7 problems

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

Fix $\alpha \in [0,1)$. Let $A=PLQ$ be an $n\times n$ random matrix, where $P$ and $Q$ are independent uniformly chosen permutation matrices and $L$ is independent of $P$ and $Q$.…

Lakshminarayan–Puchala–Życzkowski conjecture. For $k$ and $N$ positive integers, it holds that

Let $A_n$ be a random regular digraph matrix, meaning a uniformly random element of $\mathop{\mathrm{M}}_n(d)$, where $\mathop{\mathrm{M}}_n(d)$ is the set of $n\times n$ zero-one…

Cumulant conjecture. The right-hand side of this relation must be related to the $p$th cumulant of the $p$th moment of the mean spectral distribution of $M$, and this relation shou…

The Spherical Law. The spectral densities of $A_NB_N^{-1}$ converge to the uniform density on the Riemann sphere as $N$ tends to infinity.

Block-size ratio conjecture. If the limiting spectral distribution (LSD) of $\mathbf{A}_n$ exists, then