  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.
