# Decoupling quadratic terms in integrating over matrices

I need to carry out an average over a random matrix $$J$$ where its entries are gaussian random variables that are correlated by $$\tau$$. Typically the average is computed by evaluating the integral:

$$\langle\; f(J)\;\rangle\propto \int \left(\prod_{ij}\mathrm{d}J_{ij}\right)f(J)\exp\left \{ -\frac{N}{2(1-\tau^2)}\sum_{ij}J_{ij}^2+\frac{N}{2(1-\tau^2)}\sum_{ij}J_{ij}J_{ji}\right\}$$

My problem is how to decouple the $$J_{ij}J_{ji}$$ in these kind of integrals.

Usually I can express it as a square, call it $$b^2$$ and then can decouple it by adding a Gaussian integral: $$\int \frac{\mathrm{d} x}{\sqrt{2 \pi / a}} \exp \left\{-\frac{a}{2} x^{2} \pm b x\right\}=\exp \left\{\frac{b^{2}}{2 a}\right\}$$ By adding the integral, the quadratic terms indeed temporarily vanish.

However in this case I do not know how to represent $$\sum_{ij}J_{ij}J_{ji}$$ as a square of sums.

Any ideas? thanks !

Edit: To give some context I am computing the following integral:

$$$$=\int_\mathbf{J}\exp{ \left\{-\epsilon \sum_{i}\left|z_{i}\right|^{2}-|\omega|^2\sum_i|z_i|^2-\sum_{ij}z_i^*z_j\omega^*J_{ij}-\sum_{ij}z_i^*z_j\omega J_{ji}+\sum_{ijk}J_{ki}J_{kj}z_i^*z_j-\frac{N}{2(1-\tau)}\sum_{ij}J_{ij}^2+\frac{N}{2(1-\tau)}\sum_{ij}J_{ij}J_{ji}\right\}}$$$$

where $$z_i$$ and $$\omega$$ are complex numbers and $$\omega^*$$ represents the complex conjugate. As you can see, I also have trouble with decoupling $$\sum_{ijk}J_{ki}J_{kj}z_i^*z_j$$ which would be written as: $$\sum_k\left|\sum_{i}J_{ki}z_i\right|^2$$ But that would be the result of a complex gaussian integral, and $$1)$$ I don't know how to do that, $$2)$$ that would probably deserve a separate question. My current concern is how to uncouple $$J_{ij}J_{ji}$$ which is something that would be useful to know in the future. Thanks!

(To give all the context, I am trying to reproduce the results of the following paper... Sommers, H. J., Crisanti, A., Sompolinsky, H., & Stein, Y. (1988). Spectrum of large random asymmetric matrices. Physical review letters, 60(19), 1895. )

• Is the first term supposed to be $\left(\prod_{ij} dJ_{ij}\right)$? Jul 29 '19 at 17:52
• Yes indeed! Thank you for noticing it. I corrected it in an edit.
– Matt
Jul 29 '19 at 17:55
• A few other points of confusion: The exponential term does not seem to be a probability distribution, for instance it vanishes when $f(J)=1$ in the $1\times 1$ case. Are you looking for a change of variables which transforms the integral into an expectation for an uncorrelated Gaussian matrix, or something else? What, if anything, do you know about the function $f$? Jul 29 '19 at 18:09
• Sorry, I realise it is the case, I know my function $f$ but I discarded it because I thought it would bring more confusion. I have added it now, for completeness.
– Matt
Jul 29 '19 at 18:27

It may be helpful to write out the integral in matrix notation. I'll omit the prefactor $$\frac{N}{2(1-\tau)}$$. $$\int_J\exp\left\{\text{Tr}(J^2-J^T J)\right\}$$ Here, one useful trick is to change variables so that the symmetric and antisymmetric parts of $$J$$ (which I'll call $$A$$ and $$B$$ respectively) are separated. Or, to be more precise: $$A_{ij}=\frac{1}{2}(J_{ij}+J_{ji}),\ \ \ i\le j$$ $$B_{ij}=\frac{1}{2}(J_{ij}-J_{ji}),\ \ \ i $$\prod_{ij}dJ_{ij}=2^{\frac{-N(N-1)}{4}}\left(\prod_{i\le j}dA_{ij}\right)\left(\prod_{i $$J_{ij}=\begin{cases} A_{ij}+B_{ij} & ij \end{cases}$$ Wewriting the integral, $$\int_J\exp\left\{\text{Tr}(2B^2+2BA)\right\}$$ Using the cyclic property of the trace, as well as the symmetry/antisymmetry of $$A$$/$$B$$, the second term vanishes. $$\int_J\exp\left\{\text{Tr}(-2B^TB)\right\}$$ Or, in terms of elements, $$2^{\frac{-N(N-1)}{4}}\int\left(\prod_{i\le j}dA_{ij}\right)\left(\prod_{i Fortunately, the integral is completely separated; unfortunately, it diverges, since there is nothing constraining the symmetric part of $$J$$.
For your full integral, this approach wouldn't mess anything else up, but it wouldn't take care of the other quadratic term $$z^\dagger J^T Jz$$. Other tricks are needed for that one; perhaps a unitary change of variables to rotate $$z$$ onto one of the coordinate axes would do the trick.