0$ such that for all $ t \in \mathbb{R}$, \begin{align}\label{assumption_propagator} \norm{p_t}_3^3 = \sum_{\mathbf{x} \in \mathbb{Z}^d} |p_t(\mathbf{x})|^3 \leq C (1+t^2)^{-(1+\delta)/2} \,. \end{align} Furthermore, we assume also the already mentioned $ \ell_1$-clustering property (see section \ref{sec:cwdyn}) which we slightly rephrase as follows for each cumulant of order $n$: we require that \begin{align}\label{eq:defell1clnorm} \norm{\kappa_n}_1 := \sup_{\sigma \in \{\pm 1 \}^n}\sum_{x \in (\Z^{d})^n} \cf(\mathbf{x}_1=0) \big\vert \kappa(\psi(\mathbf{x}_1, \sigma_1), \ldots , \psi(\mathbf{x}_n, \sigma_n)) \big\vert < \infty \,. \end{align} We recall that the physical meaning of this condition is that the cumulants decay fast enough in space so that they are summable, once the translational invariance is taken into account. We recall that the first order non-pairing contributions in (\ref{W_expansion}) are \begin{align}\label{first_order} & -\ci\lambda \sigma \int_0^t ds \int_{(\T^{d})^3} d\mathbf{k}_1 d\mathbf{k}_2 d\mathbf{k}_3 \delta(\mathbf{k} - \mathbf{k}_1 - \mathbf{k}_2 -\mathbf{k}_3)\rme^{\ci s(\sigma \omega +\omega_1 - \sigma \omega_2 - \omega_3)} \nonumber \\ & \quad \times \kappa[ a(\mathbf{k}_1,-1); a(\mathbf{k}_2,\sigma); a(\mathbf{k}_3,1); a(\mathbf{k}', \sigma') ] \end{align} and a term which is obtained from (\ref{first_order}) by swapping $(\mathbf{k},\sigma) \leftrightarrow (\mathbf{k}',\sigma')$. As stated in (\ref{trans_inv_cum}), by translation invariance we have \begin{align} \kappa[ a(\mathbf{k}_1,-1); a(\mathbf{k}_2,\sigma); a(\mathbf{k}_3,1); a(\mathbf{k}', \sigma') ] = \delta(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}') \widehat{F}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}',\sigma, \sigma') \end{align} where $F(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3,\mathbf{x}_4, \sigma, \sigma' )=\cf(\mathbf{x}_1=0) \kappa[ a(\mathbf{x}_1,-1) ; a(\mathbf{x}_2,\sigma); a(\mathbf{x}_3,1); a(\mathbf{x}_4, \sigma') ]$. Clearly, $\norm{F}_1 \le \norm{\kappa_4}_1<\infty$ by the assumed $\ell_1$-clustering. Therefore, the term in (\ref{first_order}) is bounded by \begin{align} & \lambda \bigg\vert \int_0^{t} ds \int_{(\T^{d})^3} d\mathbf{k}_1 d\mathbf{k}_2 d\mathbf{k}_3 \delta(\mathbf{k} - \mathbf{k}_1 - \mathbf{k}_2 -\mathbf{k}_3)\rme^{\ci s(\sigma \omega +\omega_1 - \sigma \omega_2 - \omega_3)} \nonumber \\ & \qquad \times \delta(\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3+\mathbf{k}') \widehat{F}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}',\sigma, \sigma') \bigg\vert \nonumber \\ & \quad \le \lambda \delta(\mathbf{k}+\mathbf{k}') \int_0^{t} ds \bigg\vert \int_{(\T^{d})^2} d\mathbf{k}_1 d\mathbf{k}_2 \left. \rme^{\ci s(\omega_1 - \sigma \omega_2 - \omega_3)} \widehat{F}(\mathbf{k}_1,\mathbf{k}_2,\mathbf{k}_3,\mathbf{k}',\sigma, \sigma')\right|_{\mathbf{k}_3=\mathbf{k}-\mathbf{k}_1-\mathbf{k}_2} \bigg\vert \nonumber \\ & \quad \le\lambda \delta(\mathbf{k}+\mathbf{k}') \int_0^{t} ds \sum_{\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3,\mathbf{x}_4} |F(\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3,\mathbf{x}_4, \sigma, \sigma')| \nonumber \\ & \qquad \times \bigg\vert \sum_{\mathbf{y}}\rme^{-\ci 2 \pi \mathbf{k} \cdot (\mathbf{y}-\mathbf{x}_3)} p_{-s}(\mathbf{y})p_{\sigma s}(\mathbf{y}-\mathbf{x}_2)p_{s}(\mathbf{y}-\mathbf{x}_3) \bigg\vert \nonumber \\ & \quad \leq \lambda \delta(\mathbf{k}+\mathbf{k}') \norm{\kappa_4}_1 \int_0^{t} ds\, \norm{p_s}_3^3 \leq \lambda C \delta(\mathbf{k}+\mathbf{k}') \norm{\kappa_4}_1 \int_0^{t} ds(1+s^2)^{-(1+\delta)/2} \nonumber \\ & \quad \le \lambda C' \delta(\mathbf{k}+\mathbf{k}') \norm{\kappa_4}_1 \end{align} where $C'$ is a constant which depends only on $C$ and $\delta$. We have used the inverse Fourier transform of $\FT{F}$ and (\ref{eq:defpt}) in the second inequality and H\"older's inequality in the third one. Since the bound in invariant under the swap $(\mathbf{k},\sigma) \leftrightarrow (\mathbf{k}',\sigma')$, it bounds also the second non-pairing contribution in (\ref{W_expansion}). Therefore, we see that the first order contributions are $O(\lambda)$ uniformly in $t$. \addcontentsline{toc}{section}{Bibliography} \begin{thebibliography}{9} \bibitem{wick_wiki} \url{http://en.wikipedia.org/wiki/Wick_product}. Accessed 13-01-2015. \bibitem{wick_encMath} \url{http://www.encyclopediaofmath.org/index.php/Wick_product}. Accessed 13-01-2015. \bibitem{GiSu}L. Giraitis and D. Surgailis, {\it Multivariate Appell polynomials and the central limit theorem} in E. Eberlein and M. S. 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