I am attempting to understand random matrices better, in particular the measure associated the Gaussian ensembles. It seems very different from standard measure theory. I do not even know how to think of the Lebesgue measure on the strange GUE space.
For this I am currently reading: https://arxiv.org/pdf/1510.04430.pdf and I am particularly stuck on page $17$ on the section about measures:
By definition, the Gaussian Unitary Ensemble (GUE) consists of the consists of the set of complex Hermitian matrices. For the GUE with matrices of size $n$, it is stated that a Lebesgue measure $dM$ exists. Furthermore, it is stated that it is a product of all the real components of the matrix. This is done in the following way: denoting the real components of $M_{i,j}$ as $M_{i,j}^{(\alpha)}$, we may thus define the Lebesgue measure as $$ dM = \prod\limits_{i}dM_{i,i}\prod\limits_{i < j}dM_{i,j}^{(0)}dM_{i,j}^{(1)}\; \; (*)$$
Any idea of how to think of Lebesgue measure $dM$ in such a complex space GUE and dies $dM_{i,j}^{\alpha}$ simply stand for the "simple" Lebesgue measure on the complex plane?
I would be grateful for some intuition. Thanks!
A $n\times n$ Hermitian matrix, $M$, is determined by its entries along its diagonal and above its diagonal. Therefore only the entries $\left(M_{ij}\right)_{1\leq i \leq j \leq n}$ along and above the diagonal have to be specified. Furthermore since $M_{ii} = \overline{M}_{ii}$, i.e. the diagonal entries are equal to there complex conjugate, they are real. There are no further constraints on the entries thus the entries $\left(\left(M_{ii}\right)_{1\leq i \leq n}, \left(M_{ij}\right)_{1\leq i < j \leq n}\right)$ take values in $\mathbb{R}^n \times \mathbb{C}^{n(n-1)/2}$. Writing the real and imaginary parts of $M_{ij}$ as respectively, $M_{ij}^{(0)}$ and $M_{ij}^{(1)}$, $M$ is equivalent to a real vector $\left(\left(M_{ii}\right)_{1\leq i \leq n}, \left(\left(M_{ij}^{(0)}, M_{ij}^{(1)}\right)\right)_{1\leq i < j \leq n}\right)$ in $\mathbb{R}^n \times \left(\mathbb{R}^2\right)^{n(n-1)/2} \cong \mathbb{R}^{n^2}$.
$dM$ is the Lebesgue measure on $\mathbb{R}^2$ written in such a way as to make explicit the dependence on the components of $M$.