39 problems

Approximation conjecture. There exists $\delta\in(0,\infty)$ such that, for every $\Delta\in(0,\delta)$, the process above is a stationary INARMA process, and the family of point p…

Retention-probability conjecture. The function $p_{\lambda,\alpha}(\cdot)$ in the thinning representation is

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.

A high-dimensional random walk on a discrete point process is a random walk on such a point process in sufficiently large dimension. Failure conjecture. There are random walks on d…

A two-dimensional random walk on a discrete point process is a random walk in dimension $2$ whose state space is a discrete point process. Existence conjecture. There are transient…

Torquato–Stillinger conjecture. The function $g_2({\bf r})$ is a pair correlation function of a translationally invariant disordered sphere packing in $\mathbb{R}^d$ at number dens…

Let $\Lambda$ be an $\operatorname{ASL}_d(\mathbb R)$-invariant point process with distribution $\mu$. Let $\mathcal{M}(\mathbb R^d)$ denote the space on which the point-process di…

Let a rational arrival process (RAP) have joint densities on $(0,\infty)^k$ for every $k\ge 1$, and let $\bm{G}_0$ be the matrix in a RAP representation governing the evolution bet…

Matching conjecture. For all $a\geq 0$, the joint distributions of $(\hat{\mu}_a^{\infty}(i),\;i\geq 1)$ and $(\mu_a^{\infty}(i),\;i\geq 1)$ are equal. As a corollary, using the ex…

Poisson-limit conjecture. When $a\to\infty$, the rescaled point process $\mathcal{M}_a$ converges towards a Poisson point process.

Let $\mathsf{P}_{s,\beta}$ denote a one-dimensional non-singular Riesz gas with homogeneity parameter $s$, and let $\mathsf{Leb}_{\mathbb{R}}$ be Lebesgue measure on $\mathbb{R}$.…

For each $\lambda>0$, let $\Xi_{\lambda}^{(0)}$ be a simple stationary renewal process on $\mathbb{R}$ satisfying the functional strong law of large numbers, and let…

Let $\operatorname{Bessel}_{\beta,a}$ be the hard-edge point process defined by the $r=1$, $\gamma=1$ limiting operator, and consider the solvable ensembles with the explicit joint…

Let $r\geq 2$ and let $\beta=1$ or $2$. Consider the point process associated with the $r=2$, $\beta s=2$ joint density in Theorem 1, and the point process associated with the…

Poisson point-process and Gumbel conjecture. The point process

Let $n>1$ and consider the distance from a typical general vehicle to its $n$th-nearest neighbor in the PLM-PPP and in the PSP-PPP, with Rayleigh-distributed half-lengths. Nearest-…

Let $\mathbb{T}_0$ be the domain of the ordered $k$-tuples used in the paper, and let $p_k(\mathbf{x},\mathbf{y})$ denote the corresponding joint density. Sharp density conjecture.…

Let $f_n(z)=\sum_{j=0}^n\varepsilon_j z^j$, where $\{\varepsilon_j\}_j$ are iid real random variables satisfying $\mathbb{E}\,\varepsilon_1=0$ and $\mathbb{E}\,\varepsilon_1^2=1$.…

Let $(\beta_n)$ be a sequence of inverse temperatures, and consider the Coulomb gas configurations at these temperatures. Sharpness conjecture. If almost sure uniform separation ho…

GUE hypothesis. For $n\geq 1$ and any $\varphi\in\mathcal{S}(\mathbb{R}^n)$,

An MMPP is a Markovian modulated Poisson process; write $c^2$ for the squared coefficient of variation of its interarrival time. Squared coefficient-of-variation conjecture. For an…

Local-limit conjecture. There exists a rotationally symmetric, translation-invariant point process $Pi$ on $\mathbb{R}^2$ such that, for every $(x,y)$ in the open unit disk,

Let $F$ be a displacement distribution and let $N_0$ be the random counting measure constructed from the renewal and random-walk occupation measures in the paper, with distribution…

Let $F$ be an absolutely continuous displacement distribution, let $\lambda>0$, and let $B\in\mathcal{B}_b(\operatorname{\mathbb{R}})$. Let $N^{(g)}$ and $\lambda^{(g)}$ be defined…