287 problems

Let $\mathbf{X}_1,\dots,\mathbf{X}_n$ be i.i.d. matrices with common mean matrix $\mathbb{E}\mathbf{X}_1=\mathbf{M}$ and $p$th central moment matrix … For any…

Fix a subspace dimension $d$ and distortion parameters $\alpha = \beta \in (0,1)$. A random sparse matrix $\bm{\Phi}: \mathbb{R}^n \to \mathbb{R}^k$ has at most $\zeta$ nonzero entr…

Let $A\in\mathbb{R}^{n\times n}$ have independent standard Gaussian entries, and let $\rho_n(A)$ be the growth factor produced by Gaussian elimination with partial pivoting, namely…

Let $\mu_{n,T}$ be the law on $\mathrm{SO}(n)$ of Kac's walk after $T$ steps, and let $\operatorname{Haar}_n$ denote Haar measure on $\mathrm{SO}(n)$. The conjecture asks whether,…

Let $Q_n$ be a random matrix whose independent rows are chosen uniformly from the vectors in $\{0,1\}^n$ having sum exactly $\lfloor n/2\rfloor$. Nguyen's conjecture. … This model…

Let $A_k$, for $k\geq 0$, be independent $n\times n$ matrices whose entries are independent standard complex Gaussian random variables. The zeros in the unit disk of the determinan…

Uniqueness conjecture. For each $T>0$, $H(t)$ has a unique point of maximum in $[-T,T]$ almost surely.

Let $U_{NK}$ be the top-left $N\times K$ submatrix of an $M\times M$ Haar-random unitary matrix. Let $Z=Z_{N,K}$ be a random matrix distributed according to either…

Random regular graph singularity conjecture. For every $3\leq d\leq n-1$, $A_{n,d}$ is nonsingular with probability $1-o(1)$.

Positive-integrality conjecture. For $\beta\in\{1,2\}$, the expansion coefficients satisfy

Consider a determinantal point process on the real line with correlation kernel … Label its points by $\cdots<Z_{-1}<Z_0<Z_1<Z_2<\cdots$ so that $Z_{-1}<0\leq Z_0$. The sine-proces…

Let $m\in\mathbb{N}$ and $a_i\in M_m(\mathbb{Z})$ be given. Define the completely positive map … Assume that $\eta$ is rank non-decreasing. For $n$ independent GUE random matrices…

Real-eigenvalue count conjecture. With high probability, $M_n$ has $\Theta(\sqrt n)$ real eigenvalues.

Let $S_{\bullet}=\{S_N\}_{N\ge 1}$, and let $\chi^{\mu[\bullet]}=\{\chi^{\mu[N]}\}$ denote the stable irreducible character associated with a partition $\mu$. For a word $w$ in the…

Vu's distinct singular values conjecture. With probability $1-o(1)$, all the singular values of $M_n^{sym}$ are distinct.

Tracy–Widom GUE fluctuation conjecture. One has

Let $A$ and $B$ be independently chosen from the real Gaussian orthogonal ensemble $GOE_n^{\mathbb R}$, consisting of real symmetric $n\times n$ matrices with the standard GOE dist…

Wishart Gaussian product inequality conjecture.

Circular-law universality conjecture. For constant linear margins and every base measure $\mu$ satisfying the hypothesis of Corollary, the empirical eigenvalue distribution of

Let $M_n^{sym}$ be a symmetric random matrix whose entries are independent Rademacher variables. Vu's distinct singular values conjecture. With probability $1-o(1)$, all the singul…

Let $\mathcal{D}$ be the distribution of the disorder random variable, and suppose that it has finite second moment. Throughout the paper the disorder is assumed to be symmetric, w…

A real matrix is normal if it commutes with its adjoint; for real matrices this means … Let a sign matrix be a matrix whose entries belong to $\{\pm1\}$. Since every real symmetric…

Let $X=(X_{ij})_{1\leq i,j\leq n}$ be the random matrix in the setting of the section, with independent centered Gaussian entries as specified there, and let $\|X\|$ denote its ope…

Chalker–Mehlig conjecture. (i) For any two points $z_1,z_2$ such that $|z_1|<1$, $|z_2|<1$ and $|z_1-z_2|>0$,

Let $V$ be the double-well potential … where $g$ is a negative constant sufficiently close to zero, and let $\mu_t$ and $\mu_V$ denote the empirical measure process and the equilib…