A Mathematically Controlled Alternative to the Supersymmetric Sigma Model of Disorder

This paper shows how to obtain non-rigorous mathematical control over models of loosely coupled disordered grains; it provides new information about saddle point structure and perturbative corrections. Both the Wegner model and a variant due to Disertori are transformed to matrix models which are similar to the supersymmetric model of disorder, having two matrices $Q^f$ and $Q^b$ which correspond to the two bosonic sectors of the SUSY matrix. However the Grassman (fermionic) sector of the SUSY matrix is omitted, and compensated by a spectral determinant. The transformation is exact for Disertori's model, while for the Wegner model it involves an integral which can be approximated while maintaining mathematical control. Previous derivations of sigma models of disorder assumed a spatially uniform saddle point independent of the Goldstone bosons and found that corrections are well controlled in the large $N$ limit. This paper takes into account spatial fluctuations of the Goldstone bosons and finds that corrections to the sigma model approximation in extended systems are controlled by powers of the inverse conductance $1/g$. In the weak localization regime Disertori's model exhibits remarkable simplifications and is completely controlled by perturbative expansions in various small parameters. The Wegner model might also be controlled. The standard weak localization results of the supersymmetric sigma model, including anomalously localized states, are reproduced and extended.