replaced by $f_j(\kk)$ , with $f_j(\kk)$ non vanishing in $2^{j-1}\le |\kk|\le 2^{j+1}$. The presence of a minimal scale $h_L$ comes from the fact that antiperiodic boundary conditions are assumed, and therefore the momenta are of the form $\kk={2\pi\over L}({\bf n}+{1\over 2})$, so that $|k_i|\ge {\pi\over L}$. $L$ plays the role of an [*infrared cut-off*]{} while $2^N$ is the [*ultraviolet*]{} cut-off. Note that &&|g\^[(j)]{}|\_[L\_1]{}=d|g\^[(j)]{}()|C 2\^[-j]{}\
&&|g\^[(k)]{}|\_[L\_]{}=\_|g\^[(j)]{}()| C 2\^[j]{} \[fon111\] We use now the following basic property of gaussian integrations, bosonic or fermionic, called [*addition property*]{} and we get , calling the exponent in the r.h.s. of simply $V(\Psi,\h,J)$ P(d)e\^[V(,,J) ]{}=P(d\^[(N-1)]{}) P(d\^[(N)]{}) e\^[V(,,J)]{} =P(d\^[(N-1)]{})e\^[V\^[(N-1)]{}(\^[(N-1)]{},J)]{} \[all\] where $P(d\Psi^{(\le N-1)})$ and $P(d\Psi^{(N)})$ are gaussian integrations with propagator respectively $g^{(\le N-1)}(\xx)$ and $g^{(\le N)}(\xx)$ and V\^[(N-1)]{}(,,J)=\_[n=1]{}\^ \^T(V;n)with $\EE^T$ are the [*truncated expectations*]{} with respect to $P(d\Psi^{(N)})$ \^T(V();n)=[\^n\^n]{}P(d\^[(N)]{}) e\^[V(\^[(N)]{}+)]{}|\_[=0]{} When expressed in terms of Feynman graphs, the truncated expectation are written in terms of [*connected*]{} diagrams only. The multiscale analysis continues integrating the fields $\Psi^{(N-1)},..,\Psi^{(h+1)}$ obtaining e\^[\_N(J,)]{} =P(d\^[(h)]{})e\^[V\^[(h)]{}(\^[(h)]{},,J)]{}with $V^{(h)}$, called effective potential, being a sum of integral of monomials with $n\ge 0$ $\Psi,\h$ and $m\ge 0$ $J$ fields multiplied by kernels $W_{n,m}^{(h)}$; moreover $P(d\Psi^{(\le h)})$ is the integration with propagator $g^{(\le h)}(\xx)= \sum_{k \le h}
g^{(k)}(\xx)$.

