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

A Mathematically Controlled Alternative to the Supersymmetric Sigma Model of Disorder