184 problems

Let $P(U_1,\ldots,U_k,U_1^*,\ldots,U_k^*)$ be a noncommutative nonzero $*$-polynomial with complex coefficients, let $\underline{\lambda}\neq 0$ be a signature, and let…

For every $\beta>0$, the standard edge-scaled Jack–Plancherel process converges at the soft edge, in its multi-time finite-dimensional statistics (in particular, in moments), to th…

For non-Hermitian random band matrices of size $N$ and bandwidth $W_N$, the empirical spectral distribution converges to the circular law whenever $W_N\to\infty$.

Let $x_1,\ldots,x_n\in\mathbb{R}^d$ be independent standard Gaussian vectors, and let $n=n(d)$ as $d\to\infty$. The ellipsoid fitting conjecture predicts a sharp threshold at…

Let $H_N$ be an $N\times N$ random matrix from the standard Gaussian unitary ensemble or Gaussian orthogonal ensemble, and let $p_N(z)=\operatorname{Per}(zI_N-H_N)$. Define the nor…

Let $\alpha_{\mathbb{R}}(d)$ and $\alpha_{\mathbb{C}}(d)$ be the average singular value of a $d\times d$ matrix with random i.i.d. real-valued and complex-valued entries, respectiv…

Let $F_{\beta}(s,m)$ denote the distribution of the $m^{th}$ largest eigenvalue in the appropriate scaling limit for the Gaussian orthogonal ensemble ($\beta=1$) and Gaussian sympl…

Let $\lambda_{\max}^{(N)}(t)$ be the largest eigenvalue of an $N\times N$ matrix undergoing Dyson's Brownian motion with parameter $\beta=1$. Let ${\cal A}_{\rm 1}$ denote the Airy…

ASEPsc particle-position conjecture. For any $N\in\mathbb{Z}$ and every $s\in\mathbb{R}$,

Asymptotic independence conjecture. For every $x,y\in\mathbb{R}$,

Keating–Snaith moment conjecture. For every such $\boldsymbol{\kappa}$,

Let $L(s)$ be an $L$-function with nontrivial zeros written as $\rho=\frac12+i\gamma$, let $\tilde{\gamma}=\frac{\log c(L)}{2\pi}\gamma$, and for an even Schwartz function $\phi$ o…

Let $C$ be a smooth projective curve of genus $g$. Assume that $C$ is in the generic family. For each $k=1,2,\dots,g$, let $a_k(m;g)$ denote the $m$-th moment of the corresponding…

Let $\Sigma_1$ and $\Sigma_2$ be the covariance matrices, and let $H$ be their joint limiting spectral distribution. For any $n_1,n_2\in\mathbb{N}$ and every selection of non-negat…

Let $k,\gamma\in\mathbb{N}$ and $c>0$ be fixed, and define … Let $a_{k,\gamma}$ be the arithmetic constant and $g_{k,\gamma}$ the geometric constant. Bailey–Keating conjecture. … M…

Edge universality conjecture. We have

Borot–Eynard–Majumdar–Nadal conjecture. The left-tail asymptotic expansion is

Bohigas–Giannoni–Schmit conjecture. The eigenvalues are distributed like the eigenvalues of Hermitian random matrices, with the following limiting ensembles: without time-reversal…

Forrester's conjecture. As $s\to\infty$,

DMPK limiting-density conjecture. The density $\rho(x;\mathbb{M})$ is related to $\psi(x)$ by

Log-partition function conjecture. Almost surely,

Limiting-distribution conjecture. For every $a>0$ there is a Borel function $f_a:\mathbb{C}\to\mathbb{R}$ satisfying

Let $Z_X(s)$ be the truncated Hadamard factor, let $X,T\to\infty$ with $X=O((\log T)^{2-\epsilon})$, and let $k>-1/2$ be fixed. Let $G$ be the Barnes $G$-function and let $\gamma$…

Let $k$ be a positive integer, let $\theta(t)$ be the Riemann–Siegel theta function, and define Hardy's function by $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$. Let $a_k$ be the arithmetic f…

Let $Z_X(s)$ denote the truncated zeta-function used in the paper, and let $A>0$ be fixed. Truncated-zeta maximum conjecture. If $2<X<\log^A T$, then, as $T\to\infty$, … The conjec…