25 problems

Nguyen's sharp singularity conjecture. The singularity probability satisfies

Let $B=(B_{ij})$ be an $n\times n$ random matrix whose entries are independent Bernoulli random variables with … When $p$ is sufficiently small, zero rows and columns are expected…

Let $n$ be even, and let $Q_n$ be a random $n$ by $n$ $(0,1)$ matrix whose rows are independent vectors with exactly $n/2$ zero components. Fixed-row-sum singularity probability co…

Let $Q(n,1/2)$ be the adjacency matrix of an Erdős–Rényi random graph with edge probability $1/2$, and let $p_n^{\mathrm{sym}}$ be the probability that $Q(n,1/2)$ is singular. Symm…

Singularity-obstruction conjecture. If $c\in(0,1/2)$ is fixed and

Let $A_n(x)$ be an $n\times n$ random matrix whose entries are independent copies of a discrete random variable $x$. A discrete random matrix can be singular because it contains a…

Let $A=(a_{ij})_{1\leq i,j\leq n}$ be a random symmetric matrix whose entries above and on the diagonal are independent identically distributed mean-zero, unit-variance subgaussian…

Litvak–Tikhomirov conjecture. Then

Let $B_n(p)$ be an $n\times n$ random matrix with independent and identically distributed Bernoulli entries of mean $p$, and let $q_n(p)=\mathbb{P}[B_n(p)\text{ is singular}]$. Loc…

Symmetric singularity asymptotic.

Weiss's singularity conjecture.

Non-symmetric singularity conjecture.

For each $n$, let $p_n\in(0,1/2]$, and let $M_n$ be an $n\times n$ random matrix with independent identically distributed Bernoulli($p_n$) entries, meaning entries in $\{0,1\}$ wit…

For each $n$, let $B_n$ be an $n\times n$ random matrix with independent identically distributed entries taking the values $-1$ and $1$ with equal probability. Two rows or two colu…

Let $M_n$ be a random $n\times n$ symmetric matrix whose upper-diagonal entries are independent Bernoulli random variables taking the values $1$ and $-1$ with probability $1/2$ eac…

Let $M_n$ be the symmetric adjacency matrix of a uniformly random undirected $d$-regular graph on $n$ vertices. Costello–Vu conjecture. For every $3\leq d\leq n-3$, the matrix…

Let $M_n^{\mathrm{sym}}$ be a random symmetric matrix whose upper-diagonal entries are iid Bernoulli variables, and let $p_n^{\mathrm{sym}}$ denote its probability of being singula…

Let $M_n$ be an $n\times n$ random matrix with iid Bernoulli entries, and let $p_n={\mathbf P}(M_n\text{ is singular})$. Erdős–Kahn–Komlós–Szemerédi conjecture. … This is described…

Let $M=M_n$ be a random $n\times n$ Bernoulli matrix, let $S$ be the event that $M$ is singular, and let $\lambda(1),\lambda(2),\ldots$ be any enumeration of the set of novel integ…

Let $M=M_n$ be an $n\times n$ random Bernoulli matrix whose entries are independently chosen from $\{−1,1\}$, and let $S$ be the event that $M$ is singular. A null vector has templ…

Let $H$ be a symmetric random matrix satisfying the hypotheses of the source's delocalization result, and suppose its singularity-probability estimate contains the term $2e^{-n^c}$…

Let $H$ be an $n\times n$ symmetric Bernoulli matrix: the entries on and above the diagonal are independent symmetric random variables taking values in $\{\pm1\}$. The optimal sing…

Let $B$ be an $n\times n$ random Bernoulli matrix whose entries are independent symmetric random variables taking values in $\{\pm1\}$. The optimal singularity probability conjectu…

Let $A$ be an $n\times n$ random Bernoulli matrix, meaning that its entries are independent symmetric random variables taking values $\pm1$. Write … for its singularity probability…

Let $M_n$ be an $n\times n$ random matrix whose entries are independent and identically distributed Bernoulli random variables taking the values $0$ and $1$ with probability $1/2$…