Continuous vs discrete spins in the hyperbolic plane

The hyperbolic Brownian plane

We introduce and study a new random surface which we call the hyperbolic Brownian plane and which is the near-critical scaling limit of the hyperbolic triangulations constructed in arXiv:1401.3297. The law of the hyperbolic Brownian plane is obtained after biasing the law of the Brownian plane arXiv:1204.5921 by an explicit martingale depending on its perimeter and volume processes studied in arXiv:1409.4026. Although the hyperbolic Brownian plane has the same local properties as those of the Brownian plane, its large scale structure is much different since we prove e.g. that is has exponentia…

Right-angled hexagon tilings of the hyperbolic plane

We study isometry-invariant probability measures on the space $Ω$ of tilings of the hyperbolic plane with right-angled hexagons of varying shapes. We prove that, for each measure $μ$ in a certain natural family of measures on right-angled hexagons, there is an isometry-invariant measure on $Ω$ whose marginal distribution on tiles is $μ$.

通过半平面共存实现抛物-双曲分歧

Parabolic-hyperbolic dichotomy through half-plane coexistence

研究半平面共存对随机平面图的抛物性和双曲性影响，揭示URM的抛物-双曲分裂机制。

Poisson-Voronoi percolation in the hyperbolic plane with small intensities

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $πλ/3$ as $λ\to0.$ This answers a question of Benjamini and Schramm.

The critical probability for Voronoi percolation in the hyperbolic plane tends to $1/2$

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to $1/2$ as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm.

Hausdorff dimension of visibility sets for well-behaved continuum percolation in the hyperbolic plane

Let Z be a so-called well-behaved percolation, i.e. a certain random closed set in the hyperbolic plane, whose law is invariant under all isometries; for example the covered region in a Poisson Boolean model. The Hausdorff-dimension of the set of directions is determined in terms of the $α$-value of Z in which visibility from a fixed point to the ideal boundary of the hyperbolic plane is possible within Z. Moreover, the Hausdorff-dimension of the set of (hyperbolic) lines through a fixed point contained in Z is calculated. Thereby several conjectures raised by Benjamini, Jonasson, Schramm and …