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.