Schwinger-Dyson equation is given by \_[T,0,N]{}=g\_ø()+łg\_ø() dv() \_[’=,]{}\_[’=a,b]{}&lt;\_[’,’,-ø]{}()\^+\_[,ø,+]{}\^-\_[,ø,]{}&gt;\_[T,0,N]{} and using the WI &&D\_ø()&lt;\_[,,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}= \_[,]{}\_[,’]{}\_[ø,ø’]{}\[&lt;\^-\_[’,ø’,]{}\^+\_[’,ø’,]{}&gt;\_[T,0,N]{}\
&& -&lt;\^-\_[’,ø’,+]{}\^+\_[’,ø’,+]{}&gt;\] +&lt;\_[,,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}\[pp3\] where $\r_{\a,\b,\o,\xx}=\Psi^+_{\a,\b,\o,\xx}\Psi^-_{\a,\b,\o,\xx}$ and again && &lt;\_[,,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}=\
&& \_ł[14]{} D\_[-ø]{}()\_[”=,”=a,b]{} &lt;\_[”,”,-ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}+R\^[(2)]{}\_[N,,]{}(,)\[aaa1\]we get \_[T,0,N]{}=g\_ø()+łg\_ø() dv() \_[’=,,’=a,b]{} [R\^[(2)]{}\_[N,’,’]{}(,)D\_ø()]{}\[ma2\] with $\int d\pp \hat v(\pp) {R^{(2)}_{N,\a',\b'}\over D_\o}$ is $O(2^{-N})$. In a similar way one analyze the connected part of ; we write the Schwinger-Dyson equation for the four point function \_[T,0,N]{}= dv() \_[’=,’=a,b]{} &lt;\_[’,’,-ø,]{} \^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}&gt;\_[T,0,N]{}The WI for the four point function is &&D\_ø() &lt;\_[’,’,-ø,]{} \^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}&gt;\_[T,0,N]{} +\
&&&lt;\_[’,’,-ø,]{} \^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}&gt;\_[T,0,N]{} =0 with && &lt;\_[’,’,-ø,]{}\^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}&gt;\_[T,0,N]{}=\
&& \_[’]{}ł[14]{} D\_[ø]{}()\_[”,”]{}&lt;\_[”,”,ø,]{}\^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}&gt;\_[T,0,N]{}+R\^[(4)]{}\_[N,’,’]{}(\_1,\_2,\_3,\_4,)\[aaa\]and using that $\sum_{\a'=\phi,\psi} \e_{\a'}=0$ we finally obtain \_[T,0,N]{}= \_[’,’]{}dv() [R\^[(4)]{}\_[N,’,’]{}D\_[ø]{}()]{}\[ma3\] and again the r.h.s. is vanishing as $O(\l 2^{-N}))$. Therefore, by ,, the interacting average of the product $K_{\l,0,N}(\xx)$ differs from its non interacting value $K_{0,0,N}(\xx)$ by terms which are order $O(\l 2^{-N})$ for large $N$; exact universality for such quantity (and therefore for the conductivity) is achieved only in the limit of removed ultraviolet cut-off.

The non critical theory and the infinite volume limit
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We have to discuss finally the removal of the infrared cut-off and the case $E\not =0$. Again we perform a multiscale decomposition of $\Psi$ in the $E\not=0$ case and the integration of the ultraviolet (positive) scales is 