The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

Abstract

. We study the non-linear supersymmetric hyperbolic sigma model H 2 | 2 on a complete graph with hierarchical interactions. For interactions which do not decrease too fast in the hierarchical distance, we prove tightness of certain spin variables in horospherical coordinates, uniformly in the pinning and in the size of the graph. The proof relies on a reduction to an effective H 2 | 2 -model; its size is logarithmic in the size of the original model. The model deals with spin variables taking values in the H with two Grassmann components over the hyperboloid , 0 . The model has supersymmetries, which extend the generators of the Lorentz group acting on H 2 . In Disertori et al. (2010), Disertori, Spencer, and Zirnbauer examine this model over boxes in Z d , d ≥ 3 . For sufficiently small temperature, they derive moment estimates and conclude that the spins are aligned with high probability. For high temperature in any dimension d , Disertori and Spencer (2010) show exponential decay