940 problems

Let $T_r$ denote the absorption time in the discrete Whittaker process, let $x\in\mathbb R$, and let $F_1$ denote the Tracy–Widom GOE distribution. Tracy–Widom GOE conjecture. As…

Let $\mathsf{L}_N$ be the C$\beta$E field on $\mathbb{T}$, with parameter $\beta>0$, and suppose the critical relation $\beta=2s^2$ holds. Let $\mathfrak{G}_\beta$ denote the expli…

Let $\beta>0$, $k\in\mathbb{N}$, and $s\in\mathbb{R}_+$. Let $\mathsf{M}^{(\beta)}_N(k;s)$ denote the corresponding C$\beta$E field moment, and let $\mathfrak{c}^{(\beta)}(k;s)$ be…

Let $A_n$ denote the adjacency matrix of $WS_n(K_n,p_n)$, where … and suppose that $K_np_n\xrightarrow[n\to\infty]{}C$ for a constant $C\in(0,\infty)$. Finite-degree limiting spect…

Shub's conjecture. There is a constant $c_{d,k}$ such that

Let $M_n$ be drawn from the fixed-trace Gaussian Unitary Ensemble, and let $\lambda_1^n<\cdots<\lambda_n^n$ be the eigenvalues of $M_n/\sqrt n$. Define the centered empirical measu…

For a Fisher–Hartwig determinant of size $N$, the relevant singularities are expected to produce additional logarithmic terms in its asymptotic expansion. Fisher–Hartwig's conjectu…

Let $A_n$ be an $n\times n$ random matrix whose entries are independent and identically distributed random variables taking values in $\{\pm1\}$. Let $s_n(A_n)$ denote its smallest…

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…

Invertibility conjecture. $\mathfrak A_k$ is invertible for all parameters $\mu_i\in[0,1]$.

Let $(U_t)_{t\geq 0}$ and $(G_t)_{t\geq 0}$ denote the unitary and general linear multiplicative Brownian motions, respectively. The process $(U_t)_{t\geq 0}$ converges in $*$-dist…

Let $G$ be the adjacency matrix of the Erdős–Rényi graph $G(n,1/2)$. Babai's conjecture. The matrix $G$ has no repeated eigenvalues with probability $1-o(1)$. This conjecture conce…

Kahle–Newman conjecture. The distribution of the Sylow $p$-subgroup of $H_1(C_n)$ follows the Cohen–Lenstra distribution.

Nguyen's sharp singularity conjecture. The singularity probability satisfies

Let $\rho_A$ be the reduced density matrix obtained by partially tracing one factor of a random Gaussian vector in $\mathbb{C}^D\otimes\mathbb{C}^D$, with the normalization convent…

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 $p_{\frac{1}{2}}(n)$ denote the probability that an $n\times n$ signed Bernoulli matrix with independent entries, each equal to $1$ or $-1$ with probability $\frac{1}{2}$, has…

Let $A$ be the sum of $d$ independent uniformly random $n\times n$ permutation matrices, with $d$ fixed as $n\to\infty$. The oriented Kesten–McKay conjecture. The empirical spectra…

Let a quantum chaotic system have highly excited eigenfunctions, and suppose the system is sufficiently disordered or quantum chaotic. Eigenstate Thermalization Hypothesis. The hig…

An ensemble of random matrices $A^N$ has independent centered $L^2$ entries, each with variance $1/N$. Girko's circular law conjecture. The empirical measure of the eigenvalues of…

Circular law conjecture. The ESD of such matrices converges to a universal limiting law, namely the uniform distribution on the unit disk in the complex plane.

Let $A$ be a Gaussian random band matrix with bandwidth $W$, normalized so that its eigenvalues $E$ are concentrated in $[-2,2]$ and its asymptotic density of states is expected to…

Consider non-Hermitian random matrices with independent and identically distributed entries. Their microscopic eigenvalue statistics in the bulk and at the soft edge exhibit the sa…

Kahn–Komlós–Szemerédi conjecture.

Let $3\leq d\leq n-3$, and let the adjacency matrix be chosen from a uniformly random $d$-regular undirected graph on $n$ vertices. Vu's nonsingularity conjecture. With probability…