23 problems

High-dimensional spectral-distribution conjecture. For $d\to\infty$ with $t=\frac{Z}{d}$ fixed, the spectral distribution is the effective-medium approximation in the adjacency cas…

Pennington–Worah conjecture. The asymptotic spectral distribution is preserved through multiple layers only by activation functions with $\theta_2(f)=0$, and it is given by the Mar…

Let $N$ be the size of a GUE matrix, let its permanental roots lie in the complex plane with coordinate $z=x+iy$, and consider their normalized asymptotic limiting density as…

Oriented Kesten–McKay conjecture. The ESD of $M$ converges in probability, as $n\to\infty$, to the oriented Kesten–McKay law on $\mathbb{C}$ with density

Upper-edge conjecture. The values $\widetilde{w}$ and $\widehat{w}$ are the upper edges of the corresponding LSDs.

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 $Q$.…

A Lévy matrix $\mathbf{H}$ is a symmetric random matrix whose entries are random variables in the domain of attraction of an $\alpha$-stable law; let $\mu_\mathbf{H}$ denote its em…

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

Let $B_n$ be an $n\times n$ matrix with independent and identically distributed entries of zero mean and unit variance, and let the empirical spectral distribution of…

Let $\mathbf{W}_n$ be a large Wigner matrix, let $H^{\mathbf{W}_n}$ denote its eigenvector empirical spectral distribution, and let $F$ be the semicircle law. The VESD and the ESD…

Let $\hat{\Gamma}_u$ denote the non-Hermitian sample autocovariance matrices arising in the model under consideration, and let the limiting spectral distribution (LSD) mean the lim…

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…

Near-Haar spectral-distribution conjecture. The spectral distributions of $C_n$ should be close to the corresponding Haar-unitary spectral distributions when $U_n$ is distributed s…

Optimal-rate conjecture. The optimal convergence rates should be $O(N^{-1/2})$ for $H^{\mathbf{S}_n}(x)$ and $O(N^{-1})$ for $F^{\mathbf{S}_n}(x)$.

Let $X$ be a random doubly stochastic matrix of size $n$, let ${\mathbf{E}}X$ denote its expectation, and consider the empirical spectral distribution of…

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…

Let $A\Gamma_d(\eta,\Sigma)$ be an infinitely divisible positive definite matrix gamma distribution, with total mass $\omega_{d,\eta}$, and consider the limiting spectral distribut…

Let $n\geq r\geq 3$, and let $A$ be the adjacency matrix of a uniformly random oriented $r$-regular graph on $n$ vertices, meaning that every vertex has $r$ incoming and $r$ outgoi…

Let $D\in\mathcal{M}_n(\mathbb{R}_+)$ be a random diagonal matrix and let $U,V\in\mathcal{M}_n(\mathbb{C})$ be independent Haar unitary matrices, independent of $D$. Set $X=UDV^*$,…

Let the limit eigenvalue distribution of the random unit-magnitude Vandermonde matrices under consideration be given, and let an atom mean a point mass of this distribution. Atom-a…

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

Let $M$ be a Hermitian matrix of order $n$, let $A$ be a uniformly random $k\times k$ principal submatrix of $M$, and let $F_A$ be the empirical spectral distribution function of…