibit a renewal property that allows to construct a Galton-Watson process whose survival is coupled to the existence of an infinite loop. This approach enables us to show the existence of a (locally) sharp transition between the phases of finite and infinite loops, respectively. Additionally, it provides sufficient and precise criteria for both phases and all d ≥ 3. Finally, in the asymptotic regime d → ∞, we may compute the critical parameter of the phase transition up to 6th order in d−1.