Contact process on one-dimensional long-range percolation

Recently, by introducing the notion of cumulatively merged partition, M\'enard and Singh provide a sufficient condition on graphs ensuring that the critical value of the contact process is positive. In this note, we show that the one-dimensional long range percolation with high exponent satisfies their condition and thus the contact process exhibits a non-trivial phase transition.


Introduction
In this paper, we study the contact process on G s , the one-dimensional long range percolation graph with exponent s > 1, defined as follows: independently for any i and j in Z there is an edge connecting them with probability |i − j| −s . In particular, G contains Z so it is connected.
On the other hand, the contact process was introduced in an article of T. E. Harris [4] and is defined as follows: given a locally finite graph G = (V, E) and λ > 0, the contact process on G with infection rate λ is a Markov process (ξ t ) t≥0 on {0, 1} V . Vertices of V (also called sites) are regarded as individuals which are either infected (state 1) or healthy (state 0). By considering ξ t as a subset of V via ξ t ≡ {v : ξ t (v) = 1}, the transition rates are given by ξ t → ξ t \ {v} for v ∈ ξ t at rate 1, and where deg ξt (v) denotes the number of infected neighbors of v at time t. Given A ⊂ V , we denote by (ξ A t ) t≥0 the contact process with initial configuration A and if A = {v} we simply write (ξ v t ). Since the contact process is monotone in λ, we can define the critical value reproduction rate λ. Thus for integer lattices and regular trees, the critical value is positive. The behavior of the contact process on these graphs was extensively investigated, see for instance [5,7,9].
In contrast, there is a little knowledge about the sub-critical phase on unbounded degree graphs. For Galton-Watson trees, Pemantle proved in [8] that if the reproduction law B asymptotically satisfies that P(B ≥ x) ≥ exp(−x 1−ε ), for some ε > 0, then λ c = 0.
Recently, in [6], by introducing the notion of cumulatively merged partition (abbr. CMP) (see Section 2.2), the authors provided a sufficient condition on graphs ensuring that λ c > 0. As an application, they show that the contact process on random geometric graphs and Delaunay triangulations exhibits a non-trivial phase transition.
The long range percolation graph was first introduced in [10,11]. Then it gained interest in some contexts such as the graph distance, diameter, random walk, see [3] for a list of reference. The long range percolation is locally finite if and only if s > 1, so we only consider the contact process on such graphs. Moreover, it follows from the ergodicity of G s that there is a non negative constant λ c (s), such that λ c (G s ) = λ c (s) for almost all graphs G s .
It is clear that the sequence of graphs (G s ) is stochastically decreasing in s in the sense that G s1 can be coupled as a subgraph of G s2 if s 1 ≥ s 2 . Therefore λ c (s 1 ) ≥ λ c (s 2 ).
Hence, we can define s c = inf{s : λ c (s) > 0}. (1.2) We will apply the method in [6] to show that s c < +∞. Here is our main result.
There is a phase transition in the structure of the long range percolation. If s < 2, the graph G s exhibits the small-world phenomenon. More precisely, the distance between x and y is of order (log |x − y|) κ+o(1) with κ = κ(s) > 1, with probability tending to 1 as |x − y| → ∞, see for instance [2]. In contrast, if s > 2, the graph somehow looks like Z (see Section 2.1) and the distance now is of order |x − y|, see [1]. On the other hand, as mentioned above, we know that λ c (Z) > 0. Hence, we conjecture that The results in [6] can be slightly improved and thus we could get a better bound on s c , but it would still be far from the critical value 2.
The paper is organized as follows. In Section 2, we first describe the structure of the graph and show that G s can be seen as the gluing of i.i.d. finite subgraphs. Then we recall the definitions and results of [6] on the CMP. By studying the moment of the total weight of a subgraph, we are able to apply the results from [6] and prove our main theorem.
2 Proof of Theorem 1.1

Structure of the graph
We fix s > 2. For any k ∈ Z, we say that k is a cut-point if there is no edge (i, j) with i < k and j > k. (i) For all k ∈ Z P(k is a cut-point) = P(0 is a cut-point) > 0.
As a consequence, almost surely there exist infinitely many cut-points.
(ii) The subgraphs induced in the intervals between consecutive cut-points are i.i.d. In particular, the distances between consecutive cut-points form a sequence of i.i.d. random variables.
Proof. We first prove (i). Observe that P(k is a cut-point) = P(0 is a cut-point) using series integral comparison. Then the ergodic theorem implies that there are infinitely many cut-points a.s. Part (ii) is immediate, since there are no edges between different intervals between consecutive cut-points.
We now study some properties of the distance between two consecutive cut-points. Proposition 2.2. Let D be the distance between two consecutive cut-points. Then there exists a sequence of integer-valued random variables (ε i ) i≥0 with ε 0 = 1, such that (ii) T is stochastically dominated by a geometric random variable with mean e 2/(2−s) , Proof. To simplify notation, we assume that 0 is a cut-point. Set X −1 = 0 and X 0 = 1, then we define for i ≥ 1 Then ε i ≥ 0 and we define Contact process on one-dimensional long range percolation Note that X T is the closest cut-point on the right of 0, so it has the same law as D, by definition. Moreover This implies (ii). For (iii), we note that for i, ≥ 1, Since the definition of λ c is independent of the starting vertex, we can assume that the initially infected vertex is a cut-point.
It will be convenient to assume that 0 is a cut-point. Suppose that conditioned on 0 being a cut-point and infected at the beginning, we can prove that λ c > 0. Since the distribution is invariant under translations, we have λ c > 0 for the contact process starting from any cut point.
Hence, from now on we condition on the event 0 is a cut-point. Set K 0 = 0, for i ≥ 1, we call K i (resp. K −i ) the i th cut point from the right (resp. left) of 0. By Lemma 2.1 (ii), the graphs induced in the intervals [K i , K i+1 ) are i.i.d. Therefore, G s is isomorphic to the graphG s obtained by gluing an i.i.d. sequence of graphs with distribution of the graph [0, K 1 ). We have to prove that the contact process onG s exhibits a non-trivial phase transition.

Cumulatively merged partition
We recall here the definitions introduced in [6]. Given a locally finite graph G = (V, E), an expansion exponent α ≥ 1, and a sequence of non-negative weights defined on the We call cumulatively merged partition (CMP) of the graph G with respect to r and α the finest (r, α)admissible partition and denote it by C (G, r, α). It is the intersection of all (r, α)-admissible partitions of the graph, where the intersection is defined as follows: for any sequence of partitions (C i ) i∈I , As for Bernoulli percolation on Z d , the question we are interested in is the existence of an infinite cluster (here an infinite partition). For the CMP on Z d with i.i.d. weights, we have the following result. We note that in [6,Proposition 3.7], the authors only assume that E(Z γ ) < ∞ and they do not precise the dependence of β c with E(Z γ ). However, we can deduce from their proof a lower bound on β c depending only on E(Z γ ) (and only on α, γ, d if we suppose E(Z γ ) ≤ 1), see Appendix for more details. Finally, our β c (α) is a lower bound of the critical parameter λ c (α) introduced by Ménard and Singh.
Using the notion of CMP, they give a sufficient condition on a graph G ensuring that the critical value of the contact process is positive. Suppose that for some α ≥ 5/2 and ≥ 0, the partition C (G, r , α) has no infinite cluster.
Thanks to this result, Theorem 1.1 will follow from the following proposition.
Proposition 2.5. Fix s > 102. There exists a positive constant , such that the partition C (G s , r , 5/2) has no infinite cluster a.s.

Proof of Proposition 2.5
Let C 1 and C 2 be two CMPs. We write C 1 C 2 , if there is a coupling such that C 1 has an infinite cluster only if C 2 has an infinite cluster.
Contact process on one-dimensional long range percolation Lemma 2.6. We have C (G s , r , 5/2) C (Z, Z , 5/2), Then (Z i ) i∈Z is a sequence of i.i.d. random variables with the same distribution as Z , since the graphG s is composed of i.i.d. subgraphs [K i , K i+1 ). Therefore, C (Z, (Z i ), 5/2) has the same law as C (Z, Z , 5/2). Thus to prove Lemma 2.6, it remains to show that C (G s , r , 5/2) C (Z, (Z i ), 5/2).

(2.3)
For any subset A of the vertices ofG s , we define its projection We prove (2.4) by contradiction. Suppose that there exist x 0 and y 0 such that x 0 ∼ y 0 in C (G s , r , 5/2) and p(x 0 ) ∼ p(y 0 ) in C (Z, (Z i ), 5/2). Then by definition there exists C, a ((Z i ), 5/2)-admissible partition of Z, such that p(x 0 ) ∼ p(y 0 ) in C.
We define a partitionC ofG s as follows: x ∼ y inC if and only if p(x) ∼ p(y) in C.
In other words, an element inC is ∪ i∈C [K i , K i+1 ) with C a set in C. We now claim that C is (r , 5/2)-admissible. Indeed, letC andC be two different sets inC. Then by the definition ofC, we have p(C) and p(C ) are two different sets in C and Z(p(C)) := i∈p(C) On the other hand, as C is ((Z i ), 5/2)-admissible, d(p(C), p(C )) > min{Z(p(C)), Z(p(C ))} 5/2 .
Contact process on one-dimensional long range percolation Let C 0 and C 0 be the two sets in the partition C containing p(x 0 ) and p(y 0 ) respectively. Then by assumption C 0 = C 0 . We definẽ Then bothC 0 andC 0 are inC, andC 0 =C 0 . MoreoverC 0 contains x 0 andC 0 contains y 0 . Hence x 0 ∼ y 0 inC which is a (r , 5/2)-admissible partition. Therefore, x 0 ∼ y 0 in C (G s , r , 5/2), which leads to a contradiction. Thus (2.4) has been proved.
We now apply Proposition 2.3 and Lemma 2.6 to prove Proposition 2.5. To do that, we fix a positive constant β < β c (5/2) with β c (5/2) as in Proposition 2.3 with d = 1 and rewrite If we can show that there is = (β, s), such that then Proposition 2.3 implies that a.s. C (Z, Z , 5/2) has no infinite cluster. Therefore, by Lemma 2.6, there is no infinite cluster in C (G s , r , 5/2) and thus Proposition 2.5 follows.
Now it remains to prove (2.5). It follows from Proposition 2.2 (i) that  Hölder's inequality, we obtain On the other hand, Hence for all i ≥ 1 (2.10) It follows from (2.6), (2.7), (2.8) and (2.10) that since T is stochastically dominated by a geometric random variable. For any j ∈ Z and any interval I, we denote by deg I (j) the number of neighbors of j in I when we consider the original graph (without conditioning on 0 being a cut-point).
Now for any non decreasing sequence (x k ) k≥1 with x 1 ≥ 1, conditionally on ε 1 = where ≺ means stochastic domination.

Appendix: a lower bound on β c
In [6], Proposition 3.7 (our Proposition 2.3) follows from Lemmas 3.9, 3.10, 3.11 and a conclusion argument. Let us find in their proof a lower bound on β c .
At first, they define a constant c = 2αd + 1 and some sequences L n = 2 c n and R n = L 1 . . . L n and ε n = 2 −2dc n+1 .
In Lemma 3.9, the authors do not use any information on Z and β. They set a constant k 0 = [2 d+1 (c + 1)]. In Lemma 3.10, they suppose that β ≤ 1 and the information concerning Z is as follows. There exists n 0 , such that for all n ≥ n 0 , we have In fact, under the assumption E(Z γ ) ≤ 1, we can take (2.14) In Lemma 3.10, they also assume that β ≤ 1 and define a constant n 1 , such that n 1 ≥ n 0 and for all n ≥ n 1 3k α+1 0 L n+1 ≤ R n+1 20 , or equivalently, 60k α+1 0 ≤ R n .
We do not recall the definition of stable sets here. However, we notice that by the first part of Proposition 2.5 and Corollary 2.13 in [6], the event E(N ) occurs when the weights of all vertices in 1, N d are less than 1/2. Therefore P(E(N )) ≥ P r(x) ≤ 1/2 for all x ∈ 1, N d Hence, under the assumption E(Z γ ) ≤ 1, we can take 1/γ , with n 1 as in (2.15).