Small-intensity face-count conjecture for hyperbolic Poisson-Voronoi percolation

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For each dimension dd, consider Poisson-Voronoi percolation on dd-dimensional hyperbolic space Hd{\mathbb H}^d, with intensity bbbb, and let the typical cell be the typical Poisson-Voronoi cell. Let its (d−1)(d-1)-face count be the number of its (d−1)(d-1)-dimensional faces. Small-intensity face-count conjecture. For every dd, as the intensity bb↘0bb\searrow 0, the critical value is asymptotically equal to the reciprocal of the expected number of (d−1)(d-1)-faces of the typical cell. The authors expect the planar small-intensity result to generalize to all dimensions, while noting that the conjecture might be proved or disproved without determining the expected face count precisely.

References

Primary source

Benjamin T. Hansen and Tobias Müller, “Poisson-Voronoi percolation in the hyperbolic plane with small intensities”, arXiv:2111.04299 (2023).

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