The Random Loop Model on Trees

Beschreibung

We study the random loop model introduced by Ueltschi as a generalization of probabilistic representations for certain quantum spin systems. Here, loops are subsets of space and time, where space is modeled as a simple graph, and they give rise to the percolation-type question whether there are loops visiting infinitely many vertices. The random loop model builds upon several parameters that influence the answer to this question. In fact, in many cases it is conjectured that there is a phase transition. This means that there is a critical value for one parameter β beyond which loops are infinite with positive probability, while loops are finite almost surely below this critical parameter. One difficulty to establish the existence of such a phase transition is the inherent lack of monotonicity for the model on graphs like the d-dimensional cubic lattice. In this thesis we consider the case that the underlying graph is an infinite tree to circumvent this problem. Here, we relate event that a particular loop is infinite to the survival of a stochastic process that is manageable more easily. This allows us to distinguish both phases and to establish a phase transition by evaluating some monotone function of β, yielding the critical parameter as the solution to an implicit equation. Furthermore, from a more careful analysis of this function we obtain an asymptotic expansion for the critical parameter. A second challenge is that the relevant probability measure depends on the number of loops. On the one hand, this induces highly non-local correlations. On the other hand, we have to start our analysis with finite graphs and then take an infinite-volume limit, where the uniqueness of the limit for the corresponding probability measures is in general not secured. Hence and again for certain trees, we are going to show that such a unique limit exists, enabling our aforementioned investigation of the phase transition and its critical parameter in the first place.

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Johannes Ehlert

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