Matthias Irlbeck

Interpretation

The diameter and mean width are each asymptotic to $2R$, while the largest inscribed ball has only about half the enclosing ball’s radius. Uniformly over all vertices, their distances from the origin divided by $R$ tend to one in probability.

Yet the whole cell remains far from spherical: with probability tending to one, its Hausdorff distance from every ball is at least a fixed positive fraction of $D$.

The facets also vary greatly in size. For every fixed $\varepsilon>0$, the proportion, by number, of facets with diameter larger than $\varepsilon D$ tends to zero in probability. Nevertheless, with probability tending to one, some facets have diameter at least $cD$ for a fixed $c>0$. The preprint develops this picture further for faces of other dimensions.

Interpretation

The constant comes from continuum AB percolation: two independent Poisson point clouds in $\mathbb{R}^d$, each of intensity $\lambda$, with edges between opposite types at Euclidean distance at most one. If $\lambda_{\mathrm{AB}}(d)$ is its critical intensity and $\sigma_d$ is the surface area of $S^d$, then

Applying the antipodal map $x\mapsto-x$ to the points in one of two small antipodal caps and rescaling gives a local approximation by this model. The proof joins local connections above its percolation threshold to create a global odd cycle on the sphere, the obstruction to two-colouring.

Interpretation

The ambient geometry makes a decisive difference. D’Achille, Grebík, Khezeli, Recke and Wilkens show that the uniqueness threshold tends to zero as the site intensity tends to zero on the Riemannian product $\mathbb{H}^2\times\mathbb{H}^2$. Our result rules out this behaviour in $\mathbb{H}^d$ for $d\geq3$.

The proof bounds the probability that two distant points belong to the same black cluster. At low intensities, it extracts reduced paths from black-cell connections and bounds their expected counts for paths first reaching a distant region. The comparison uses paths with no guaranteed edge between nonconsecutive vertices in an auxiliary model whose edge choices are independent once point locations are fixed. A separate estimate handles larger intensities. Together these estimates rule out uniqueness at small $p$.

Interpretation

Exploring a black cluster means repeatedly discovering new black neighbours. A branching-process approximation treats those discoveries as independent and predicts survival when their mean exceeds one, suggesting the threshold $p\,m_d\approx1$.

For every fixed $\varepsilon\in(0,1)$ and all sufficiently large $d$, there is almost surely no unbounded black cluster at $p=(1-\varepsilon)/m_d$, whereas an unbounded cluster exists with positive probability at $p=(1+\varepsilon)/m_d$.

Making this prediction rigorous requires handling unbounded random degrees and spatial dependencies. For the upper bound, we explore only selected connections, couple these finite explorations to branching random walks, and join them through comparison with site percolation on $\mathbb{Z}^2$. For the lower bound, we count paths in a more generous adjacency model that includes every actual Voronoi connection. The complete proof appears in Chapter 3 of my PhD thesis.