Phase Transition in Long-Range Percolation on Bipartite Hierarchical Lattices

Abstract

We propose a family of bipartite hierarchical lattice of order N governed by a pair of parameters ℓ and 𝛾. We study long-range percolation on the bipartite hierarchical lattice where any edge (running between vertices of unlike bipartition sets) of length k is present with probability 𝑝_𝑘=1−exp(−𝛼𝛽^−𝑘), independently of all other edges. The parameter 𝛼 is the percolation parameter, while 𝛽 describes the long-range nature of the model. The model exhibits a nontrivial phase transition in the sense that a critical value 𝛼_𝑐∈(0,∞) if and only if ℓ≥1,1≤𝛾≤𝑁−1, and 𝛽∈(𝑁,𝑁²). Moreover, the infinite component is unique when 𝛼>𝛼_𝑐.

1. Introduction

For an integer 𝑁≥2, the hierarchical lattice of order 𝑁 is defined by Ω_𝑁=𝐱=𝑥₁,𝑥₂,…∶𝑥_𝑖∈{0,1,…,𝑁−1},_𝑖𝑥_𝑖<∞. (1) The hierarchical distance 𝑑 on Ω_𝑁 is defined by 𝑑(𝐱,𝐲)=0,if𝐱=𝐲,max𝑖∶𝑥_𝑖≠𝑦_𝑖,if𝐱≠𝐲, (2) which satisfies the strong (non-Archimedean) triangle inequality: 𝑑(𝐱,𝐲)≤max{𝑑(𝐱,𝐳),𝑑(𝐳,𝐲)}, (3) for any 𝐱,𝐲,𝐳∈Ω_𝑁. This means that (Ω_𝑁,𝑑) is an ultrametric space. Roughly speaking, this corresponds to the leaves of an infinite 𝑁-ary tree, with metric distance half the graph distance.

Some stochastic models based on hierarchical lattices have been studied. The asymptotic long-range percolation on Ω_𝑁 is analyzed in [1] for 𝑁→∞. To the best of our knowledge, this is the first paper devoted to (Ω_𝑁,𝑑). For different purpose, the works [2–4] study the long-range percolation on Ω_𝑁 for fixed 𝑁 by using different connection probabilities. The contact process and perturbation analysis on Ω_𝑁 for finite 𝑁 have been studied in [5, 6], respectively. Random walks on hierarchical lattices have been examined in [7, 8].

In this paper, we study percolation on a class of bipartite hierarchical lattices, where edges always run between vertices of unlike type. Bipartite graphs have been studied intensively in the literature (see e.g., [9, 10]) and bipartite structure is popular in many social networks including sexual-contact networks [11] and affiliation networks [12], but we have not seen the setup that we consider here. For two integers ℓ≥1 and 1≤𝛾≤𝑁−1, consider a partition of Ω_𝑁 into two sets: Ω¹_𝑁=Ω¹_𝑁=𝑥ℓ,𝛾𝐱=₁,𝑥₂,…∈Ω_𝑁∶𝑥_ℓ∈,Ω0,1,…,𝛾−1_2𝑁=Ω²_𝑁=𝑥ℓ,𝛾𝐱=₁,𝑥₂,…∈Ω_𝑁∶𝑥_ℓ∈𝛾,𝛾+1,…,𝑁−1. (4) Vertices in Ω¹_𝑁 and Ω²_𝑁 are said to have types 1 and 2, respectively. For each 𝑘≥1, the probability of connection between two vertices 𝐱 and 𝐲 of unlike type such that 𝑑(𝐱,𝐲)=𝑘 is given by 𝑝_𝑘−𝛼=1−exp𝛽^𝑘, (5) where 0≤𝛼<∞ and 0<𝛽<∞, all connections being independent. Vertices of the same type cannot be connected with each other, and hence the resulting graph is a class of random bipartite graph.

In the above bipartite hierarchical lattice, denoted by (Ω¹_𝑁,Ω²_𝑁,𝑑), vertices of both types are countable and the shortest distance between vertices in Ω¹_𝑁 and Ω²_𝑁 is ℓ. The vertices in (Ω¹_𝑁,Ω²_𝑁,𝑑) can be represented by the leaves at the bottom of an infinite regular tree, where 𝑁 branches emerge from each inner node, see Figure 1. The distance between two vertices (leaves at level 0) is the number of levels from the bottom to their most recent common ancestor. The partition of types for vertices is determined by their ancestors at level ℓ; in other words, we need to track back at least ℓ levels to find the most recent common ancestor of two vertices of unlike type.

Figure 1 

An illustration of bipartite hierarchical lattice (Ω1𝑁(ℓ,𝛾),Ω2𝑁(ℓ,𝛾),𝑑) of order 𝑁=3, ℓ=2, and 𝛾=2. Vertices of type 1 are represented by solid points while those of type 2 hollow points. The distances between three vertices 𝟎=(0,0,0,…)∈Ω1𝑁, 𝐱=(1,0,0,…)∈Ω1𝑁, and 𝐲=(0,2,0,…)∈Ω2𝑁 are 𝑑(𝟎,𝐱)=1 and 𝑑(𝟎,𝐲)=𝑑(𝐱,𝐲)=2.

Two vertices 𝐱,𝐲∈(Ω¹_𝑁,Ω²_𝑁,𝑑) are in the same component if there exists a finite sequence 𝐱=𝐱₀,𝐱₁,…,𝐱_𝑛=𝐲 such that each pair of vertices 𝐱_𝑖−1 and 𝐱_𝑖 has different types and shares an edge for 𝑖=1,…,𝑛. In our model, the parameter 𝛽>0 describes the long-range nature, while we think of 𝛼≥0 as a percolation parameter. We are interested in studying when there is a nontrivial percolation threshold in (Ω¹_𝑁,Ω²_𝑁,𝑑), namely, the critical percolation value 𝛼_𝑐∈(0,∞). Our results for phase transition are analogous to those in the monopartite counterpart (Ω_𝑁,𝑑) (see [3]). The similar (comparable) behavior of phase transitions in bipartite and corresponding monopartite networks has also been observed in other percolation contexts (see the discussion in Section 2).

The rest of the paper is organized as follows. The results are stated and discussed in Section 2, and the proofs are given in Section 3.

2. Results

Let |𝑆| be the size of a vertex set 𝑆. The connected component containing the vertex 𝐱 is denoted by 𝐶(𝐱). By definition, the origin 𝟎∈Ω¹_𝑁(ℓ,𝛾) for all ℓ≥1 and 1≤𝛾≤𝑁−1. Since, for all 𝐱∈Ω¹_𝑁(ℓ,𝛾) and 𝐲∈Ω²_𝑁(ℓ,𝑁−𝛾), |𝐶(𝐱)| and |𝐶(𝐲)| have the same distribution, it suffices to consider only |𝐶(𝟎)| without loss of generality. The percolation probability is defined as 𝜃ℓ,𝛾,𝛼,𝛽=𝑃(|𝐶(𝟎)|=∞), (6) and the critical percolation value is defined as 𝛼_𝑐=𝛼_𝑐ℓ,𝛾,𝛽=inf𝛼≥0∶𝜃ℓ,𝛾,𝛼,𝛽>0, (7) which is nondecreasing in 𝛽 for any given ℓ and 𝛾.

Theorem 1. Assume that ℓ≥1, 1≤𝛾≤𝑁−1 and that 0<𝛽<∞. One has the following: (i)If 𝛽≤𝑁, then 𝛼_𝑐=0.(ii)If 𝛽≥𝑁², then 𝛼_𝑐=∞.(iii)If 𝑁<𝛽<𝑁², then 0<𝛼_𝑐<∞. Moreover, there is almost surely at most one infinite component when 𝛼>𝛼_𝑐.

Remark 2. The critical value 𝛼_𝑐=𝛼_𝑐(ℓ,𝛾,𝛽) turns out to be a function of only 𝛽 irrespective of the values of ℓ and 𝛾. Koval et al. [3] showed the same behavior of 𝛼_𝑐 for percolation in the monopartite lattice Ω_𝑁. This analogy of phase transition has been recognized in other percolation problems in statistical physics. An example is the 𝐴𝐵 percolation introduced by Mai and Halley [13] for the study of gelation processes. In this model, each vertex of an infinite connected graph 𝐺 is assigned one of two states, say 𝐴 and 𝐵, with probability 𝑝 and 1−𝑝, respectively, independently of all other vertices. Edges with two end-vertices having unlike states (called 𝐴𝐵 bonds) are occupied. Thus, the 𝐴𝐵 percolation can be viewed as a bond percolation with occupation probability 2𝑝(1−𝑝) (although some dependence is involved, namely, no odd path of 𝐴𝐵 bonds exists). Appel and Wierman [14] proved that 𝐴𝐵 percolation does not occur for any value of 𝑝∈[0,1] on a bipartite square lattice with bipartition 𝑉₁={𝑣=(𝑣_𝑥,𝑣_𝑦)∈ℤ²∶𝑣_𝑥−𝑣_𝑦isodd} and 𝑉₂={𝑣=(𝑣_𝑥,𝑣_𝑦)∈ℤ²∶𝑣_𝑥−𝑣_𝑦iseven} such that ℤ²=𝑉₁∪𝑉₂. In other words, the bond percolation cannot occur on the above bipartite square lattice for occupation probability 2𝑝(1−𝑝)≤1/2. This is consistent with the classical result which says that bond percolation on ℤ² does not occur when occupation probability ≤1/2 (see, e.g., [15, 16]). Other comparable 𝐴𝐵 percolation thresholds for monopartite and bipartite high-dimensional lattices can be found in [17].

Another example is the biased percolation [18, 19] on infinite scale-free networks with a power-law degree distribution 𝑃(𝑘)∝𝑘^−𝛾. In this model, an edge between vertices with degrees 𝑘₁ and 𝑘₂ is occupied with probability proportional to (𝑘₁𝑘₂)^−𝛼. By using generating function method, Hooyberghs et al. [9] showed that biased percolation on a bipartite scale-free network with two bipartition sets following degree distributions 𝑃^𝐴(𝑘)∝𝑘^−𝛾𝐴 and 𝑃^𝐵(𝑘)∝𝑘^−𝛾𝐵, respectively, has the same critical behaviors with biased percolation on a monopartite scale-free network when 𝛾_𝐴=𝛾_𝐵=𝛾.

Remark 3. The uniqueness of the infinite component holds here for the same reason as the uniqueness result for the percolation graph of Ω_𝑁 (see [3, Theorem 2]). Note that our graph resulting from (Ω¹_𝑁,Ω²_𝑁,𝑑) can be viewed as a spanning subgraph of that from (Ω_𝑁,𝑑).

We consider Theorem 1 as an intermediate step towards the study of percolation on bipartite hierarchical lattices. In particular, one may explore the connectivity at the critical regime 𝛽=𝑁² and the graph distance (chemical distance) between 𝟎 and a vertex 𝐱. It is also interesting to study the mean field percolation (𝑁→∞) and compare it with that on Ω_𝑁 [1]. Directed percolation [20] and other meaningful colorings on Ω_𝑁 (other than the 2-coloring addressed in this paper) are possible.

3. Proofs

We start with some notations. Then we prove Theorem 1.

For a vertex 𝐱∈(Ω¹_𝑁,Ω²_𝑁,𝑑), define 𝐵_𝑟(𝐱) as the ball of radius 𝑟 around 𝐱; that is, 𝐵_𝑟(𝐱)={𝐲∶𝑑(𝐱,𝐲)≤𝑟}. We make the following observations. Firstly, for any vertex 𝐱, 𝐵_𝑟(𝐱) contains 𝑁^𝑟 vertices. In particular, if 𝑟<ℓ, all vertices in the ball have the same type. Secondly, 𝐵_𝑟(𝐱)=𝐵_𝑟(𝐲) if 𝑑(𝐱,𝐲)≤𝑟. Finally, for any 𝐱, 𝐲, and 𝑟, we have either 𝐵_𝑟(𝐱)=𝐵_𝑟(𝐲) or 𝐵_𝑟(𝐱)∩𝐵_𝑟(𝐲)=∅.

For a set 𝑆 of vertices, denote by 𝑆=Ω_𝑁⧵𝑆 its complement. Let 𝐶_𝑛(𝐱) be the component of vertices that are connected to 𝐱 by a path using only vertices within 𝐵_𝑛(𝐱). For disjoint sets 𝑆₁,𝑆₂⊆Ω_𝑁, we denote by 𝑆₁↔𝑆₂ the event that at least one edge joins a vertex in 𝑆₁ to a vertex in 𝑆₂. 𝑆₁↮𝑆₂ means the event that such an edge does not exist. By definition, if 𝑆₁,𝑆₂⊆Ω^𝑖_𝑁 for 𝑖=1 or 2, then 𝑆₁↮𝑆₂ occurs with probability 1. Let 𝐶^𝑚_𝑛(𝐱) be the largest component in 𝐵_𝑛(𝐱). If there are more than one such components, just take any one of them as 𝐶^𝑚_𝑛(𝐱). It is clear that |𝐶^𝑚_𝑛(𝐱)|=max_{𝐲∈𝐵𝑛(𝐱)}|𝐶_𝑛(𝐲)| [3].

Proof of (i). Let 𝐴_𝑘 be the event that the origin 𝟎∈Ω¹_𝑁 connects by an edge to at least one vertex at distance 𝑘 in Ω²_𝑁. By construction, for 𝑘<ℓ, 𝑃(𝐴_𝑘)=0. For 𝑘=ℓ, there are ((𝑁−𝛾)/(𝑁−1))(𝑁−1)𝑁^𝑘−1 vertices in Ω²_𝑁 at distance 𝑘 from 𝟎. Hence, 𝑃𝐴_ℓ=1−1−𝑝_ℓ^{(𝑁−𝛾)𝑁𝑘−1}−𝛼=1−exp𝛽^ℓ𝑁−𝛾𝑁^ℓ−1, (8) by using (5). For 𝑘>ℓ, there are ((𝑁−𝛾)/𝑁)(𝑁−1)𝑁^𝑘−1 vertices in Ω²_𝑁 at distance 𝑘 from 𝟎. Similarly, we obtain 𝑃𝐴_𝑘−𝛼)=1−exp𝑁−𝛾(𝑁−1𝛽^𝑘𝑁𝑁^𝑘−1, (9) for 𝑘>ℓ.
Since all the events {𝐴_𝑘}_𝑘≥1 are independent and ^∞_𝑘=1𝑃𝐴_𝑘−𝛼≈1−exp𝛽^ℓ𝑁−𝛾𝑁^ℓ−1+𝛼𝑁−𝛾𝑁^∞_𝑘=ℓ+1𝑁𝛽^𝑘 (10) diverges for any 0<𝛽≤𝑁, 1≤𝛾≤𝑁−1, and 𝛼>0, infinitely many of 𝐴_𝑘 occur with probability 1 by the second Borel-Cantelli lemma. Thus, 𝜃(ℓ,𝛾,𝛼,𝛽)=1 for any ℓ>1, 1≤𝛾≤𝑁−1, 𝛼>0, and 0<𝛽≤𝑁. The result then follows.

Proof of (ii). We only need to show 𝛼_𝑐(ℓ,𝛾,𝑁²)=∞ by virtue of the monotonicity. Note that, for 𝑗≥ℓ, there are (𝛾/𝑁)𝑁^𝑗 vertices in 𝐵_𝑗(𝟎)∩Ω¹_𝑁 and ((𝑁−𝛾)/𝑁)𝑁^𝑗 vertices in 𝐵_𝑗(𝟎)∩Ω²_𝑁. Hence, by the comments in the proof of (i) and taking 𝛽=𝑁², for any 𝑗≥ℓ, we obtain 𝑃𝐵_𝑗(𝟎)⟷𝐵_𝑗(𝟎)=1−^∞_𝑘=𝑗+11−𝑝_𝑘^{((𝑁−𝛾)/𝑁)(𝑁−1)𝑁𝑘−1(𝛾/𝑁)𝑁𝑗}⋅^∞_𝑘=𝑗+11−𝑝_𝑘^{(𝛾/𝑁)(𝑁−1)𝑁𝑘−1((𝑁−𝛾)/𝑁)𝑁𝑗}−()=1−exp2𝛼𝑁−𝛾𝑁−1𝛾𝑁^𝑗𝑁⁴×^∞_𝑘=1𝑁^{𝑘+𝑗−1}𝑁^{2(𝑘+𝑗−1)}−𝛾=1−exp2𝛼𝑁−𝛾𝑁³, (11) which is strictly less than 1 for any finite 𝛼≥0.
Let 𝑛_ℓ=0 and 𝑛_𝑖+1=inf{𝑛≥𝑛_𝑖∶𝐵_𝑛𝑖(𝟎)↮𝐵_𝑛(𝟎)}. We have {|𝐶(𝟎)|=∞}⊆^∞_𝑖=ℓ𝐵_𝑛𝑖(𝟎)⟷𝐵_𝑛𝑖(𝟎). (12) Since the events {𝐵_𝑛𝑖(𝟎)↔𝐵_𝑛𝑖(𝟎)} are independent and all have the same probability strictly less than 1, 𝑃𝐵_𝑛𝑖(𝟎)⟷𝐵_𝑛𝑖(𝟎),forany𝑖≥ℓ=0. (13) Consequently, there exists an 𝑖 such that 𝐵_𝑛𝑖(𝟎)↮𝐵_𝑛𝑖(𝟎) with probability 1. It follows from (12) that 𝜃(ℓ,𝛾,𝛼,𝑁²)=0 for all 𝛼≥0. This implies 𝛼_𝑐(ℓ,𝛾,𝑁²)=∞.

Proof of (iii). The positivity of 𝛼_𝑐 is a direct consequence of the proof of Theorem 1(b) in [3]. Since the percolation graph of (Ω¹_𝑁,Ω²_𝑁,𝑑) can be viewed as a spanning subgraph of that of (Ω_𝑁,𝑑), the percolation cluster 𝐶(𝟎) is almost surely finite; namely, 𝜃(ℓ,𝛾,𝛼,𝛽)=0, for 𝛼 small enough.

Now we turn to the proof of finiteness of 𝛼_𝑐. The main technique to be used is an iteration involving the tail probability of binomial distributions [3, 21]. Since 𝛽<𝑁², we choose an integer 𝐾 and a real number 𝛿 such that 𝛽<𝛿≤𝑁^𝐾−1^1/𝐾. (14) Clearly, 1<𝛿<𝑁. For 𝑛≥1, let 𝑎_𝑛=𝑃𝐶^𝑚_𝑛𝐾(𝟎)∩Ω¹_𝑁≥𝛾𝑁𝛿^𝑛𝐾,𝐶^𝑚_𝑛𝐾(𝟎)∩Ω²_𝑁≥𝑁−𝛾𝑁𝛿^𝑛𝐾, (15) and analogously, 𝑏_𝑛=𝑃𝐶_𝑛𝐾(𝟎)∩Ω¹_𝑁≥𝛾𝑁𝛿^𝑛𝐾,𝐶_𝑛𝐾(𝟎)∩Ω²_𝑁≥𝑁−𝛾𝑁𝛿^𝑛𝐾. (16) Here, 𝑎_𝑛 is the probability that the largest component of a ball of radius 𝑛𝐾 contains at least (𝛾/𝑁)𝛿^𝑛𝐾 vertices in Ω¹_𝑁 and at least ((𝑁−𝛾)/𝑁)𝛿^𝑛𝐾 vertices in Ω²_𝑁. Such a ball is said to be good. We set 𝑎₀=𝑏₀=1 by convention. It is clear that, for 𝛼>0, all 𝑎_𝑛 and 𝑏_𝑛 are positive, since 𝑛𝐾 is a finite number and the connection probability in (5) is positive.

In what follows, we will prove 𝛼_𝑐<∞ in two steps.

Step 1. We show that there exists some 𝛼>0 such that 𝑎_𝑛 converges to 1 exponentially fast; namely, 1−𝑎_𝑛≤exp(−𝑐𝑛) for some 𝑐>0.

Step 2. We show that there exists some 𝛼>0 such that liminf_𝑛→∞𝑏_𝑛>0.

We start with Step 1. To this end, denote by ℕ the nonnegative integers. We can naturally label the vertices in Ω_𝑁 via the map 𝑓∶Ω_𝑁→ℕ as 𝑓∶𝐱=𝑥₁,𝑥₂,…⟼^∞_𝑖=1𝑥_𝑖𝑁^𝑖−1. (17) This order agrees with the depiction in Figure 1. A ball of radius 𝑛𝐾 is said to be very good if it is good and its largest component connects by an edge to the largest component of the first (as per the aforementioned order) good subball in the same ball of radius (𝑛+1)𝐾. Clearly, the first good subball of radius 𝑛𝐾 in a ball of radius (𝑛+1)𝐾 is very good. Condition (14) implies that (𝑁^𝐾−1)𝛿^𝑛𝐾≥𝛿^(𝑛+1)𝐾. Thus we assert that the ball 𝐵_(𝑛+1)𝐾(𝟎) is good if (a) it contains 𝑁^𝐾−1 good subballs of radius 𝑛𝐾, and (b) all these good subballs are very good.

The number of good subballs of radius 𝑛𝐾 in a ball of radius (𝑛+1)𝐾 has a binomial distribution Bin(𝑁^𝐾,𝑎_𝑛) with parameters 𝑁^𝐾 and 𝑎_𝑛. Given the collection of good subballs, the probability that the first such good subball is very good equals to 1. Fix any of the other good subballs, say 𝐵; the probability that 𝐵 is not very good is upper bounded by 1−𝑝_(𝑛+1)𝐾^{𝑎𝑑+𝑏𝑐}≤1−𝑝_(𝑛+1)𝐾^{(2𝛾(𝑁−𝛾)/𝑁2)𝛿2𝑛𝐾}=exp−2𝛼𝛾𝑁−𝛾𝛽^𝐾𝑁²𝛿²𝛽^𝑛𝐾∶=𝜀_𝑛, (18) where 𝑎 and 𝑏 are the number of vertices in the largest component of the first good subball in Ω¹_𝑁 and Ω²_𝑁, respectively; likewise, 𝑐 and 𝑑 are the number of vertices in the largest component of the subball 𝐵 in Ω¹_𝑁 and Ω²_𝑁, respectively. By definition, we have 𝑎,𝑐≥(𝛾/𝑁)𝛿^𝑛𝐾, 𝑏,𝑑≥((𝑁−𝛾)/𝑁)𝛿^𝑛𝐾, and the distance between two vertices in a ball of radius (𝑛+1)𝐾 is at most (𝑛+1)𝐾.

Therefore, the probability for any of the other good subballs 𝐵 to be very good is at leat 1−𝜀_𝑛. Thus, the number of very good subballs is stochastically larger than a random variable obeying a binomial distribution Bin(𝑁^𝐾,𝑎_𝑛(1−𝜀_𝑛)). From the above comments (a) and (b) and the definition of 𝑎_𝑛, it follows that 𝑎_𝑛+1≥𝑃Bin𝑁^𝐾,𝑎_𝑛1−𝜀_𝑛≥𝑁^𝐾−1. (19) In general, we have the following inequality for the tail of binomial random variable: 𝑃𝑛2Bin𝑛,𝑝≥𝑛−1≥1−1−𝑝². (20) By (19), (20), and writing 𝜉_𝑛=1−𝑎_𝑛, we obtain 𝜉_𝑛+1=1−𝑎_𝑛+1≤𝑁^𝐾21−𝑎_𝑛+𝑎_𝑛𝜀_𝑛²≤𝑁^𝐾21−𝑎_𝑛+𝜀_𝑛²=𝑁^𝐾2𝜉_𝑛+𝜀_𝑛². (21) We can choose 𝑐>0 large enough so that 4^𝑁𝐾₂≤exp(𝑐), and then we choose 𝛼 large enough so that (c) 𝜀_𝑛≤exp(−𝑐(𝑛+1)) and (d) 𝜉₁≤exp(−2𝑐) hold. To see (c), note that 𝛽<𝛿² and then 𝜀_𝑛=exp−2𝛼𝛾𝑁−𝛾𝛽^𝐾𝑁²𝛿²𝛽^𝑛𝐾≤𝛽𝛿^{22𝐾𝛼𝛾(𝑁−𝛾)/𝛽𝑘𝑁2𝑛}. (22) To see (d), note that lim_𝛼→∞𝜀₀=0, 𝜉₀=0 and by (21) we obtain 𝜉₁=1−𝑎₁≤𝑁^𝐾2𝜉₀+𝜀₀², (23) which also approaches 0.

According to our above choice of 𝑐 and 𝛼, we have inductively, if 𝜉_𝑛≤exp(−𝑐(𝑛+1)), then 𝜉_𝑛+1≤𝑁^𝐾2𝜉_𝑛+𝜀_𝑛²𝑁≤4^𝐾2((((((,exp−2𝑐𝑛+1))≤exp−𝑐2𝑛+1))≤exp−𝑐𝑛+2)) (24) which implies that 𝜉_𝑛≤exp(−𝑐(𝑛+1))≤exp(−𝑐𝑛) for all 𝑛∈ℕ. We then finish the proof of Step 1.

For Step 2, recalling the definition of 𝑏_𝑛, we claim that 𝑏_𝑛+1≥𝑏_𝑛⋅𝑃Bin𝑁^𝐾−1,𝑎_𝑛1−𝜀_𝑛≥𝑁^𝐾−2. (25) In deed, if |𝐶_𝑛𝐾(𝟎)∩Ω¹_𝑁|≥(𝛾/𝑁)𝛿^𝑛𝐾 and |𝐶_𝑛𝐾(𝟎)∩Ω²_𝑁|≥((𝑁−𝛾)/𝑁)𝛿^𝑛𝐾, then 𝐵_𝑛𝐾(𝟎) is the first good subball in the derivation above. If this component is connected to at least 𝑁^𝐾−2 other large components in 𝐵_(𝑛+1)𝐾(𝟎) as above, then the component containing the origin in 𝐵_(𝑛+1)𝐾(𝟎) has (𝛾/𝑁)𝛿^𝑛𝐾(𝑁^𝐾−1)≥(𝛾/𝑁)𝛿^(𝑛+1)𝐾 vertices in Ω¹_𝑁 and ((𝑁−𝛾)/𝑁)𝛿^𝑛𝐾(𝑁^𝐾−1)≥((𝑁−𝛾)/𝑁)𝛿^(𝑛+1)𝐾 vertices in Ω²_𝑁. Thus, the inequality (25) follows.

A simple coupling gives 𝑃Bin𝑁^𝐾−1,𝑎_𝑛1−𝜀_𝑛≥𝑁^𝐾𝑁−2≥𝑃Bin^𝐾,𝑎_𝑛1−𝜀_𝑛≥𝑁^𝐾−1. (26) Hence, we derive that the right-hand side of (26) converges to 1 exponentially fast by exploiting (20) and the fact that 𝑎_𝑛(1−𝜀_𝑛) converges to 1 exponentially fast. It then follows from (25) that 𝑏_𝑛+1≥𝑏_𝑛1−exp(−𝑐𝑛), (27) for some 𝑐>0. Hence, 𝑏_𝑛+1≥𝑏₁^𝑛_𝑘=11−exp(−𝑐𝑘). (28) It is direct to check that ln^𝑛_𝑘=11−exp(−𝑐𝑘)≥ln1−^𝑛_𝑘=1(()exp−𝑐𝑘)≥ln1−2exp−𝑐1−exp(−𝑐)>−∞, (29) and hence ∏^𝑛_𝑘=1(1−exp(−𝑐𝑘))>0 for all 𝑛. Since 𝑏₁>0, inequality (28) yields liminf_𝑛→∞𝑏_𝑛>0, (30) as desired.