Many texts (e.g. 1-2-3) on random matrices start with some variation of the identity:

$$\rho_1(\lambda) = \frac{1}{\pi} \text{Im}\{\langle\text{Tr}(\mathbf{X}-\lambda\mathbf{I})^{-1}\rangle\}$$

where the brakets denote the average over the ensemble of eigvenvalues and 'Tr' is the trace of the resolvant matrix $(\mathbf{X}-\lambda\mathbf{I})^{-1}$, where $\lambda\in\mathbb{C}\backslash\mathbb{R}$. The left side is the 1 point correlation function, i.e. the density, which is also defined as

$$\rho_1(\lambda)= \big\langle\sum_i \delta(\lambda-\lambda_i)\big\rangle $$

(same ensemble). How does one prove the above identity?


I provided a partial proof on Math Overflow, and Carlo Beenakker corrected some mistakes related to complex integration and clashing definitions of the marginal probability density. I consider this question answered.


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