Fourier transform of eigenvalue distribution of GUE matrices

Question

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

$p_{\mathrm{GUE}(d)}(\lambda_1,\ldots,\lambda_d) = N_d \prod_{1\leq i<j\leq d} (\lambda_i-\lambda_j)^2 \exp(-\frac{d}{2}\sum_{i=1}^d \lambda_i^2)$ ,

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 $H\sim\mathrm{GUE}(d)$).

Answer 1

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 $f_{\rm GUE(d)}(k,...,k)=e^{-k^2/2}$.

Answer 2

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 $1-k^2/4$.}