Working with a random matrix $\mathbf{J}$ whose elements follow a gaussian distribution, I need to evaluate the following integral:
$$ \int \prod_{i, j=1}^{N} \sqrt{\frac{N}{2 \pi}} d J_{i j}\exp\left (-\frac{1}{2} \sum_{i, j, k} J_{k i} A_{i j} J_{k j}+\sum_{k, j} B_{k j} J_{k j} + \frac{\tau}{\gamma}\sum_{ij}J_{ij}J_{ji}\right)$$
When $\tau=0$, the integral can be computed by shifting coordinates and then calculating the determinant of $\mathbf{A}$. However I do not know how to handle the terms dependent on $\tau$.
Any ideas or observations are always welcome. Thank you!