Spectral density of non-Hermitian random matrix

Question

I have a question regarding the computation of the spectral density of some non-Hermitian random $N\times N$ matrix $A$. Following Rogers and Castillo, 2009, we denote the collection of eigenvalues of $A$ by $\{\lambda_i^A: i = 1,\ldots,N\}$, and for a point in the complex plane $z = x+iy$ we can therefore write the spectral density as $$ \varrho_A(z,\bar{z}) = \frac{1}{N} \sum_i \delta(x-\Re(\lambda_i^A)) \delta(y-\Im( \lambda_i^A)), $$ with $\Re(\ldots)$ and $\Im(\ldots)$ the real and imaginary parts respectively.

They proceed by following previous works and define a matrix $$ H = H(z,\bar{z}; \kappa) = \begin{pmatrix} \kappa I_N & i(z I_N - A) \\ i(z I_N - A)^\dagger & \kappa I_N \end{pmatrix}, $$ with $I_N$ the $N\times N$ identity matrix, and $(\ldots)^\dagger$ the conjugate-transpose. They do not specify $\kappa$, but I assume $\kappa \in \mathbb{R}$, as they later take the limit $\kappa \rightarrow 0$. Then they define the Wirting derivatives $$ \partial_z = \frac{1}{2}\left( \partial_x - i\partial_y\right), \quad \partial_{\bar{z}} = \frac{1}{2}\left( \partial_x + i\partial_y\right), $$ and state that the spectral density can thus be written as $$ \varrho_A(z,\bar{z}) = -\frac{1}{\pi N} \lim_{\kappa \rightarrow 0} \partial_{\bar{z}}\partial_z \log \det H. $$

I simply cannot figure out why this is true. One of the example references by Feinberg, Zee, 1997 notes that $\partial_z (\bar{z})^{-1} = \pi \delta(x)\delta(y)$ (I can also not figure this out), and that from this (and other relations) follows that $\partial_z\partial_{\bar{z}} \log(z\bar{z}) = \pi \delta(x)\delta(y)$ and hence the spectral distribution can be written as $$ \varrho_A(z,\bar{z}) = \frac{1}{\pi N} \partial_z \partial_\bar{z} \text{Tr} \log (z - A)(\bar{z} - A^\dagger) $$ Note that this has a very high abuse of notation, but I assume the trace is taken over the log of the product.

How is the form of the spectral density $\varrho_A(z,\bar{z})$ derived in both cases? And why is $\partial_z\partial_{\bar{z}} \log(z\bar{z}) = \pi \delta(x)\delta(y)$? Any help is greatly appreciated.

Answer 1

Your questions about the delta function via the Wirtinger calculus are partially addressed here: Dirac delta distribution in the complex plane

In more detail, if we set $ g= \log( z\overline z +\epsilon)$ then the Wirtinger derivative rules tell us that

(i) $g_{\overline z}= \frac{ z}{z\overline z+ \epsilon}$ and from there one next proceeds as in the referenced link to deduce

$g_{z\overline z} $ tends to a delta function at the origin as the parameter $\epsilon\to 0$.