, Volume 300, Issue 2, pp 435-486,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 31 Aug 2010

Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model


We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.

Communicated by M. Salmhofer