44 problems

Let $p$ be a prime, and consider the sequence of residue classes … More generally, these terms arise from sequences of values of a power, polynomial, or rational function modulo…

Shepp–Vanderbei's conjecture. Identified with its counting measure, this set converges as $n\to\infty$ to a Poisson point process. In particular, the closest root is typically at d…

Optimality conjecture. When $d>1$, these rates are also optimal. The rate is known to be optimal for $d=1$ in several related models, whereas the source leaves the higher-dimension…

Exponential-moment conjecture. In the supercritical case $alpha>1$, the random variable $(R^*)^d$ has a finite exponential moment; equivalently, there exists $c>0$ such that

Conjecture C.2. For every such $l$ and collection of constants,

Conjecture C.1. The point process on $\mathbb R_+^{k-1}$

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…

Triple-sum normal approximation conjecture. Under these modified locality and stabilization definitions, it should be possible to control the difference operators associated with t…

Let $bb^E_c(L)$ and $bb^E_u(L)$ denote the critical and uniqueness values for the Poisson stick process in $bb^d$, defined analogously to the corresponding hyperbolic quantities. E…

Let $bb^E_c(L)$ and $bb^E_u(L)$ denote the critical and uniqueness values for the Poisson stick process in $bb^d$, defined analogously to the corresponding hyperbolic quantities. E…

Let $f\in L^2_{\operatorname{imp}}$, let $(A_k)$ be pairwise disjoint Borel subsets of $\mathbb R_+$, and define $f_k=f\mathbf 1_{A_k}$. Let $(S_n)$ be the arrival times of a Poiss…

Let $\mathscr C$ denote the class of functions for which the Poisson series under consideration has the required convergence properties, and let an affine transformation of the var…

Let $N$ be a Poisson random measure and let $L^2_{\operatorname{imp}}$ be the space of bounded, square-integrable functions for which the relevant improper integrals exist. For a f…

Let $q^\gamma$ satisfy an evolution equation with two Poisson noises, … under the hypothesis … Here $\gamma_1\to\infty$ while $\gamma_2$ remains of order $1$, and the first Poisson…

Let $(q_t^\gamma)_{t\geq 0}$ be a piecewise one-dimensional Markov process evolving according to … where $\mathcal N^\gamma$ is a Poisson process with conditional intensity…

Let $F^{(m)}_r$ denote the $m$-th intersection functional of the Poisson $k$-flat process in hyperbolic $d$-space, with $2k>d+1$. A standard central limit theorem means convergence…

Let $N$ be the Poisson process with jump points $\sigma_i$ and let $(\sigma_i,\sigma_{i+1})$ be an interval containing the free interval. Let $M_t$ denote the martingale term appea…

Let $V$ be a set with one Poisson particle source for each element, and let a degree-$r$ arrival experiment receive particles in successive batches of $r$, silencing each source af…

Let $\Phi(r)$ be the increasing function defined by the $L^1$ limit for maximal decreasing subsets of a homogeneous Poisson point process in the sloped rectangle: for $r\ge 0$, if…

Consider a Poisson process of $k$-flats in hyperbolic space $\mathbb{H}^d$, where $d\geq 4$ and $k\in\{0,1,\ldots,d-1\}$. For $r>0$ and $m\in\mathbb{N}$ satisfying … let…

The paper considers approximating deterministic decision times in the classical discrete-time model by Poisson arrival times, thereby using the Gittins index policy from the Poisso…

The Poisson equivalence-relation conjecture. The group $G$ is sofic if and only if the measured equivalence relation $\mathcal{R}$ is sofic. This is presented as a locally compact…

The paper studies random matrices whose limiting behavior is governed by Poisson statistics. Poisson covariance conjecture. A phenomenon arising from the covariance formula for Poi…

Let ${\mathcal L}$ be a fixed lattice, let {\mathcal P}={\mathcal L}^*+{\text{\boldmathalpha}}, and let {\text{\boldmathalpha}} be a translation parameter. For $n\geq 1$,…

Next-order asymptotic conjecture. The next term in the asymptotic expansion of $\nu(t)$ is of order $t^{-1/2}$.