285 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 $n,d\geq 1$, and let $x_1,\ldots,x_n$ be independent random vectors with … For an ellipsoid $\mathcal{E}$, say that it is an ellipsoid fit to $(x_i)_{i=1}^n$ if every point…

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

Four-root-of-unity anti-concentration conjecture. As $n\to\infty$,

Fix $\beta>0$. Let $g:\mathbb{R}\to\mathbb{R}$ be $2\pi$-periodic and belong to $C^{1,q}$ for some $0<q\leq 1$, let $t\in\mathbb{R}$, and let $I$ denote the integral whose normaliz…

Global rigidity lower-bound conjecture.

The large-$n$ expansion of the Toeplitz determinant is organized by the remainder $m$ of dividing $n$ by $2r+1$, and let $F^{\{0\}}$ denote the corresponding $O(1)$ coefficient as…

Let $\lambda^1(N),\lambda^2(N)\in\widehat U(N)$, for $N=1,2,\dots$, be two sequences of signatures satisfying the technical assumption of Definition $\mathrm{regularity}$, and supp…

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…

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

Let $P_d$ be the probability that a random $0/1$-matrix of size $(d+1)\times(d+1)$ is singular; equivalently, in the associated $\pm1$ model, let $P_d$ denote the probability that…

Let $M$ be an indecomposable, doubly stochastic matrix indexed by an alphabet of size $k$, satisfying … where $k+1\equiv1$. A word of length $N$ is generated by the Markov chain wi…

Let $M$ be a $\{-1,1\}$ matrix. Call $M$ good if every matrix obtained by independently switching any subset of its diagonal entries is nonsingular. Let $M_n$ be a random…

For a $\{-1,1\}$ matrix $M$, let $\operatorname{Res}(M)$ be the minimum number of entries that must be switched from $1$ to $-1$ or vice versa in order to make $M$ singular. Let…

Let $Q_n$ be the random symmetric matrix whose upper-diagonal entries are independent Bernoulli random variables. Symmetric determinant conjecture. Almost surely, … This is the sym…

Maximal determinant conjecture. Almost surely,

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…

Let $Q_n$ be the random symmetric matrix whose entries take values $1$ and $-1$, as in the stated model. Determinant growth conjecture. Almost surely, … This is the symmetric-matri…

Let $n$ be a large integer, and let $M_n$ be a random $n$ by $n$ matrix whose entries are independent Bernoulli random variables, each equal to $1$ or $-1$ with probability $1/2$.…

Dominant-pair conjecture.

Let $P(\mathcal{X}_n)$ and $P(\mathcal{D}_n)$ be the matrices associated with the random geometric graph and its deterministic comparison model, and let $\lambda_i(\mathcal{X}_n)$…

Real sample covariance conjecture. For real sample covariance, Theorem 1 should still hold, with different limiting distributions but the same scaling. In particular, the critical…

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

Asymptotical freeness conjecture. The sequences