The range of the disorder ($\k=1$ in ) provides a natural [*momentum scale*]{} separating the scales $j$ in [*ultraviolet scales*]{}, between $0$ and $N$, and [*infrared scales*]{}, namely between $h_L$ and $-1$. Let us consider first the integration of the ultraviolet scales. The [*scaling dimension*]{} in the case of $\d$-correlated disorder is the same in the ultraviolet and infrared region and equal to $D=2-n/2-m$, that is greater or equal to zero in the case $n,m=(2,0), (4,0), (2,1)$. Therefore there are in general ultraviolet divergences and this requires that the ultraviolet $N\to\io$ limit can be taken only choosing properly the bare parameters to $N$-dependent and possibly singular value. For instance, in the case of the Thirring model with a local $\d$-like interaction, the $N\to\io$ limit can be taken only choosing the bare wave function renormalization vanishing as $2^{-\h N}$ with $\h>0$.

In the case of short-ranged correlated disorder the situation is different; the non locality of the disorder induces an [*improvement*]{} in the scaling dimension, and indeed no ultraviolet divergences are present; the kernels of the effective potential are bounded uniformly in the ultraviolet cut-off $N$. Consider for instance $W^{(h)}_{2,0}$, $h\ge 0$, with scaling dimension $D=1$. We can decompose $W^{(h)}_{2,0}$, using 