28 problems

Let $d$ be the dimension, let $u$ be a probability measure on positive radii with finite $d$th moment, and let … where $\lambda^c_d(\nu)$ is the critical intensity of the Boolean…

Let $\alpha_c^{(k)}(d)$ denote the critical value of the parameter $\alpha$ for percolation in the GN$_k(d,\alpha)$ model in dimension $d$, with integer $k\geq 1$. High-dimensional…

Consider the GN$(1,\underline{\alpha})$ model on a Poisson process on $\mathbb{R}$, where $\alpha_i$ are the connection parameters and clusters are formed according to the model's…

Ashkin–Teller magnetisation-interface percolation conjecture. The law of the outermost interfaces of the Ashkin–Teller magnetisation field is the law of the outermost outer boundar…

Two-edge conjecture. Theorem remains valid for $epsilon=0$; consequently,

Phase-coincidence conjecture. The subcritical and quantitatively subcritical phases agree.

Percolation criterion. There are an $\epsilon>0$ and an $N \in \mathbb{N}$ such that, whenever $\mathbb{P}$ is an automorphism-invariant positively associated percolation measure o…

Let $I_c>0$ denote the critical intensity in the phase transition theorem for $I=(\mu+\mu)\cdot\lambda$, and consider the Boolean model generated by a homogeneous Poisson point pro…

Let $bb>0$ and $bc_x>0$, and let $bc_y(bc_x,bb)>0$ denote a street intensity. Write $Xi(bc_x,bc_y,bb)$ for the Manhattan grid model (MGM), and let $bb_c(bc_x,bc_y)$ be its critical…

Spherical hyperplate conjecture. Among all convex hyperplates, spherical hyperplates have the largest percolation thresholds for any fixed $d\geq 3$.

Percolation-threshold extremal conjecture. Overlapping spheres possess the maximal value of $\eta_c$ among all identical nonzero-volume convex overlapping bodies, randomly or unifo…

Let $\mathcal P^\circ=\{x_n^\circ\}_n$ be a Poisson process, and let $\mathcal P=\{x_n\}_n$ be one of the Matérn I, II, or III hardcore processes obtained from it by the thinning r…

Let $\mathcal P=\{x_n\}_n$ be a Poisson process in $\mathbb R^d$, with associated Voronoi tessellation $\mathcal V=\{V_n\}_n$. Write $r_n$ for the diameter of $V_n$, and define … F…

Consider the continuum model formed by a Poisson cloud of Brownian motion paths or Wiener sausages, truncated at time $t$, with intensity parameter $\lambda>0$. Let continuous rand…

Let $z>0$ be the activity and let $\beta\geq 0$ be the inverse temperature of the continuum Widom–Rowlinson (area-interaction) model. The model has a unique Gibbs measure in the un…

Large-activity monochromaticity conjecture. For activities $z$ large enough, every Widom–Rowlinson measure is monochromatic. The paper proves the existence of polychromatic phases…

Consider the symmetric Widom–Rowlinson model in dimension $d\geq 2$, with radius distribution $Q$ satisfying the integrability assumption and with $Q(\{0\})<1$. Small-atom extensio…

Let $WR(\boldsymbol{z},\boldsymbol{Q})$ denote the set of Widom–Rowlinson measures in the non-integrable setting, with $q$ ordered phases. Large-activity ordered-phase conjecture.…

Consider continuum percolation models formed from convex shapes, and let the percolation threshold measure the critical volume fraction for percolation. Torquato's hypersphere-maxi…

Large-activity uniqueness conjecture. There exists $z_1>0$ such that, for all $z>z_1$, there exists a unique stationary $\operatorname{CRCM}(z,Q,q)$, and this measure is…

Maximum-threshold conjecture. The threshold among all such systems is maximized by hyperspheres:

Let $R\geq r_1$, and let $\mu_-(\beta;R)$ and $\mu_+(\beta;R)$ denote, respectively, the lower and upper grand-canonical chemical-potential thresholds for the existence of an infin…

Let $\Phi$ be a homogeneous sub-Poisson point process with intensity $a>0$, and let $\Phi_a$ be the homogeneous Poisson point process of the same intensity. Write $r_c(\Phi)$ for t…

Shape-noise-sensitivity conjecture. If $S$ is bounded and simply connected, then the Poisson Boolean model for $S$ is noise sensitive at criticality.

Random-radius noise-sensitivity conjecture. For every $R>0$ and $\mu_R$, the Poisson Boolean model with random radii chosen according to $\mu_R$ is noise sensitive at criticality.