\^+\_[’,-ø,]{}\^-\_[’,-ø,]{}\[67\] with $\int d\xx=a^2\sum_\xx$ and $P(d\psi)$ and $P(d\phi)$ are the fermionic and bosonic integration with propagator $\d_{\a,\a'} g(\xx)$ with g() =[1L\^2]{}\_e\^[i]{}()

[1a]{} \[ia k\_1-a k\_2\]& -E\
-E & [1a]{}\[ia k\_1+a k\_2\]

\^[-1]{}\[pro\] and we have used that the normalization of the bosonic and fermionic integration are one inverse to the other. The fermionic sector of the above functional integral coincides with a massless Thirring model with a [*non local*]{} current-current interaction.

In the same way we can rewrite the averaged product of two functions as &&K\_[ł,E,N]{}()= {\_a e\^[-(\_a\^+,A \^-\_a)-(\_a\^+,A \^-\_a)- ]{} \^-\_[a,ø,]{}\^+\_[a,ø,0]{}\
&& \_b e\^[-(\_b\^+, B\_b\^-) -(\_b\^+, B\^-\_b)]{} \^-\_[b,ø,]{}\^+\_[b,ø,0]{}}\[89\] with $A=i H- E$, $B=i H+ E$, and averaging over the disorder K\_[ł,E,N]{}()=P(d\_a) P(d\_b)\^ \^-\_[a,ø,]{}\^+\_[a,ø,0]{} \^-\_[b,ø,]{}\^+\_[b,ø,0]{}\]\[24\] where = -łddv(-)\_[,=,,’=a,b]{}\_[ø=]{}\^+\_[,,ø,]{}\_[,,ø,]{}\^+\_[’,’,-ø,]{}\_[’,’,-ø,]{}

The critical theory
===================

The averaged two point function
-------------------------------

We define the [*generating function*]{} as e\^[\_N(J,)]{}= P(d) e\^[()+d]{}\[al\] where $\r_{\o,\a,\xx}=\Psi^+_{\o,\a,\xx}\Psi^-_{\o,\a,\xx}$ and we define, for $\a=(\psi,\phi)$, $\o=\pm$, the truncated correlations &&&lt;\^-\_[,ø,]{}\^+\_[,ø,]{}&gt;\_[T,E,N]{} =[\^2\^+\_[,ø,]{}\^-\_[,ø,]{}]{}\_N(J,)|\_[0]{} \[zaz\]\
&&&lt;\_[’,ø’,]{}\^-\_[,ø]{}\^+\_[,ø]{}&gt;\_[T,E,N]{} =[\^3J\_[’,ø’,]{}\^+\_[,ø,]{}\^-\_[,ø,]{}]{}\_N(J,)|\_[0]{}\
where $<AB>_T=<AB>-<A><B>$ and $<\Psi^-_{\psi,\o,\xx}\Psi^+_{\psi,\o',0}>_{T,E,N}\equiv <\psi^-_{\o,\xx}\psi^+_{\o',0}>_{T,E,N}
\equiv G_{\l,E,N;\o,\o'}(\xx)$ defined by .

Using a smooth decomposition of the unity, we write the propagator as sum of single scale propagators at $E=0$ g()=\_[j=h\_L]{}\^N g\^[(j)]{}() where $h_L\sim -\log L$ and $g^{(j)}(\xx)$, the single scale propagator, is similar to $g(\xx)$ with $\hat\chi(\kk)$ 