# Fourier transform of eigenvalue distribution of GUE matrices

I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $$X\sim \mathrm{GUE} (d)$$, where $$\mathrm{GUE} (d)$$ stands for Gaussian Unitary Ensemble in dimension $$d$$.

The expression for the joint distribution of eigenvalues $$=\lambda_1,\ldots,\lambda_d$$ of matrices in this ensemble is well-known

$$p_{\mathrm{GUE}(d)}(\lambda_1,\ldots,\lambda_d) = N_d \prod_{1\leq i ,

where $$N_d$$ is a normalisation constant.

However, nowhere in the literature, I could find information about the Fourier transform of $$p_{\mathrm{GUE}(d)}$$ i.e

$$f_{\mathrm{GUE}(d)}(k_1,\ldots,k_d) = \int_{\mathbb{R}^d}d\lambda_1 \ldots d\lambda_d \exp(i\sum_{j=1}^d k_j \lambda_j )p_{\mathrm{GUE}(d)}(\lambda_1,\ldots,\lambda_d)$$ .

This was surprising to me since characteristic functions seem a rather natural object to study and GUE is one of the basic enables appearing in random matrix theory.

I am especially interested in understanding the behaviour (decay) of $$f_{\mathrm{GUE}(d)}(k_1,\ldots,k_d)$$ as a function of $$|k|=\sqrt{\sum_{i=1}^d k_i^2}$$ for large $$d$$.

My motivation to study this problem comes from some considerations at the intersection of quantum chaos and quantum computing (particularly, the problem of "complexity growth" in unitary evolution $$\exp(it H)$$, where $$H\sim\mathrm{GUE}(d)$$).

Complementing the answer by Carlo, if you take all $$k$$'s equal you have $$f_{\rm GUE(d)}(k,...,k)\propto \int dX e^{ik{\rm Tr}(X)}e^{-\frac{d}{2}{\rm Tr}(X^2)}.$$ Taking $$x$$ to be any real diagonal element from $$X$$, this is $$f_{\rm GUE(d)}(k,...,k)\propto \left(\int dx e^{ikx}e^{-\frac{d}{2}x^2}\right)^d.$$

I think in the end you have simply $$f_{\rm GUE(d)}(k,...,k)=e^{-k^2/2}$$.

• you seem to assume that a diagonal element of $X^2$ is the square of a diagonal element of $X$ ... that seems wrong... Apr 13, 2021 at 15:01
• @CarloBeenakker No, I just write $e^{-{\rm Tr}(X^2)}=\prod_{ij}e^{-|x_{ij}|^2}$. The integral over the non diagonal elements is trivial Apr 13, 2021 at 15:11
• right, my mistake. Apr 13, 2021 at 15:18

The Fourier transform of the marginal distribution of a single eigenvalue in the GUE is known, $$f_{{\rm GUE}(d)}(k,0,0,\ldots,0)=e^{-\tfrac{1}{2}k^2/d}\sum_{j=0}^{d-1}(-1)^jk^{2j}\frac{(d-1)(d-2)\cdots(d-j)}{j!(j+1)!d^j},$$ see these lecture notes.

A curiosity: the $$d=2$$ result is given in this publication in terms of the confluent hypergeometric function $$M(-1,1/2,k^2/8)$$, without the realization that this is simply $$1-k^2/4$$.

• Thanks for the answer. However, I am interested in the Fourier transform of the full distribution, not just the marginals. Apr 13, 2021 at 14:34
• I would be surprised if there is a closed form expression for that, you would have a closed form expression for correlators of arbitrary order... Apr 13, 2021 at 14:45