A08 – Nonlinear sigma models
Nonlinear sigma models (NLSM) can be seen as extensions of the classical Ising model to the case of continuous symmetry or, more generally, as generalizations of Gaussian measures where the integration domain is restricted to some nonlinear manifold. Their properties depend then heavily on the geometrical features of the corresponding symmetry group. They appear in classical and quantum statistical mechanics, as well as in quantum field theory. They are key ingredients in the analysis of phase transitions and critical phenomena, since they arise as effective description for a large variety of models. In this project we plan to investigate features and relations between certain types of NLSMs appearing in (a) random band matrices and random Schrödinger operators related to quantum diffusion for disordered materials and (b) history-dependent stochastic processes.
To study these problems we need a mixture of tools from analysis (convexity bounds, complex saddle point methods), probability, geometry (internal supersymmetries and Ward identites), and mathematical physics (rigorous renormalization group techniques, cluster expansions).