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

TL;DR

This paper investigates a non-linear supersymmetric hyperbolic sigma model on complete graphs with hierarchical interactions, demonstrating tightness of spin variables and reducing the model to an effective smaller version.

Contribution

It provides a novel analysis of the $H^{2|2}$ model on hierarchical graphs, establishing tightness results and a reduction technique for large systems.

Findings

Proves tightness of spin variables in the model.

Develops a reduction to an effective $H^{2|2}$ model of logarithmic size.

Shows the model's behavior under certain 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.