Uniqueness of the Infinite Component for Percolation on a Hierarchical Lattice

We study a long-range percolation in the hierarchical lattice Ω𝑁 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.


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 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 Ω 2 along with its metric generating tree.Such a distance d satisfies the strong triangle inequality 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 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 x 0 , x 1 , . . ., x n y of vertices such that each pair x i−1 , x i , i 1, . . ., n, of vertices presents an edge.

Main results
Let N be the nonnegative integers including 0, and denote by : 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 and the critical percolation value is defined as The following theorem characterizes the phase transition for this model.
The uniqueness of infinite component is established in the following result.

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 B 1 0 uniformly at random among all N possible collections of N consecutive integers containing the origin 0 in Z.Given the choice of B 1 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 B 2 0 is a union of N balls of radius 1 and contains B 1 0 .Since any ball of radius 2 is a collection of N 2 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 B 2 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 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, η 1 η 3 0 and η 2 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

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.
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 k N −k 1 δ with δ > −1 and p k 1 − exp −β −k α , respectively.