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

We study the non-linear supersymmetric hyperbolic sigma model H2|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 H2|2 model; its size is logarithmic in the size of the original model. 4 5

[1]  Supersymmetric Hyperbolic $$\sigma $$-Models and Bounds on Correlations in Two Dimensions , 2021, Journal of Statistical Physics.

[2]  T. Spencer,et al.  Anderson Localization for a Supersymmetric Sigma Model , 2009, 0910.3325.

[3]  M. Zirnbauer.  Fourier analysis on a hyperbolic supermanifold with constant curvature , 1991 .

[4]  S. Rolles,et al.  A supersymmetric approach to martingales related to the vertex-reinforced jump process , 2015, 1511.07157.

[5]  Tyler Helmuth,et al.  Spin systems with hyperbolic symmetry: a survey , 2021, 2109.02566.

[6]  Luca Fresta,et al.  Supersymmetric Cluster Expansions and Applications to Random Schrödinger Operators , 2020, 2004.00145.

[7]  P. Tarres,et al.  Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model , 2011, 1111.3991.

[9]  T. Spencer,et al.  Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model , 2009, 0901.1652.

[10]  Tyler Helmuth,et al.  The geometry of random walk isomorphism theorems , 2021, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[11]  P. Tarres,et al.  The Vertex Reinforced Jump Process and a Random Schr\"odinger operator on finite graphs , 2015, 1507.04660.

[12]  Polynomial localization of the 2D-Vertex Reinforced Jump Process , 2019, 1907.07949.

[13]  C. Sabot,et al.  A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs , 2015, Journal of the American Mathematical Society.

[14]  G. Kozma,et al.  Power-law decay of weights and recurrence of the two-dimensional VRJP , 2019, Electronic Journal of Probability.

[15]  Andrew Swan.  Superprobability on Graphs , 2021 .