The random-matrix Markov–Krein correspondence conjecture

Let $M_N$ be a sequence of unitarily invariant random matrices, let $\lambda=(\lambda_1,\ldots,\lambda_N)$ be the eigenvalue vector of $M_N$, and let $\mu=\pi_{N,N-1}\lambda$ be the eigenvalue vector of a principal $(N-1)\times(N-1)$ submatrix. Suppose that the spectral measures

$$\frac{1}{N}\sum_{i=1}^N\delta_{\lambda_i}$$

converge weakly in probability to a deterministic measure $\mathbf{m}_{\mathrm{RMT}}$. Random-matrix Markov–Krein correspondence conjecture. The random signed measures

$$\sum_{i=1}^N\delta_{\lambda_i}-\sum_{i=1}^{N-1}\delta_{\mu_i}$$

converge weakly in probability to a signed measure $\mathbf{d}_{\mathrm{RMT}}$ satisfying

$$\exp\left(\sum_{k=1}^\infty\frac{z^k}{k}\int_{\mathbb R}x^k\,\mathbf{d}_{\mathrm{RMT}}(dx)\right)=\sum_{k=0}^\infty z^k\int_{\mathbb R}x^k\,\mathbf{m}_{\mathrm{RMT}}(dx).$$

The statement is presented as a folk theorem in random matrix theory and expresses the limiting relationship between the empirical spectral measure and the eigenvalue measure of consecutive principal minors; the supplied text gives no resolution status.