equals the cut-offs scales $N$. Therefore, the second term in Fig. 3, which contributes to $R_{N,\a}$ in (32), can be bounded as with the difference that one of the scales of the propagator attached to the back dot have scale $N$; therefore one obtains the bound &&|ł| |v|\_[L\^]{} | W\^[(h)]{}\_[4,1]{}|\_[L\^1]{}\_[hi’iN]{}|g\^[(N)]{}|\_[L\^1]{} |g\^[(i)]{}|\_[L\^1]{}|g\^[(i’)]{}|\_[L\^]{}\[xaxa\]\
&&C\_1ł\^2 2\^[-2h]{}(N-h)2\^[-N+h]{}C\_2ł\^2 2\^[-2h]{}2\^[-(N-h)/2]{}leading to the vanishing of this contribution for $N\to\io$. On the other hand the non-connected contributions, that is the first term in Fig. 3, is now non vanishing and contribute to the first term in (32) ; the bubble is now given by \_\_[C\_[N]{}(,)D\_[-ø]{}()]{} g\_ø()g\_ø(+)=\_+O(2\^[-N]{}) Remarkably, the anomalies cancel out in the WI for the total density due to supersymmetry (that is, due to ) &&D\_ø()\_[=,]{}&lt;\_[,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}&gt;\_[T,0,N]{}=\_[ø,ø’]{} \[&lt;\^-\_[’,ø’,]{}\^+\_[’,ø’,]{}&gt;\_[T,0,N]{}\
&& -&lt;\^-\_[’,ø’,+]{}\^+\_[’, ø’,+]{}&gt;\_[T,0,N]{}\] +\_[=,]{} R\_[N,]{}(,)\[pp1\] We can write the Schwinger-Dyson equation \_[T,0,N]{}=g\_ø()+łg\_[ø]{}() dv() \_[’=,]{}&lt;\_[’,-ø]{}()\^+\_[,ø,+]{}\^-\_[,ø,]{}&gt;\_[T,0,N]{} \[sd\]and inserting we obtain \_[T,0,N]{}=g\_ø()+łg\_ø() dv() \_[’=,]{} [R\_[N,’]{}(,)D\_ø()]{}\[sd1\] It can be shown, by an analysis similar to the one for $R_{N,\a}$ , that that $\int d\pp \hat v \sum_{\b=\phi,\psi} {R_{N,\a}\over D_\o}$ is smaller than $2^{-N}$, that is vanishing when the ultraviolet cut-off is removed. Therefore at $E=0$ the averaged 2-point function is equal to the free one up to corrections which are vanishing [*only*]{} when the 