127 problems

High-dimensional realizability conjecture. The known standard non-negativity conditions on $g_2$ should be sufficient to ensure the existence of point processes in all space dimens…

Hard-core pair-correlation existence conjecture. It is possible to find a point process with density $\rho>0$ whose pair correlation function is this $g(r)$. This conjecture asks w…

Let $\beta>0$, and let $s$ be a random particle-generated Nevanlinna function satisfying the bulk condition in or the edge condition in. Under either condition, the poles of $s$ de…

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 $\Lambda$ be an $\operatorname{ASL}_d(\mathbb R)$-invariant point process. A cut-and-project alternative conjecture. Either $\Lambda$ is a Poisson point process, or there exist…

Let $H_n$ be a random Hamiltonian with an exponentially large collection of energy levels, and suppose the levels are properly rescaled. A REM universality conjecture asserts that,…

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.

For the Sine$_\beta$ point process at $\beta=2$, the two-point truncated correlation function is … A Berezinskii–Kosterlitz–Thouless transition conjecture. The critical value…

Let $M_{Q,t}/D/\infty$ denote the potentially coupled point-process sampler with deterministic service times, let $M_{Q,t}/M_t/\infty$ denote the corresponding potentially coupled…

Consider a wireless signal-propagation model with randomly positioned signal sources and reflections from walls, and suppose the model has sufficient randomness and regular walls.…

Spiking-process convergence conjecture. As $\varepsilon\to 0$, the point processes $(\mathbb Q^\varepsilon)_{\varepsilon>0}$ converge to the decorated Poisson point process…

Bulk Wigner eigenvector bead-process conjecture. The sequence $\left(\Theta_N^{(s)},\Upsilon_N^{(s)}\right)_{0\leq s\leq K}$ defines a coupled Markov chain, which converges in dist…

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 $\mathbb{R}$.…

Let $\mu$ be either the sine$_2$ point process or the Airy$_2$ point process, and let $\nu$ be a probability measure on the configuration space $\boldsymbol\Upsilon$. Suppose that…

Let $\mathrm{\oldXi}$ be a spatio-temporal point process satisfying Assumption … is sharp, with errors of order … This conjecture concerns the effect of temporal dependence on empi…

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 $d\geq 2$, and consider a polynomial perturbation of the lattice in dimension $d$, including perturbations whose displacement tails decay arbitrarily slowly. A translation inva…

Domain-extension conjecture. The same conclusions should hold for more general domains, say Lipschitz domains.

Maillard–Mallein conjecture. Any point measure $\zeta$ satisfying this invariance relation has the law of a randomly shifted decorated Poisson point process.

Let a Poisson point process (PPP) be given, and consider its minimal spanning forest (MSF), the infinite-volume analogue of the minimal spanning tree for random points. An Aldous–S…

Let a Poisson point process (PPP) be given in Euclidean space of dimension $d$, and construct the $\mathrm{CHN}^2$ EFF phylogenetic tree from it. A connectivity conjecture states t…

Let $\mathbb{A}$ be the underlying measurable space, and let $\mu$ be an s-finite valuation on $\mathbb{A}$. A Poisson point process with expectation measure $\mu$, denoted…

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…