Sharpness of phase transition for Voronoi percolation in hyperbolic space

Hyperbolic $d-$space ($d\geq 2$), denoted by $\mathbb{H}^{d}$, is the maximally symmetric, simply connected, $d-$dimensional Riemannian manifold with a constant negative sectional curvature. Typically, people study hyperbolic space by constructing models in order to utilize the Euclidean space to describe the hyperbolic space. In our following arguments, we choose the Poincaré ball model to represent $\mathbb{H}^{d}$ and now we are going to provide a brief summary of its definitions.

The Poincaré ball model can be constructed through equipping the open unit ball $\mathcal{B}^{d}=\{x\in\mathbb{R}^{d}:\|x\|<1\}$ with a hyperbolic distance $d_{p}$ and a metric tensor. Here, $\|\cdot\|$ represents the Euclidean norm.

For points $x,y\in\mathcal{B}^{d}$, the hyperbolic distance is defined by

where $\operatorname{arsinh}$ is the inverse function of hyperbolic sine.

The metric tensor in the Poincaré ball model is

In this paper we will consider both Euclidean balls and hyperbolic balls, and thus in the following context we will use subscripts to distinguish them. For example, we use $\operatorname{B}_{p}(x,r)$ and $\operatorname{B}_{e}(x,r)$ respectively to represent the hyperbolic ball and the Euclidean ball centered at $x$ with radius $r$. Similarly, the hyperbolic sphere centered at $x$ with radius $r$ is denoted by $\operatorname{\partial B}_{p}(x,r)$. Also notice that every hyperbolic sphere centered at the origin is also an Euclidean sphere centered at the origin. More precisely,

And we denote the hyperbolic annulus centered at the origin with inner radius $r$ and outer radius $R$ while containing the inner boundary by

For a countable set of points $\mathcal{Z}$ in the hyperbolic plane, we denote the corresponding hyperbolic Voronoi cell of $z\in\mathcal{Z}$ by

A homogeneous Poisson point process with constant intensity $\lambda$ on $\mathbb{H}^{d}$ can be viewed as an inhomogeneous Poisson point process on the ordinary Euclidean space with intensity

A gentle introduction to point process and Poisson point process can be found in [LP17] and in the rest of this paper we will usually use $\mathcal{Z}$ to denote a homogeneous Poisson point process with constant intensity $\lambda$ on $\mathbb{H}^{d}$.

After presenting the definition of Poisson point process on $\mathbb{H}^{d}$, we attach to each point in $\mathcal{Z}$ a randomly chosen colour independently. Precisely, every point in $\mathcal{Z}$ is coloured black with probability $p$ and white with probability $1-p$, and the colours of different points in $\mathcal{Z}$ are independent. By using $\mathcal{Z}_{b}$ (resp. $\mathcal{Z}_{w}$) to denote the black points (resp. white points) after the colouring process, we can also view $\mathcal{Z}_{b}$ and $\mathcal{Z}_{w}$ as independent Poisson point processes on $\mathbb{H}^{d}$ with respective intensities $\lambda\cdot p$ and $\lambda\cdot(1-p)$. Let $\mathbb{P}_{\lambda,p}$ denote the law of $(\mathcal{Z}_{b},\mathcal{Z}_{w})$, or equivalently the law of $\mathcal{Z}$ with its colours, then the measure $\mathbb{P}_{\lambda,p}$ induces a colouring $\omega$ of the hyperbolic space $\mathbb{H}^{d}$ defined as follows:

From now on we will use $\mathbb{P}_{p}$ for $\mathbb{P}_{\lambda,p}$ when it does not cause misunderstanding since we are going to prove Theorem 1 for every fixed $\lambda>0$.

For $x,y\in\mathbb{H}^{d}$, let $\{x\leftrightarrow y\}$ denote the event that there exists a continuous path of black points connecting $x$ to $y$, and let $\{0\leftrightarrow\infty\}$ denotes the event that the origin belongs to a infinite-volume connected component of black points. For two subsets $A$ and $B$ of $\mathbb{H}^{d}$, we use $\{A\leftrightarrow B\}$ to denote $\{\exists\ x\in A,\ y\in B\text{\ such\ that }x\leftrightarrow y\}$, and briefly write $\{x\leftrightarrow B\}$ for $\{\{x\}\leftrightarrow B\}$. Also for $p\in[0,1]$, $\lambda>0$, $n\geq 0$, define

Then we can abbreviate $\{0\ \text{is\ contained\ in\ an\ unbounded\ black\ connected\ component}\}$ as $\{0\leftrightarrow\infty\}$ and $\{0\ \text{is\ contained\ to\ distance\ }n\ \text{by\ a\ black\ path}\}$ as $\{0\leftrightarrow\operatorname{\partial B}_{p}(0,n)\}$.

An event $A$ is said to be $increasing$ if for every configurations $(\mathcal{Z}_{b},\mathcal{Z}_{w})$ and $(\mathcal{\tilde{Z}}_{b},\tilde{\mathcal{Z}}_{w})$,

An event $W$ is said to be $decreasing$ if its complement $W^{c}$ is increasing.

As in Bernoulli percolation model, we also have FKG inequality for Voronoi percolation in $\mathbb{H}^{d}$ listed in the following context. We will avoid the proof and refer the reader to [BR06b, Chapter 8] for more information about this lemma.

For an increasing event $A$ and a configuration $(\mathcal{Z}_{b},\mathcal{Z}_{w})$, we define the set of $pivotal\ points$ by

And define an increasing event $A$ $local$ if there exists $n\geq 0$ such that $A$ is measurable with respect to the $\sigma$-field generated by $\{\omega(x)\}_{x\in\operatorname{B}_{p}(0,n)}$.

The main idea of this lemma can be seen in [[DCRT19a], Lemma 4]. Here, we supplement the details for the proof of the integrability of $|\mathsf{Piv}A|$ and the continuity of $\mathbb{E}_{s}(|\mathsf{Piv}_{A}|)$ which is not mentioned in the former paper.

The OSSS inequality was introduced in [OSSS05] and states that for a given boolean function $f$ and a randomized algorithm $\operatorname{\mathsf{T}}$ determining $f$, the variance of this boolean function can be controlled by the revealments of each coordinate during the process of $\operatorname{\mathsf{T}}$ and the influences of coordinates.

Assume that $I$ is a countable set, $(\Omega^{I},\pi^{\otimes I})$ is a product probability space and $f$ is a function mapping $\Omega^{I}$ to $\{0,1\}$. A decision tree $\operatorname{\mathsf{T}}$ determining $f$ is an algorithm that takes $\omega=(\omega_{i})_{i\in I}\in\Omega^{I}$ as an input, and reveals the coordinates of $\omega$ step by step based on the values of $\omega$ revealed so far. The algorithm stops when the possible values of $f$ no longer depend on the values of the coordinates of $\omega$ that have not been revealed.

The revealment of the $i$-th coordinate is defined by

and the influence of the $i$-th coordinate is given by

where $\tilde{\omega}$ represents the random element in $\Omega^{I}$ which is the same as $\omega$ in every coordinate while has the $i$-th coordinate resampled independently.

The OSSS inequality is stated as follows.

Before turning back to the proof of our main theorem, we first introduce an appropriate product space to encode the measure of Voronoi percolation on the Poincaré disk model so that we are able to use Lemma 3. Here, to present the idea clearly, we will only focus on the case for $d=2$. And as for higher dimensions, using the polar coordinate to denote the volume of a measurable set and cutting the angles appropriately just like what we are going to do for $d=2$, we can also derive Lemma 4 and give the proof of the main theorem.

First note that for $d=2$ and every measurable subset $B\subset\mathbb{H}^{2}$, the area is given by

Now we are going to divide the hyperbolic space $\mathbb{H}^{2}$ up into countable disjoint hyperbolic annuli centered at the origin and cut each annulus into several sectors while making sure that the areas of these sectors are uniformly bounded.

Precisely, for a given $\epsilon>0$, we first regard $\mathbb{H}^{2}$ as unions of annuli of the form $\operatorname{Ann_{p}}(2k\epsilon,2(k+1)\epsilon)$, $k\geq 0$. Define $N_{k}^{\epsilon}=[\frac{\sinh((2k+1)\epsilon)}{\sinh\epsilon}]+1$. For each $k\geq 0$, we then divide $\operatorname{Ann_{p}}(2k\epsilon,2(k+1)\epsilon)$ into $N_{k}^{\epsilon}$ annulus sectors of the form

while making sure that each two of these sectors are disjoint and congruent.

Therefore, there exists a determined constant $C$ such that for every $\epsilon>0$ and each $k\geq 0$,

In order to denote annulus sectors clearly, we introduce the definition of representative points:

1. (1)

   For $k=0$, let $x_{0}^{\epsilon}=0$ be the representative point of the degenerate annulus $\operatorname{Ann_{p}}(0,2\epsilon)=\operatorname{B}_{p}(0,2\epsilon)$.

2. (2)

   For $k\geq 1$, define the representative point of $\operatorname{N_{p}}(2k\epsilon,2(k+1)\epsilon,\theta_{1},\theta_{2})$ as the exact point $re^{i\theta_{1}}$ satisfying $d_{p}(re^{i\theta_{1}},0)=2k\epsilon$. And denote the representative points contained in $\operatorname{Ann_{p}}(2k\epsilon,2(k+1)\epsilon)$ by $x_{k,l}^{\epsilon}$, $l=1,2,\cdots,N_{k}^{\epsilon}$.

Let $K_{\epsilon}$ be the set of all representative points, i.e., $K_{\epsilon}=\cup_{k=1}^{\infty}\cup_{l=1}^{N_{k}^{\epsilon}}\{x_{k,l}^{\epsilon}\}\cup\{x_{0}^{\epsilon}\}.$ Then for every annulus sector $\operatorname{N_{p}}(2k\epsilon,2(k+1)\epsilon,\theta_{1},\theta_{2})$, there exists a unique representative point, say, $x_{k,l}^{\epsilon}$ contained in this annulus sector. We then denote this region by $\operatorname{N}_{x_{k,l}^{\epsilon}}$ for simplicity.

For every $x\in K_{\epsilon}$, define

Let $(\Omega_{x},\pi_{x})$ be the space associated with the random variable $\mathcal{Z}_{x}=(\mathcal{Z}_{x}^{b},\mathcal{Z}_{x}^{w})$, and it is obvious that the random variable $\mathcal{Z}_{x}$ are independent for different $x$. Then the product space $(\prod_{x\in K_{\epsilon}}\operatorname{N}_{x},\otimes_{x\in K_{\epsilon}}\pi_{x})$ agrees with the original space and we will thus use $\mathbb{P}_{p}$ to denote the probability measure for the product space.

For every $x\in K_{\epsilon}$ and every increasing event $A$, we define

where $\mathcal{Z}=(\mathcal{Z}_{x})_{x\in K_{\epsilon}}$ has law $\otimes_{x\in K_{\epsilon}}\pi_{x}$, and $\tilde{\mathcal{Z}}$ is a random variable which equals to $\mathcal{Z}$ except on $x-$coordinate which is resampled independently.

Based on the space $(\prod_{x\in K_{\epsilon}}\operatorname{N}_{x},\mathbb{P}_{p})$ introduced in Section 3.1, we are now going to construct an algorithm $\operatorname{\mathsf{T}}_{k}$ determining $\mathbf{1}_{0\leftrightarrow\operatorname{\partial B}_{p}(0,n)}$ with its revealment having the following upper bound.

Before proving Theorem 1, we first introduce a lemma that converts the proof of Theorem 1 to showing a family of differential inequalities.

The proof of this lemma can be found in [[DCRT19a], Lemma 3.1].