# Multivariate Gaussian with correlated elements

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!

• another user asked this and then once again Mar 9 '20 at 5:19
• see here too Mar 11 '20 at 12:49