Sub-dominant eigenvalue and spectral gap conjecture for the Dirichlet Markov Ensemble

Let $\mathbf{M}$ be the random matrix from theorem $\mathrm{th:sym}$, with eigenvalues ordered so that $\lambda_1(\mathbf{M})$ is the dominant eigenvalue and $\lambda_2(\mathbf{M})$ is the sub-dominant eigenvalue. Let $(a_n)$ and $(b_n)$ be deterministic sequences, and let $\mathcal{G}$ be a probability distribution on $\mathbb{R}$. Sub-dominant eigenvalue and spectral gap conjecture. One has $\lambda_1(\mathbf{M})=1$ and, almost surely,

$$\lim_{n\to\infty}\sqrt{n}\,|\lambda_2(\mathbf{M})|=1.$$

In particular, the spectral gap $1-|\lambda_2(\mathbf{M})|$ is of order $1-1/\sqrt{n}$ for large $n$. Moreover, there exist $(a_n)$, $(b_n)$ and $\mathcal{G}$ such that

$$b_n\bigl(|\lambda_2(\mathbf{M})|-a_n\bigr)\overset{\mathrm{d}}{\underset{n\to\infty}{\longrightarrow}}\mathcal{G}.$$

The conjecture is motivated by simulations and analogy with the Complex Ginibre Ensemble. It predicts both the asymptotic size of the second eigenvalue and a limiting fluctuation law, but the centering, scaling and distribution are not specified.