10 problems

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…

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…

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

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.

Let $\Pi_\mathrm{L}$ be any transitive point process of intensity $1$, let $\Pi_\mathrm{GAF}$ be the zero set of a Gaussian analytic function with intensity $1$, and let…

Fix a point $o\in\mathbb{H}^2$. For each $r>0$, let $p_r$ be the least $p\in[0,1]$ such that every random closed set $\mathcal Z\subset\mathbb{H}^2$ with isometry-invariant law and…