I am trying to compute the averaged partition function for some system (with very large $N$) and I reach this point: \begin{equation} \left < Z\right > =\int \prod_i^N \left (\frac{\mathrm{d}x_i}{\sqrt{2\pi/i}} \right )\exp{\left \{-\frac{i}{2}\sum_i^N\lambda_\epsilon x_i^2 - \frac{\left ( \sum_{i}^Nx_i^2\right ) ^2}{4N} \right\} } \end{equation} I simply have to evaluate the integral, but the annoying part is to first correctly uncouple the $x_i$ and $x_j$ which I cannot seem to do correctly. One way is to use the method of gaussian linearisation, which consists in saying that: \begin{equation} \exp{\left \{ \frac{\left ( \sum_{i}^Nx_i^2\right ) ^2}{4N} \right\} }=\exp{\{\frac{b^2}{4a}\}}=\int\sqrt\frac{a}{\pi}\exp\{-am^2+bm\}\mathrm{d}m \end{equation} Here is my working out: \begin{equation} \left < Z\right >_M =\int \prod_i \left (\frac{\mathrm{d}x_i}{\sqrt{2\pi/i}} \right )\exp{\left \{-\frac{i}{2}\sum_i\lambda_\epsilon x_i^2 - \frac{\left ( \sum_{i}x_i^2\right ) ^2}{4N} \right\} } \end{equation} \begin{equation} \left < Z \right >_M =\int_m \int \prod_i \left (\frac{\mathrm{d}x_i}{\sqrt{2\pi/i}} \right )C_1\exp{\left \{-\frac{i}{2}\sum_i\lambda_\epsilon x_i^2 +i \sum_{i}x_i^2m - Nm^2 \right\}}\mathrm{d}m \end{equation} \begin{equation} \left < Z\right >_M =\int_m C_1 \prod_i \left (\int \frac{\mathrm{d}x_i}{\sqrt{2\pi/i}} \exp{\left \{-\frac{i}{2}\lambda_\epsilon x_i^2 +i x_i^2m \right\} }\right ) \exp{\left \{ - Nm^2 \right\}}\mathrm{d}m \end{equation} \begin{equation} \left < Z\right >_M =C_1\int_m \prod_i \left (\frac{1}{\sqrt{\lambda_\epsilon - 2m}} \right ) \exp{\left \{ - Nm^2 \right\}}\mathrm{d}m \end{equation} \begin{equation} \left < Z\right >_M =C_1\int_m \exp{\left \{ N(-m^2 -\frac{1}{2}\ln(\lambda_\epsilon - 2m) \right\}}\mathrm{d}m \end{equation} Unfortunately this last integral cannot be computed via Laplace method, and anyway does not make sense with the results I am supposed to recover. I tried different methods, notably enforcing a delta function in order to uncouple the variables but it gets really messy. Any hint or advice about where it goes wrong?
Thanks a lot! Any input is more than welcome.
