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