29 problems

Let the tridiagonal random matrix models of Dumitriu and Edelman be scaled appropriately so that they approximate random stochastic operators. Edelman–Sutton conjecture. Scaling li…

Macdonald's conjecture. The integral identity

Let $\beta>0$, and let the Gaussian beta ensemble G$\beta$E have joint density … Let $t^n_{k,\beta}$ be its $k$-th smallest gap. Small-gap conjecture. There exists a constant…

Let $\Lambda_n\sim \mathrm{Laguerre}_{n,\beta,a_n}$ be as in Theorem … liminflimits{ntoinfty}frac{an}{n}0 … should remain valid without assuming … The conjecture concerns the regim…

Let $\mathsf{Ht}_\mathcal N$ be the random height function of the model at scale $\mathcal N$, let $\mathbb E[\mathsf{Ht}_\mathcal N]$ be its expectation, let $\pounds$ denote the…

Conjectured orders. The first probability has order $O\Bigl(N\frac{(\ln k)^2}{k}\Bigr)$, while the second has order $O\Bigl(N\frac{\ln k}{k^{c^2}}\Bigr)$ for $0<c<1$. These orders…

Let $\beta$ be the inverse temperature of a Gaussian $\beta$-ensemble with $n$ particles, and let the separation be the smallest distance between two distinct particles. The cited…

Let $\beta>0$ and let $\gamma$ be a Jordan curve with logarithmic capacity $\text{cap}(\gamma)$. For a function $g$ on $\gamma$, let $D_{n,\beta}[e^g]$ be the corresponding beta-en…

Stochastic Airy universality conjecture. The suitably rescaled characteristic polynomial of a $b2$-ensemble with a soft edge converges to $a3Ai$ in a neighborhood of a spectral edg…

Let $\beta>0$. In the two limiting results for the largest eigenvalues of the beta-Gaussian corners process and Dyson Brownian motion, respectively, the limits are obtained by firs…

Randomized semidefinite-relaxation sampling conjecture. Semidefinite relaxations of multivariate moment problems with properly randomized constraints should lead to efficient sampl…

Let $(\boldsymbol{y}^{(T)})_{T\in\mathbb{Z}_+}$ be distributed as in the paper's local-fluctuation setup, and consider the rescaled point process … Here $\widehat{T}>0$ is rescaled…

For arbitrary $\beta>0$, consider the right-edge fluctuation point process of the beta-Jacobi product process in the regime where time $T$ grows linearly with $N$, with rescaled ti…

Let $\beta>0$, let $\boldsymbol{x}$ be fixed, and let $\boldsymbol{y}$ be a generally distributed vector for which the Jack-function relation … holds, where $\mathbb{Y}_N$ is the s…

Let the circular $\beta$-ensemble have Dyson parameter $\beta>0$, and consider the extreme values of its characteristic polynomial as the matrix size $N$ tends to infinity. The asy…

Exotic edge-operator conjecture. After the appropriate scaling, $T_n$ converges to the random operator $\mathcal{S}_{\beta,k}$. The source presents this as a conjecture and refers…

Let $c$ be the constant in the corollary giving a lower bound for the distance between neighbouring particles, under the regularity and probability assumptions stated there. Optima…

Let $\{z_j\}_1^n$ be the system of particles in the $\beta$-ensemble, and suppose the conditions of the corollary hold: $c<1/(8\sqrt{e})$, together with the regularity condition…

Intensity–density conjecture. The intensity $\theta_E$ should agree with the density of states $\bar\mu_\alpha(E)$.

The Wishart $beta$-ensemble has number variance in the edge regime described by the edge scaling variable $s=(L-4)(N/4)^{2/3}$. The limits $s\to\pm\infty$ are expected to match the…

The paper considers Dyson beta ensembles and related canonically quantized Painlevé or Garnier equations, with Dyson parameter $\beta$ and special regimes $\beta=2$, $\beta=0$, and…

Generalized Witte–Forrester conjecture. For $n\geq 1$ and even $g\geq 0$,

Witte–Forrester conjecture. For even $l\leq 2$,

Large-argument hypergeometric asymptotic conjecture. For large values of the arguments $\{s_j\}_{j=0,\dots,n}$, the generalized hypergeometric function has the asymptotic form give…

Hard-edge gap-ratio asymptotic conjecture. As $s\to\infty$,