Uniqueness of the Infinite Component for Percolation on a Hierarchical Lattice

Abstract

We study a long-range percolation in the hierarchical lattice Ω_𝑁 of order 𝑁 where probability of connection between two nodes separated by distance 𝑘 is of the form min{𝛼𝛽^−𝑘,1}, 𝛼≥0 and 𝛽>0. We show the uniqueness of the infinite component for this model.

1. Introduction

Percolation theory in the Euclidean lattice ℤ^𝑑 started with the work of Broadbent and Hammersley in 1957. The infinity of the space of sites (or vertices) and its geometry are principal features of this model, see for example [1, 2]. Some questions of percolation in other non-Euclidean infinite systems are formulated in [3]. The study of long-range percolation on ℤ^𝑑 traces back to [4] and leads to a range of interesting results in probability theory and statistical physics [5–9]. On the other hand, hierarchical structures have been used in applications in the physics, genetics, and social sciences thanks to the multiscale organization of many natural objects [10–13].

Recently, long-range percolation is studied on the hierarchical lattice Ω_𝑁 of order 𝑁 (to be defined below), where classical methods for the usual lattice break down. The asymptotic long-range percolation on Ω_𝑁 is addressed in [14] for 𝑁→∞. The works [15–17], analyze the phase transition of long-range percolation on Ω_𝑁 for finite 𝑁 using different connection probabilities and methodologies. The contact process on Ω_𝑁 for fixed 𝑁 has been investigated in [18]. In this paper, we investigate the question of uniqueness of infinite component in percolation on Ω_𝑁 for fixed 𝑁. The form of the connection probabilities used here follows from a prior work [17].

For an integer 𝑁≥2, we define the set Ω_𝑁∶=𝑥𝐱=𝑥₁,𝑥₂,…∶𝑥_𝑖∈{0,1,…,𝑁−1},𝑖=1,2,…,_𝑖≠0onlyforfinitelymany𝑖,(1.1) and define a metric 𝑑 on it: 𝑑(𝐱,𝐲)=0m,ax𝑖∶𝑥_𝑖𝐱=𝐲,≠𝑦_𝑖,𝐱≠𝐲.(1.2) The pair (Ω_𝑁,𝑑) is called the hierarchical lattice of order 𝑁, which may be thought of as the set of leaves at the bottom of an infinite regular tree without a root, where the distance between two nodes is the number of levels (generations) from the bottom to their most recent common ancestor. Figure 1 shows the lattice Ω₂ along with its metric generating tree.

Figure 1 

An illustration of hierarchical lattice Ω2 of order 2. The distances between three nodes 𝟎=(0,0,0,…), 𝐱=(1,0,0,…) and 𝐲=(0,1,0,…) are 𝑑(𝟎,𝐱)=1 and 𝑑(𝟎,𝐲)=𝑑(𝐱,𝐲)=2.

Such a distance 𝑑 satisfies the strong triangle inequality 𝑑(𝐱,𝐲)≤max{𝑑(𝐱,𝐳),𝑑(𝐳,𝐲)},(1.3) for any triple 𝐱,𝐲,𝐳∈Ω_𝑁. Hence, (Ω_𝑁,𝑑) is an ultrametric (or non-Archimedean) space [19]. From its ultrametricity, it is clear that for every 𝐱∈Ω_𝑁 that there are (𝑁−1)𝑁^𝑘−1 nodes at distance 𝑘 from it.

Now consider a long-range percolation on Ω_𝑁. For each 𝑘≥1, the probability of connection between 𝐱 and 𝐲 such that 𝑑(𝐱,𝐲)=𝑘 is given by 𝑝_𝑘=min𝛼𝛽^𝑘,1,(1.4) where 0≤𝛼<∞ and 0<𝛽<∞, all connections being independent. Two vertices 𝐱,𝐲∈Ω_𝑁 are in the same component if there exists a finite sequence 𝐱=𝐱₀,𝐱₁,…,𝐱_𝑛=𝐲 of vertices such that each pair (𝐱_𝑖−1,𝐱_𝑖), 𝑖=1,…,𝑛, of vertices presents an edge.

The rest of the paper is organized as follows. In Section 2, we provide the uniqueness result and Section 3 is devoted to the proof.

2. Main results

Let ℕ be the nonnegative integers including 0, and denote by ℓ∶=min{𝑘∈ℕ∶𝛼≤𝛽^𝑘+1}. Let |𝑆| be the size of a set 𝑆. The connected component containing the node 𝐱∈Ω_𝑁 is denoted by 𝐶(𝐱). Since, for every node 𝐱, |𝐶(𝐱)| has the same distribution, it suffices to consider only |𝐶(𝟎)|. The percolation probability is defined as 𝜃(𝛼,𝛽)∶=𝑃||𝐶(𝟎)||=∞,(2.1) and the critical percolation value is defined as 𝛼_𝑐(𝛽)∶=inf{𝛼≥0∶𝜃(𝛼,𝛽)>0}.(2.2) The following theorem characterizes the phase transition for this model.

Theorem 2.1 (see [17]). (i) If 𝛽≤𝑁, then 𝛼_𝑐(𝛽)=0.(ii) If 𝑁<𝛽<𝑁², then 0<𝛼_𝑐(𝛽)<∞.(iii) If 𝛽≥𝑁², then 𝛼_𝑐(𝛽)=∞.

The uniqueness of infinite component is established in the following result.

Theorem 2.2. For 0≤𝛼<∞ and 0<𝛽<∞, there is at most one infinite component almost surely.

3. Proof of Theorem 2.2

For any node 𝐱∈Ω_𝑁, define 𝐵_𝑟(𝐱) the ball of radius 𝑟 around 𝐱, that is, 𝐵_𝑟(𝐱)={𝐲∶𝑑(𝐱,𝐲)≤𝑟}. From this definition, we make the following observations. Firstly, for any 𝐱∈Ω_𝑁, 𝐵_𝑟(𝐱) contains 𝑁^𝑟 vertices. Secondly, 𝐵_𝑟(𝐱)=𝐵_𝑟(𝐲) if 𝑑(𝐱,𝐲)≤𝑟. Finally, for any 𝐱, 𝐲, and 𝑟, we either have 𝐵_𝑟(𝐱)=𝐵_𝑟(𝐲) or 𝐵_𝑟(𝐱)∩𝐵_𝑟(𝐲)=∅.

The proof of Theorem 2.2 follows the idea in [16, Theorem 1.2] and is based on several lemmas.

Lemma 3.1 (see [20]). Consider long range percolation on ℤ^𝑑 with the properties (i)the model is translation-invariant(ii)the model satisfies the positive finite energy condition. Then there can be at most one infinite component almost surely.

Lemma 3.2. The original metric generating tree (as shown in Figure 1) can be embedded into ℤ in a stationary way.

Proof. We will prove this lemma in two steps. (i)Construct a new metric generating tree, which is isomorphic to the original metric generating tree.(ii)The new metric generating tree is stationary on ℤ.
To show step (i), we first describe the construction roughly and then provide the formal construction. The new metric generating tree embeds into ℤ in such a way that for every 𝑟∈ℕ, (a) any ball of radius 𝑟 will be represented by 𝑁^𝑟 consecutive integers and (b) the collection of balls of radius 𝑟 partitions ℤ.
We choose 𝐵₁(𝟎) uniformly at random among all 𝑁 possible collections of 𝑁 consecutive integers containing the origin 0 in ℤ. Given the choice of 𝐵₁(𝟎), all other balls of radius 1 can be defined, although not specified at this point, by the above criteria (a) and (b). Next, the ball 𝐵₂(𝟎) is a union of 𝑁 balls of radius 1 and contains 𝐵₁(𝟎). Since any ball of radius 2 is a collection of 𝑁² consecutive integers, there are 𝑁 possible ways to achieve this. We choose one of the 𝑁 possible ways to do this with probability 1/𝑁 each. Once we have chosen 𝐵₂(𝟎), all other balls of radius 2 are determined for the same reason as above. We continue this procedure to obtain the new metric generating tree, which is embedded in ℤ. To get a picture of this, we illustrate in Figure 2 a possible implementation for 𝑁=2.
Now we formalize the above construction. We choose the probability space as the unit interval [0,1] with Borel sigma field and Lebesgue measure. For 𝜂∈[0,1], denote by 𝜂=0⋅𝜂₁𝜂₂⋯ the 𝑁-adic expansion. In other words, 𝜂=^∞_𝑘=1𝜂_𝑘𝑁^−𝑘,(3.1) where we assume that the expansion for 𝜂 is unique without loss of generality. In the above construction, for each 𝑟, 𝐵_𝑟−1(𝟎) is one of the balls of radius 𝑟−1 among the balls making up 𝐵_𝑟(𝟎). The new metric generating tree corresponding to 𝜂 is obtained as follows. We let 𝐵_𝑟(𝟎) be such that 𝐵_𝑟−1(𝟎) is the (𝜂_𝑟+1)-st ball in 𝐵_𝑟(𝟎) from left to right. In Figure 2, we can see, for example, 𝜂₁=𝜂₃=0 and 𝜂₂=1. It is clear that this construction formalize the informal description given earlier. By first identifying the 𝟎 in Figure 1 and 𝟎 in Figure 2, and then building up the balls 𝐵_𝑟(𝟎) for 𝑟=1,2,…, in that order, we can see that the new metric generating tree is isomorphic to the original one.
Next, we move to step (ii). Let 𝑓 be the map that assigns to each 𝜂 a new metric generating tree as before. The map 𝑓 is invertible on a set of complete Lebesgue measure. Denote by 𝑙 the left-shift transformation, which translates the edges over one unit to the left on the new metric generating trees. Let the transformation 𝑔∶[0,1]→[0,1] correspond to the left-shift transformation 𝑙 on the space of new metric generating trees in the sense that 𝑓∘𝑔=𝑙∘𝑓 (see Figure 3), hence 𝑙=𝑓𝑔𝑓⁻¹. Let 𝐴(𝜂)∶=min{𝑘∶𝜂_𝑘≠𝑁−1}, and then we can see that the 𝑖th digit in 𝑔(𝜂), 𝑔(𝜂)_𝑖, is given by 𝑔(𝜂)_𝑖=⎧⎪⎨⎪⎩𝜂𝜂0,+1,𝑖𝑖=<𝐴𝐴((𝜂𝜂)),_𝑖,_𝑖,𝑖>𝐴(𝜂).(3.2) Furthermore, Lebesgue measure is invariant under the action of 𝑔, which implies that the construction of new random metric generating tree is stationary on ℤ.

Figure 2 

A realization of a new metric generating tree for 𝑁=2. 𝐵1(𝟎) corresponds to {0,1}, 𝐵2(𝟎) corresponds to {−2,−1,0,1}, and 𝐵3(𝟎) corresponds to {−2,−1,0,1,2,3,4,5}.

Figure 3 

The commutative diagram.

Now we are at the stage to prove Theorem 2.2.

Proof of Theorem 2.2. We assign a uniformly-[0,1] distributed random variable 𝛿_𝑒 to each edge 𝑒 in such a way that the collection is independent. Given a new metric generating tree, an edge 𝑒 is said to be open if 𝛿_𝑒≤min{𝛼𝛽^−|𝑒|,1}, where |𝑒| denotes the length of 𝑒. This gives a realization of the percolation process with the correct distribution and shows that the whole long-rang percolation process on the hierarchical lattice can be embedded as a stationary percolation process on ℤ involving Lemma 3.2.
For any 0<𝛼<∞ and 0<𝛽<∞, every pair of vertices are connected by an edge with positive probability, irrespective of the presence or absence of other edges. Therefore, the positive finite energy condition is satisfied. The result then follows from Lemma 3.1. As for 𝛼=0, the result is immediate.

To conclude the paper, we mention that the uniqueness of infinite component has also been proved in [15, 16] for connection probabilities 𝑝_𝑘=𝑐_𝑘𝑁^{−𝑘(1+𝛿)} with 𝛿>−1 and 𝑝_𝑘=1−exp(−𝛽^−𝑘𝛼), respectively.

Acknowledgment

The author would like to thank the referees who provided valuable suggestions and helpful comments.