On percolation in one-dimensional stable Poisson graphs

Equip each point x of a homogeneous Poisson point process P onR withDx edge stubs, where theDx are i.i.d. positive integer-valued random variables with distribution given by μ. Following the stable multi-matching scheme introduced by Deijfen, Häggström and Holroyd [1], we pair off edge stubs in a series of rounds to form the edge set of a graph G on the vertex set P. In this note, we answer questions of Deijfen, Holroyd and Peres [2] and Deijfen, Häggström and Holroyd [1] on percolation (the existence of an infinite connected component) in G. We prove that percolation may occur a.s. even if μ has support over odd integers. Furthermore, we show that for any ε > 0, there exists a distribution μ such that μ({1}) > 1− ε, but percolation still occurs a.s..


Introduction
In this paper, we study certain matching processes on the real line.Let D be a random variable with distribution µ supported on the positive integers.Generate a set of vertices P by a Poisson point process of intensity 1 on R. Equip each vertex x ∈ P with a random number D x of edge stubs, where the (D x ) x∈P are i.i.d.random variables with distribution given by D. Now form edges in rounds by matching edge stubs in the following manner.In each round, say that two vertices x, y are compatible if they are not already joined by an edge and both x and y still possess some unmatched edge stubs.Two such vertices form a mutually closest compatible pair if x is the nearest y-compatible vertex to y in the usual Euclidean distance and vice-versa.For each such mutually closest compatible pair (x, y), remove an edge stub from each of x and y to form the edge xy.Repeat the procedure indefinitely.This matching scheme, known as stable multi-matching, was introduced by Deijfen, Häggström and Holroyd [1], who showed that it a.s.exhausts the set of edge stubs, yielding an infinite graph G = G(µ) with degree distribution given by µ.Note that the graph G(µ) arising from our multi-matching process is stable a.s.; for any pair of distinct points x, y ∈ P, either xy ∈ E(G) or at least one of x, y is incident to no edge in G of length greater than |x − y|.The concept of stable matchings was introduced in an influential paper of Gale and Shapley [3]; in the context of spatial point processes its study was initiated by Holroyd and Peres, and by Holroyd, Pemantle, Peres and Schramm [4,5].
Their proof of part (i) of Theorem 1.1 relies on a comparison of the d-dimensional stable multi-matching process with dependent site percolation on Z d .In particular, since the threshold for percolation in Z is trivial, their argument cannot say anything about percolation in the 1-dimensional Poisson graph G = G 1 .
Related to part (ii) of Theorem 1.1, Deijfen, Häggström and Holroyd asked the following question.
In subsequent work on G = G 1 , Deijfen, Holroyd and Peres [2] observed that simulations suggested percolation might not occur when µ({3}) = 1, and asked whether the presence of odd degrees kills off infinite components in general.
Question 2 (Deijfen, Holroyd and Peres).Is it true that percolation in G = G 1 occurs a.s., if and only if, µ has support only on the even integers?
In this paper we prove the following theorem: for all but finitely many i, then a.s. the one-dimensional stable Poisson graph G = G 1 (µ) will contain an infinite path.
Since Theorem 1.2 does not assume anything about µ besides its heavy tail, our result implies a negative answer to both Question 1 and Question 2: Corollary 1.3.For any ε > 0, there exist degree distributions µ with µ({1}) > 1 − ε such that the one-dimensional stable Poisson graph G = G 1 (µ) a.s.contains an infinite connected component.
x i 0=z 0.1d i 0.2d i x i+1 z+0.4d i -0.1d i at most 0.3d i nodes at most 0.3d i nodes at most 0.3d i nodes We note however that the degree distributions µ satisfying the assumptions of Theorem 1.2 have unbounded support; it would be interesting to find a distribution µ with bounded support only that still gives a negative answer to Questions 1 and 2 (see the discussion of this problem in Section 3).

Proof of Theorem 1.2
To prove Theorem 1.2, we construct a degree distribution µ for which G 1 (µ) a.s.contains an infinite path, and then show that for any degree distribution µ stochastically dominating µ, G 1 (µ ) also a.s.contains an infinite path.
The idea underlying our construction of µ is to set µ({d i }) = 1/2 i for a sharply increasing sequence of integers (d i ) i∈N .Suppose that we are given a vertex x i with degree D xi = d i .By choosing d i large enough we can ensure that with probability close to 1, there exists some vertex x i+1 with D xi+1 = d i+1 that is connected to x i by an edge of G. Let U i , i ≥ 1, be the event that a given vertex x i of degree d i is connected to some vertex x i+1 of degree d i+1 .Starting from a vertex x 1 of degree d 1 , we see that if ∞ i=1 U i occurs, then there is an infinite path x 1 x 2 x 3 . . . in G.If the events (U i ) i∈N were independent of each other, then P( ∞ i=1 U i ) = i∈N P(U i ), which we could make strictly positive by letting the sequence (d i ) i∈N grow sufficiently quickly, ensuring in turn that percolation occurs a.s.. Of course the events (U i ) i∈N as we have loosely defined them above are highly dependent.We circumvent this problem by working with a sequence of slightly more restricted events, for which we do have full independence.
Before we begin the proof, let us introduce the following notation.Given x ∈ P, let B(x, r) be the collection of all vertices in P within distance at most r of x.We say that a pair of vertices (x, y) with degrees Observe that if a pair of vertices (x, y) is strongly connected, then, by the stability property of the multi-matching scheme, there will a.s.be an edge of G(µ) joining x and y.
Proof of Theorem 1.2.Set d i = 20 • 3 i and µ({d i }) = 1 2 i for each i ∈ N. Let z ∈ R be arbitrary.Suppose that we condition on a particular vertex x i of degree d i belonging to the point process P and lying inside the interval [z, z + 0.1d i ], and further condition on there being at most 0.3d i points of P in the interval of length 0.2d i centered at z. Write F i (z) for the event that we are conditioning on.By the standard properties of Poisson point processes, conditioning on F i (z) does not affect the probability of any event defined outside the interval [z − 0.1d i , z + 0.1d i ].
Let A i (z) be the event that there is a vertex x i+1 ∈ P with degree d i+1 such that 0.1d i < x i+1 − z < 0.2d i .Viewing P as the union of two thinned Poisson point processes, one of intensity 2 −(i+1) giving us the vertices of degree d i+1 and another of intensity 1 − 2 −(i+1) giving us the rest of the vertices, we see that P( i . If A i (z) occurs, let x i+1 denote the a.s.unique vertex of degree d i+1 which is nearest to x i among those degree d i+1 vertices lying at distance at least 0.1d i to the right of z.
Let B i (z) be the event that there are at most 0.3d i vertices x ∈ P with 0.1d i < |x−z| < 0.2d i .Furthermore, let C i (z) be the event that there are at most 0.3d i vertices x ∈ P lying in the interval [z + 0.2d i , z + 0.4d i ].A quick calculation (using the Chernoff bound, see e.g., [6]) yields that P(B i (z) occurs, then the vertices x i and x i+1 are strongly connected, since our initial assumption F i (z) together with B i (z) tells us that (see Figure 1).This last inequality (together with the fact that x i+1 ∈ [z + 0.1d i , z + 0.2d i ]) also gives our initial conditioning F i (z) with i replaced by i + 1 and z replaced by z + 0.1d By the union bound, we have i (1 + o(1)).
Selecting i 0 sufficiently large and some arbitrary vertex z i0 = x i0 of degree d i0 as a starting point, we may define events E i0 (z i0 ), E i0+1 (z i0+1 ), E i0+2 (z i0+2 ), . . .inductively, each conditional on its predecessors, with z i+1 = z i + 0.1d i for all i ≥ i 0 , and Thus, from any vertex x i0 ∈ P of degree d i0 there is, with strictly positive probability, an infinite sequence of vertices from P, x i0 , x i0+1 , . .., with increasing degrees d i0 , d i0+1 , . .., such that (x i , x i+1 ) is strongly connected for every i ≥ i 0 .By the stability property of our multi-matching scheme, there is a.s.an infinite path in G through these vertices.It follows that G a.s.contains an infinite path.We now only need to make two remarks about the proof to obtain the full statement of Theorem 1.2.
Remark 2.1.The pairs (x i0 , x i0+1 ), (x i0+1 , x i0+2 ), . . .remain strongly connected if we increase the degrees.Also, our proof of Theorem 1.2 does not use any information about d i for i < i 0 .Thus, for any measure µ which agrees with (or stochastically dominates) µ on {n ∈ N : n ≥ d i0 }, G 1 (µ ) will percolate a.s.. Remark 2.2.Note that we could replace the distribution in the proof of Theorem 1.2 by any distribution µ such that µ({x : x ≥ d i }) ≥ 2 −i .Instead of obtaining a (strongly connected) sequence x i such that x i has exactly degree d i , we get a (strongly connected) sequence x i such that x i has at least degree d i .The distribution µ we construct in Theorem 1.2 has unbounded support, and the expected degree of a vertex in G(µ) is infinite.We believe however that the answer to Questions 1 and 2 should still remain negative if µ is required to have bounded support.Indeed we conjecture the following: Conjecture 3.1.For every ε > 0, there exists k = k(ε) such that if µ({n ∈ N : n ≥ k}) > ε, then percolation occurs a.s. in G = G 1 (µ).

Concluding remarks
One might expect that there is a critical value d of the expected degree for percolation.We believe however that no such critical value exists: Conjecture 3.2.There is no critical value d , such that if E(D) < d , then a.s.percolation does not occur, while if E(D) > d , then a.s.percolation occurs in the stable multi-matching scheme on R.
Let us give some motivation for this conjecture.By [1, Theorem 1.2 b)], for any µ with support on {1, 2} and µ({1}) > 0, G 1 (µ) a.s.does not percolate.So any putative critical value must satisfy d ≥ 2. Now, pick ε > 0 and choose δ d .Let µ be a degree distribution with support on {1, δ}, such that the expected degree satisfies E(D) < d − ε.By the definition of d this would imply that G(µ) a.s.does not percolate.Assign degrees independently at random to the vertices of G(µ).Perform the first δ/2 stages of the stable multi-matching process.By then most degree 1 vertices have been matched (and in fact matched to other degree 1 vertices).Now force the remaining degree 1 vertices to match to their future partners.Consider the vertices that had originally been assigned δ edge stubs.A number of these edge stubs will have been used up by the process so far, and the number of edge stubs left at each vertex is not independent; nevertheless we expect most degree δ vertices will have at least δ/4 edge stubs left, and that the number of stubs left will be almost independently distributed.Thus, we believe that the stable multi-matching scheme on the remaining edge stubs of the degree δ vertices will contain as a subgraph the edges of a stable multi-matching scheme on a thinned Poisson point process on R corresponding to the degree δ vertices, and with degrees given by some random variable D with E(D ) > δ/4 d .Since rescaling a Poisson point process does not affect the stable multi-matching process, this would imply that G(µ) a.s.percolates (by definition of d ), a contradiction.
, Proposition 1.1), an infinite component in G, if it exists, is almost surely unique.Taking the Poisson point process in R d for some d ≥ 1 and applying the stable multi-matching scheme mutatis mutandis, we obtain the d-dimensional Poisson graph G d .Deijfen, Häggström and Holroyd proved the following result on percolation in G d : Theorem 1.1.(Deijfen, Häggström and Holroyd [1, Theorem 1.2]) (i) For all d ≥ 2 there exists k = k(d) such that if µ({n ∈ N : n ≥ k}) = 1, then a.s.G d percolates.

Figure 1 :Corollary 1 . 4 .
Figure 1: Restrictions on the number of nodes in various intervals when the event E i (z) occurs.

Remark 3 . 1 .
The existence of degree distributions that a.s.result in an infinite component in dimensions d ≥ 2 was established in [1, Theorem 1.2 a)].Our proof of Theorem 1.2 for G = G 1 (µ) easily adapts to higher dimensions d ≥ 2 (with d-dimensional balls and annuli replacing intervals and punctured intervals, and the sequence (d i ) i∈N being scaled accordingly), giving a different approach to the construction of examples in that setting.