Circle law for the Dirichlet Markov Ensemble

Let $\mathbf{M}$ be the symmetric random matrix from theorem $\mathrm{th:sym}$, with eigenvalues $\lambda_1(\sqrt{n}\,\mathbf{M}),\ldots,\lambda_n(\sqrt{n}\,\mathbf{M})$. Let $\mathcal{C}_1$ denote the uniform probability distribution on the unit disc

$$\{z\in\mathbb{C}:|z|\leqslant 1\}.$$

Circle law for the Dirichlet Markov Ensemble. Almost surely, the empirical spectral distribution converges weakly to $\mathcal{C}_1$:

$$\mathbb{P}\left(\frac{1}{n}\sum_{k=1}^n\delta_{\lambda_k(\sqrt{n}\,\mathbf{M})}\overset{\mathrm{w}}{\underset{n\to\infty}{\longrightarrow}}\mathcal{C}_1\right)=1.$$

The conjecture concerns the limiting eigenvalue distribution of a non-normal random matrix with dependent, non-identically distributed entries. The paper notes that the problem is difficult because eigenvalues of non-normal matrices are sensitive to perturbations and the usual singular-value methods do not apply; only partial results are available.