general properties of truncated expectations (or the fact that they are expressed in terms of connected diagrams), as in Fig. 1. Note that the first and third contributions are vanishing by parity considerations (remember that $E=0$ here); regarding the second, we can use the following bound &&|d\_1 d\_2 d\_3 v(\_1-\_2) g\^[\[h,N\]]{}(\_1-\_3) W\_[2,1]{}\^[(h)]{}(\_2;\_3,0)|\
&&|g\^[\[h,N\]]{}|\_[L\_1]{} |v|\_[L\_]{} d\_2 d\_3 |W\_[2,1]{}\^[(h)]{}(\_2;\_3,0)|C 2\^[-h]{}, where we have inductively bounded $|W_{2,1}^{(h)}|_{L_1}$ by a constant, as its dimension is $D=0$. Note the crucial role played by the non locality of the disorder; in the case of $\d$-correlated disorder one needs to integrate over the wiggly lines instead than over the propagator (as $|v|_{L_\io} $ is unbounded) so that in the above bound one gets $|g^{[h,N]}|_{L_\io} |v|_{L_1}$ instead of $|g^{[h,N]}|_{L_1} |v|_{L_\io}$ and the resulting bound would be diverging as $N\to\io$ as $2^N$. Similar considerations could be done for $W_{0, 2}^{(h)}$ which can be decomposed as in the r.h.s. Fig. 2; the second term can again be bounded by && |v|\_[L\^]{} |W\^[(k)]{}\_[2,2]{}|\_[L\^1]{}\_[hi’jiN]{}|g\^[(j)]{}|\_[L\^1]{} |g\^[(i)]{}|\_[L\^1]{}|g\^[(i’)]{}|\_[L\^]{}\
&& C\_1 ł2\^[-2h]{}\_[hiN]{} (i-h)2\^[-i+h]{}C\_2 ł2\^[-2h]{}\[aza\] This argument again cannot be repeated for the first term in Fig. 2, but the local part vanishes since the local part of the bubble graph is zero by symmetry \_\_[N]{}()[k\_0\^2- k\^2+2 