multiplication but at the same time preserving the chiral symmetry. The role of the lattice cut-off is just to make the functional integrals appearing below well defined and it will removed first.

Supersymmetric formalism
------------------------

It is well known, see for instance [@Mu1], that the average of the two-point function and the average of the product of two functions can be represented in terms of a supersymmetric functional integral in the chiral basis. One introduces a finite set of Grassmann variables $\psi^+_{\o,\xx},\psi^-_{\o,\xx}$ with $\o=\pm$ and defines, if $\e=\pm$, the Grassmann integration by d\^\_[ø,]{} \^\_[ø,]{}=1d\^\_[ø,]{}=0 where $d\psi^\e_{\o,\xx}$ is another set of Grassmann variables. Therefore if $
\DD\psi=\prod_{\xx,\o=\pm}d\psi^+_{\o,\xx}d\psi^-_{\o,\xx}$ we can write |\_[ø,-ø’]{}=[e\^[-(\^+, A \^-) ]{} \^-\_[,ø]{}\^+\_[,ø’]{}e\^[-(\^+, A \^-) ]{}]{} \[aaxx\]where $A=\s_1( i H- E)$. Note that the denominator of is equal to ${\rm Det } A$, and that $\int \DD\phi  e^{-(\phi^+, A \phi^-)}={1\over {\rm Det }A}$, for any $n\times n$ invertible matrix $A$ with $Re A>0$ and $\phi^+,\phi^-$ complex numbers with $(\phi^+)^*=\phi^-$. Therefore one obtains the following representation of the average of the two point function G\_[ł,E,N;ø,ø’]{}() =\[ e\^[-(\^+, A \^-)-(\^+, A \^-)]{} \]\[88\]and integrating over the disorder, calling $\Psi^+=(\psi^+_+,\psi^+_-,\phi^+_+,\phi^+_-)$ and $\Psi^-=(\psi^-_+,\psi^-_-,\phi^-_+,\phi^-_-)$, $\phi^+=(\phi^-)^*$ G\_[ł,E,N;ø,ø’]{}()=P(d) P(d) e\^ \^-\_[ø,]{}\^+\_[ø’,0]{}\[66\] where, if $\l=2g^2$ =- łddv(-)\_[,’=,]{}\_[ø=]{}\^+\_[,ø,]{}\^-\_[,ø,]{} 