High-intensity hyperbolic Voronoi crossing limits

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

In the Poincare disc $\mathbb D$ with hyperbolic metric, sample a Poisson point process of intensity $\lambda>0$ per unit hyperbolic area. Partition $\mathbb D$ into its hyperbolic Voronoi cells and color each cell black with probability $1/2$, independently. Four distinct ideal-boundary points $a,b,c,d$ in cyclic order determine open arcs $A=(a,b)$ and $C=(c,d)$. Let $q_\lambda(a,b,c,d)$ be the annealed probability, averaging points and colors, that a connected black crossing joins these two arcs, meaning that a connected black set has closure in $\overline{\mathbb D}$ meeting both arcs. Let $x\in(0,1)$ be their conformal cross-ratio, characterized by a conformal map from the disc to the upper half-plane taking $(a,b,c,d)$ to $(0,x,1,\infty)$. Cardy's crossing function is \[F(x)=\frac{\Gamma(2/3)}{\Gamma(1/3)\Gamma(4/3)}x^{1/3}\,{}_2F_1(1/3,2/3;4/3;x),\] where $\Gamma$ and ${}_2F_1$ are the gamma and Gauss hypergeometric functions.

Statement recorded in the supplied source.

Prove that $q_\lambda(a,b,c,d)\to F(x)$ as $\lambda\to\infty$. In particular, prove existence and conformal invariance of the limit and that for each nondegenerate choice of arcs it lies strictly between $0$ and $1$. [S2]

Short English statement

Show that high-density hyperbolic Voronoi crossings converge to Cardy's conformally invariant probability.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000055 (AMR-099-0055). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Percolation and Coarse Conformal Uniformization, June 2015; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Section 3, boundary-crossing construction and limit conjectures, pp. 4--5. https://arquivo.pt/noFrame/replay/20201231041529id_/http://www.wisdom.weizmann.ac.il/~itai/resistance_percolation.pdf.