not receive contribution by correlations involving pairs of points in space-time such that $\gamma_{|\x-\x'|^{-1}}|t-t'|>\rho$, but this is expected to be true from the exponential decay of the time correlations$]$.

The interaction terms in the perturbation expansion are obtained Taylor expanding the kernels $\gamma$, ${\bf G}$ and $h$ ($H$ working with the field action). The result for $\gamma$ is, for instance: $$\gamma(\x-\y+\delta\z(t|\x,\y))=\gamma(\x-\y)+\sum_{n=1}^\infty\lambda_{\gamma_n}
\gamma_n^{i_1...i_n}(\x-\y)\delta z_{i_1}(t|\x,\y)...\delta z_{i_n}(t|\x,\y)
\eqno(2.15)$$ with $\lambda_{\gamma_n}=1$ a coefficient which may scale when carrying on power counting. Similar coefficients $\lambda_{G_n}$ and $\lambda_{H_n}$ are introduced in the Taylor expansion for $G$ and $H$. The theory is thus characterized by an infinite number of interactions involving vertices, which, to $\O(\rho^{-n})$, have up to $2+n$ legs.

To check for divergences at large $k$ in the perturbation expansion, we use power counting directly in Eqns. (2.11-13) [@zinn-justin]. Rescaling coordinates and times as: $$x\to\Lambda x\quad{\rm and}\quad t\to\Lambda^\frac{2}{3}t,
\eqno(2.16)$$ Eqns. (2.9-11) remain invariant in form provided we rescale the various fields and interactions $A=q^\smalL,\delta z$, $\lambda_{\gamma_n},\lambda_{G_n},\lambda_{H_n}$: as $A\to \Lambda^{[A]}A$, with $$[q^\smalL]=-\frac{2}{3}\qquad [\delta\z]=1\qquad$$ $$[\lambda_{\gamma_n}]=[\lambda_{H_n}] =[\lambda_{G_n}] =0.
\eqno(2.17)$$ This leads to expect logarithmic divergences at large $k$, meaning renormalizability of the field theory and the possibility of logarithmic correction to scaling, produced by renormalization of the parameters in 