Insertion and Deletion Tolerance of Point Processes

We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching.


Introduction
Let Π be a point process on R d . Point processes will always be assumed to be simple and locally finite. Let ≺ denote absolute continuity in law; that is, for random variables X and Y taking values in the same measurable space, X ≺ Y if and only if P(Y ∈ A) = 0 implies P(X ∈ A) = 0 for all measurable A. Let B denote the Borel σ-algebra on R d and let L be Lebesgue measure. We say that Π is insertion-tolerant if for every S ∈ B with L(S) ∈ (0, ∞), if U is uniformly distributed on S and independent of Π, then where δ x denotes the point measure at x ∈ R d .
Let M denote the space of simple point measures on R d . The support of a measure µ ∈ M is denoted by [µ] := y ∈ R d : µ({y}) = 1 .
A Π-point is an R d -valued random variable Z such that Z ∈ [Π] a.s. A finite subprocess of Π is a point process F such that F (R d ) < ∞ and [F ] ⊆ [Π] a.s. We say that Π is deletion-tolerant if for any Π-point Z we have Π − δ Z ≺ Π.
For S ∈ B we define the restriction µ| S of µ ∈ M to S by µ| S (A) := µ(A ∩ S), A ∈ B.
We will prove the following equivalences for insertion-tolerance and deletion-tolerance. if U i is a uniformly random point in S i , with U 1 , . . . , U n and Π all independent, then Π + n i=1 δ U i ≺ Π. (iii) If (X 1 , . . . , X n ) is a random vector in (R d ) n that admits a conditional law given Π that is absolutely continuous with respect to Lebesgue measure a.s., then Π + n i=1 δ X i ≺ Π. In fact we will prove a stronger variant of Theorem 2, in which (ii),(iii) are replaced with a condition involving the insertion of a random finite number of points.
We say that a point process is translation-invariant if it is invariant in law under all translations of R d . In this case further equivalences are available as follows.

Proposition 3. A translation-invariant point process Π on R d is insertion-tolerant if and only if there exists S ∈ B
with L(S) ∈ (0, ∞) such that, if U is uniformly distributed in S and independent of Π, then Π + δ U ≺ Π.
Let Π be a translation-invariant point process with finite intensity; that is, EΠ([0, 1] d ) < ∞. We let Π * be its Palm version. See Section 4 or [13,Chapter 11] for a definition. Informally, one can regard Π * as the point process Π conditioned to have a point at the origin. Condition (1) below appears to be the natural analogue of Theorem 4 (ii) for deletion-tolerance. However, it is only sufficient and not necessary for deletion-tolerance.
Theorem 5. Let Π be a translation-invariant point process of finite intensity on R d and let Π * be its Palm version. If then Π is deletion-tolerant.
In Section 2, Example 3 shows that a deletion-tolerant process need not satisfy (1), while Example 6 shows that the natural analogue of Proposition 3 fails for deletion-tolerance.

Remark 1 (More general spaces). Invariant point processes and their
Palm versions can be defined on more general spaces than R d . See [3,12,14,16,17] for more information. For concreteness and simplicity, we have chosen to state and prove Theorems 1, 2, 4 and 5 in the setting of R d , but they can easily be adapted to any complete separable metric space endowed with: a group of symmetries that acts transitively and continuously on it, and the associated Haar measure. We will make use of this generality when we discuss Gaussian zero processes on the hyperbolic plane in Proposition 13. ♦ Next we will illustrate some applications of insertion-tolerance and deletion-tolerance in the contexts of continuum percolation and stable matchings. We will prove generalizations of earlier results.
The Boolean continuum percolation model for point processes is defined as follows (see [19]). Let · denote the Euclidean norm on The proof of Theorem 6 is similar to the uniqueness proofs in [19,Chapter 7] which in turn are based on the argument of Burton and Keane [1].
Next we turn our attention to stable matchings of point processes (see [9] while in the two-colour case we say that a matching scheme is stable if almost surely there do not exist x ∈ [R] and y ∈ [B] satisfying (2). These definitions arise from the concept of stable marriage as introduced by Gale and Shapley [5]. It is proved in [9] that stable matching schemes exist and are unique for point processes that satisfy certain mild restrictions, as we explain next. Let µ ∈ M. We say that µ has a descending chain if there exist x 1 , x 2 , . . . ∈ [µ] with We say that µ is non-equidistant if for all x, y, u, v ∈ [µ] such that {x, y} = {u, v} and x = y we have x − y = u − v . The following fact are proved in [9,Proposition 9]. Suppose that R is a translationinvariant point process on R d with finite intensity that almost surely is non-equidistant and has no descending chains. Then there exists a one-colour stable matching scheme which is an isometry-equivariant factor of R; this matching scheme may be constructed by a simple procedure of iteratively matching, and removing, mutually-closest pairs of R-points; furthermore, any two one-colour stable schemes agree almost surely [9,Proposition 9]. In this case we refer to the above-mentioned scheme as the one-colour stable matching scheme. Similarly, in the twocolour case, let R and B be point processes on R d of equal finite intensity, jointly invariant and ergodic under translations, and suppose that R+B is a simple point process that is almost surely non-equidistant and has no descending chains. Then there exists an almost surely unique two-colour stable matching scheme, which is an isometry-equivariant factor and may be constructed by iteratively matching mutually-closest R / B pairs. Homogeneous Poisson process are non-equidistant and have no descending chains (see [7]). Descending chains are investigated in detail in [2], where it is shown in particular that they are absent in many well-studied point processes.
In this paper, our interest in stable matching lies in the typical distance between matched pairs. Let M be the one-colour stable matching scheme for R. Consider the distribution function (3) As in [9], let X be a random variable with probability measure P * and expectation operator E * such that P * (X ≤ r) = F (r) for all r ≥ 0. One may interpret X as the distance from the origin to its partner under the Palm version of (R, M) in which we condition on the presence of an R-point at the origin; see [9] for details. For the two-colour stable matching scheme of point processes R, B we define X, P * , and E * in the same way.
Theorem 7 (One-colour stable matching). Let R be a translation-invariant ergodic point process on R d with finite intensity that almost surely is non-equidistant and has no descending chains. If R is insertion-tolerant or deletion-tolerant, then the one-colour stable matching scheme satisfies E * X d = ∞.
Theorem 8 (Two-colour stable matching). Let R and B be independent translation-invariant ergodic point processes on R d with equal finite intensity such that the point process R + B is non-equidistant and has no descending chains. If R or B is deletion-tolerant or insertion-tolerant, then the two-colour stable matching scheme satisfies E * X d = ∞.
Theorems 7 and 8 strengthen the earlier results in [9] in the following ways. In [9], Theorem 7 is proved in the case of homogeneous Poisson processes, but the same proof is valid under the condition that R is both insertion-tolerant and deletion-tolerant. Similarly, in [9], Theorem 8 is proved in the Poisson case, but the proof applies whenever R or B is insertion-tolerant. Related results appear also in [8,Theorems 32,33] The following complementary bound is proved in [9] for Poisson processes, but again the proof given there applies more generally as follows.
Theorem 9 ([9, Theorem 5]). Let R be a translation-invariant ergodic non-equidistant point process on R d with no descending chains, and unit intensity. The one-colour stable matching scheme satisfies P * (X > r) ≤ Cr −d for all r > 0, for some constant C = C(d) that does not depend on R.
Thus, Theorems 7 and 9 provide strikingly close upper and lower bounds on X for the one-colour stable matching schemes of a wide range of point processes. For two-colour stable matching, even in the case of two independent Poisson processes, the correct power law for the tail of X is unknown in dimensions d ≥ 2; for d = 1 the bounds E * X 1 2 = ∞ and P * (X > r) ≤ Cr −1/2 hold. See [9] for details. The rest of the paper is organized as follows. In Section 2 we present examples. In Section 3 we prove some of the simpler results including Theorems 1 and 2. Despite the similarities between insertion-tolerance and deletion-tolerance, the proof of Theorem 2 relies on the following natural lemma, whose analogue for deletion-tolerance is false (see Example 5).
Lemma 10 (Monotonicity of insertion-tolerance). Let Π be a point process on R d and let S ∈ B have finite nonzero Lebesgue measure. If Π is insertion-tolerant, and U is uniformly distributed in S and independent of Π, then Π + δ U is insertion-tolerant. Section 4 deals with Theorems 4 and 5. In Sections 5 and 6 we prove the results concerning continuum percolation and stable matchings. Section 7 provides proofs relating to some of the more elaborate examples from Section 2.

Examples
First, we give examples of (translation-invariant) point processes that possess various combinations of insertion-tolerance and deletiontolerance. We also provide examples to show that certain results concerning insertion-tolerance do not have obvious analogues in the setting of deletion-tolerance. Second, we give examples to show that the conditions in the results concerning continuum percolation and stable matching are needed. Finally, we provide results on perturbed lattice processes and Gaussian zeros processes on the Euclidean and hyperbolic planes.

Elementary examples.
Example 1 (Poisson process). The homogeneous Poisson point process Π on R d is both insertion-tolerant and deletion-tolerant. This follows immediately from Theorem 4 (ii) and Theorem 5 and the relation It is also easy to give an direct proof of insertion-tolerance and to prove deletion-tolerance via Theorem 1 (iii). ♦ Example 5 (Non-monotonicity of deletion-tolerance). We show that in contrast with Lemma 10, deleting a point from a deletion-tolerant process may destroy deletion-tolerance. Let (N i ) i∈Z be i.i.d., taking values 0, 1, 2 each with probability 1/3, and let Π have exactly N i points in the interval [i, i + 1), for each i ∈ Z, with their locations chosen independently and uniformly at random in the interval. It is easy to verify that Π is deletion-tolerant using Theorem 1 (iii). Consider the Π-point Z defined as follows. If the first integer interval [i, i + 1) to the right of the origin that contains at least one Π-point contains exactly two Π-points, then let Z be the point in this interval that is closest to the origin; otherwise, let Z be the closest Π-point to the left of the origin. The point process Π ′ = Π − δ Z has the property that the first interval to the right of the origin that contains any Πpoints contains exactly one Π-point.
Let Z ′ be the first Π ′ -point to the right of the origin. The process Π ′′ := Π ′ − δ Z ′ has the property that with non-zero probability the first interval to the right of the origin that contains any Π ′′ -points contains exactly two Π ′′ -points. Thus Π ′ is not deletion-tolerant.
If desired, the above example can be made translation-invariant by applying a random shift U as before. ♦ Example 6 (One set S satisfying Π| S c ≺ Π does not suffice for deletion-tolerance). Let Λ be a randomly shifted lattice in d = 1 (as in Example 2) and let Π be a Poisson point process on R of intensity 1 that is independent of Λ.

Example 7 (A point process that is neither insertion-tolerant nor deletion-tolerant and has infinitely many unbounded clusters). Let
and let U be uniformly distributed in [0, 1] 2 and independent of W . Consider the point process Λ with support U + W . Thus Λ is a randomly shifted lattice with columns randomly deleted. As in Example 2, Λ is neither insertion-tolerant nor deletion-tolerant. In the continuum percolation model with parameter R = 2, the occupied region O(Λ) has infinitely many unbounded clusters. ♦ Example 8 (A point process that is not insertion-tolerant, but is deletion-tolerant and has infinitely many unbounded clusters). Let Λ be a randomly shifted super-critical site percolation in d = 2, as in Ex- Thus Γ is a point process in R 3 , obtained by stacking independent copies of Λ. Clearly, the point process Γ is deletion-tolerant, but not insertion-tolerant. With R = 2, the occupied region O(Γ) has infinitely many unbounded clusters. ♦ Example 9 (One-colour matching for two perturbed lattices). Let It is easy to verify that R is neither insertion-tolerant nor deletiontolerant, and that R has no descending chains and is non-equidistant. The one-colour stable matching scheme satisfies x − M(x) < 1 2 for all x ∈ [R] (in contrast with the conclusion in Theorem 7). ♦ Example 10 (Two-colour matching for randomly shifted lattices). Let R and B be two independent copies of the randomly shifted lattice Z in d = 1 as defined in Example 2. Although R + B is not nonequidistant, it is easy to verify that there is an a.s. unique two-colour stable matching scheme for R and B, and it satisfies 3. Perturbed lattices and Gaussian zeros. The proofs of the results stated below are given in Section 7.
Consider the point process Λ given by Note that Λ is invariant and ergodic under shifts of Z d . It is easy to see that (for all dimensions d) if Y 0 has bounded support, then Λ is neither insertion-tolerant nor deletion-tolerant. Indeed, in this case we have Λ(B(0, 1)) ≤ M for some constant M < ∞, so, by Theorem 2 (ii), Λ is not insertion-tolerant (otherwise we could add M + 1 random points in B(0, 1)). Also, Λ(B(0, N)) ≥ 1, for some N < ∞, so Theorem 1 (iii) shows that Λ is not deletion-tolerant. ♦ For dimensions 1 and 2 we can say more.
Question 1. Does there exists a distribution for the perturbation Y 0 such that the resulting perturbed lattice is insertion-tolerant? In particular, in the case d = 1, does this hold whenever Y 0 has infinite mean? What are the possible combinations of insertion-tolerance and deletiontolerance for perturbed lattices? Allan Sly has informed us that he has made progress on these questions.
Perturbed lattice models were considered by Sodin and Tsirelson [21] as simplified models to illustrate certain properties of Gaussian zero processes (which we will discuss next). Our proof of Proposition 11 is in part motivated by their remarks, and similar proofs have also been suggested by Omer Angel and Yuval Peres (personal communications).
The Gaussian zero processes on the plane and hyperbolic planes are defined as follows (see [11,21] for background). Let {a n } ∞ n=0 be i.i.d. standard complex Gaussian random variables with probability density π −1 exp(−|z| 2 ) with respect to Lebesgue measure on the complex plane. Firstly, consider the entire function The set of zeros of f forms a translation-invariant point process Υ C in the complex plane. Secondly, consider the analytic function on the unit disc D := {z ∈ C : |z| < 1} given by The set of zeros of g forms a point process Υ D . We endow D with the hyperbolic metric |dz|/(1 − |z| 2 ) and the group of symmetries G given by the maps z → (az + b)/(bz +ā), where a, b ∈ C and |a| 2 − |b| 2 = 1. Then Υ D is invariant in law under action of G.
The following two facts were suggested to us by Yuval Peres, and are consequences of results of [21] and [20] respectively. Proposition 12. The Gaussian zero process Υ C on the plane is neither insertion-tolerant nor deletion-tolerant.
Proposition 13. The Gaussian zero process Υ D on the hyperbolic plane is both insertion-tolerant and deletion-tolerant.

Basic results
In this section we prove elementary results concerning insertiontolerance and deletion-tolerance. The following simple application of Fubini's theorem will be useful. Recall that L denotes Lebesgue measure.
Remark 2. Let Π be a point process on R d . If S ∈ B is a set of positive finite measure and U is uniformly distributed S and independent of Π, Thus A x is the set of point measures for which adding a point at x results in an element of A.
Proof of Lemma 10. Let Π be insertion-tolerant. We first show that for almost all x ∈ R d the point process Π + δ x is insertion-tolerant. The proof follows from the definition of A x . Let V be uniformly distributed in S ′ ∈ B and independent of Π. Suppose A ∈ M is such that Next, let U be uniformly distributed in S ∈ B and independent of (Π, V ). Let P(Π + δ U ∈ A) = 0, for some A ∈ M. By Remark 2, With Lemma 10 we prove that insertion-tolerance implies the following stronger variant of Theorem 2 in which we allow the number of points added to be random. If (X 1 , . . . , X n ) is a random vector in (R d ) n with law that is absolutely continuous with respect to Lebesgue measure, then we say that the random (unordered) set {X 1 , . . . , X n } is nice. A finite point process F is nice if for all n ∈ N, conditional on F (R d ) = n, the support [F ] is equal in distribution to some nice random set; we also say that the law of F is nice if F is nice.
Corollary 14. Let Π be an insertion-tolerant point process on R d and let F be a finite point process on R d . If F admits a conditional law given Π that is nice, then Π + F ≺ Π.
Consider the events Let {U r,k } n r=1 i.i.d. random variables uniformly distributed in B(0, k) and independent of (Π, U). Let F ′ n,k := n r=1 δ U r,k . By applying Lemma 10, n times, we see that Π + F ′ n,k ≺ Π; thus it suffices to where s i are the elements of S in lexicographic order. For each n ≥ 0, let g n : (R d ) n × M → R be a measurable function such that g n (·, π) is the probability density function (with respect to n-dimensional Lebesgue measure) of [f (π, U)] , conditional on f (π, U)(R d ) = n. Let Q be the law of Π and let A ∈ M. Thus On the other hand, If P(Π + f (Π, U) ∈ A) > 0, then there exist n, k ≥ 0 such that P(Π + f (Π, U) ∈ A, E n,k ) > 0; moreover from (6) and (7), we deduce that The proof of Theorem 1 relies on the following lemma.
Let C be the collection of all unions of finitely many rational balls. Clearly C is countable. We will show that there exists S ∈ C satisfying (8). Since Π is locally finite, it follows that there exists a C-valued random variable S such that Π| S = F a.s. Since With Lemma 15 we first prove the following special case of Theorem 1.
Lemma 16. Let Π be a point process on R d . The following conditions are equivalent.
We show by induction on the number of points of the finite subprocess that (i) implies (ii). Assume that Π is deletion-tolerant. Suppose that (ii) holds for every finite subprocess F of Π such that F (R d ) ≤ n.
Thus we assume without loss of generality that Let A S := {µ + δ x : µ ∈ A, x ∈ S}, so that by the definition of A S and (9), we have P(Π − F ∈ A S ) > 0. By the inductive hypothesis, P(Π ∈ A S ) > 0.
Observe that if Π ∈ A S , there is an x ∈ [Π] ∩S such that Π−δ x ∈ A. Define a Π-point R as follows. If Π ∈ A S , let R be the point of [Π] ∩ S closest to the origin (where ties are broken using lexicographic order) such that Π − δ R ∈ A, otherwise let R be the Π-point closest to the origin. Hence Assume that (iii) holds and that for some Π-point Z and some A ∈ M we have P(Π − δ Z ∈ A) > 0. By Lemma 15, P(Π| S c ∈ A) > 0 for some S ∈ B, with finite Lebesgue measure. From (iii), P(Π ∈ A) > 0. Thus (i) holds and Π is deletion-tolerant.
Assume that (i) holds. Let F be a finite subprocess of Π and suppose for some A ∈ M we have P(Π − F ∈ A) > 0. Define F n as follows. Take F n = F if F (R d ) = n, otherwise set F n = 0. Note that for some n, we have P(Π − F n ∈ A) > 0. Since Π is deletion-tolerant, by Lemma 16, P(Π ∈ A) > 0. Thus (ii) holds.
Clearly (ii) implies (iii), since for any set S ∈ B with finite measure, the point process with support [Π] ∩ S is a finite subprocess of Π.
For a translation θ of R d and a point measure µ ∈ M, we define θµ ∈ M by (θµ)(S) := µ(θ −1 S) for all S ∈ B; for A ∈ M, we write θA := {θµ : µ ∈ A}. For x ∈ R d let θ x be the translation defined by θ x (y) := y + x for all y ∈ R d .
Proof of Proposition 3. Let U, V be uniformly distributed on S, T ∈ B respectively and let U, V, Π be independent. Assume that Π + δ U ≺ Π and let A ∈ M be such that P(Π + δ V ∈ A) > 0. We will show that P(Π ∈ A) > 0.

Palm equivalences
In this section, we discuss insertion-tolerance and deletion-tolerance in the context of Palm processes. We begin by presenting some standard definitions and facts. Let Π be a translation-invariant point process with finite intensity λ. The Palm version of Π is the point process Π * such that for all A ∈ M and all S ∈ B with finite Lebesgue measure, we have where #B denotes the cardinality of a set B. Sometimes (10) is called the Palm property. By a monotone class argument, a consequence of (10) is that for all see [13,Chapter 11].
Without loss of generality, take A to be a set that does not care whether there is a point at 0; that is if µ ∈ A, then µ ′ ∈ A, provided µ, µ ′ agree on R d \ {0}. By translation-invariance, 0 < c := P(Π + δ 0 ∈ A) = P(Π ∈ A) = P(Π ∈ θ x A) for every x ∈ R d . Hence the translation-invariant random set G := {x ∈ R d : Π ∈ θ x A} has intensity EL([0, 1] d ∩ G) = c. Moreover, if U is uniformly distributed in [0, 1] d and independent of Π, then P(U ∈ G) = c. Therefore defining the set we deduce that P(Π + δ U ∈ A ′ ) > 0. (Recall that A does not care whether there is a point at 0.) On the other hand by the Palm property (10) we have Thus Π is not insertion-tolerant.
The following observations will be useful in the proof that (ii) implies (i) in Theorem 4.

Lemma 17. Let Π be a translation-invariant point process on R d with finite intensity. If Y is any R d -valued random variable, and U is uni-
Lemma 18. Let Π be a translation-invariant point process on R d with finite intensity. There exists a Π-point Z such that Π * ≺ θ −Z Π.
Since Π is translation-invariant and U is independent of Π we have Since we assume that Π + δ 0 ≺ Π * and U is independent of (Π, Π * ) we deduce from (15) Proof of Lemma 17. Let Q be the joint law of Π and Y . Since U is independent of (Π, Y ), by Fubini's theorem, for all A ∈ M, we have Lemma 18 is an immediate consequence of a result of Thorisson [22], which states that there exists a shift-coupling of Π and Π * ; that is, a Π-point Z such that Π * d = θ −Z Π. In fact, Holroyd and Peres [10] prove that such a Z may be chosen as a deterministic function of Π. Since Lemma 18 is much weaker result, we can give the following simple self-contained proof.
Proof of Lemma 18. Let {a i } i∈N = [Π] be an enumeration of the Πpoints. Let K be a random variable with support N; also assume that K is independent of (a i ) i∈N . Define the Π-point Z := a K . We will show that Π * ≺ θ −Z Π.
Let A ∈ M be so that P(Π * ∈ A) > 0. By the Palm property (10), there exists a Π-point Z ′ = Z ′ (A) such that P(θ −Z ′ Π ∈ A) > 0; moreover, there exists i ∈ N such that P(θ −Z ′ Π ∈ A, Z ′ = a i ) > 0. Since K is independent of (a i ) i∈N , it follows from the definition of Z that   Proof of Theorem 6. From Lemma 19, it suffices to show that there can not be infinitely many unbounded clusters; this follows from Theorem 21 and Lemma 20.

Lemma 19. For a translation-invariant ergodic insertion-tolerant point process, the number of unbounded clusters is a fixed constant a.s. that is zero, one, or infinity.
For r > 0, let rZ d := rz : z ∈ Z d .
Proof of Lemma 19. Let Π be a translation-invariant ergodic insertiontolerant point process. Let the occupied region be given by a union of balls of radius R > 0. By ergodicity, if K(Π) is the number of unbounded clusters, then K(Π) is a fixed constant a.s. Assume that K(Π) < ∞. It suffices to show that P(K(Π) ≤ 1) > 0. Since K(Π) < ∞, there exists N > 0 so that every unbounded cluster intersects B(0, N) with positive probability. Consider the finite set S := (R/4)Z d ∩ B(0, N). For each x ∈ S, let U x be uniformly distributed in B(x, R) and assume that the U x and Π are independent. Let F := x∈S δ Ux . Since B(0, N) ⊂ ∪ x∈S B(U x , R), we have that Proof of Lemma 20. The proof is similar to that of Lemma 19. Let Π be an insertion-tolerant point process with infinitely many unbounded clusters. Let the occupied region be given by a union of balls of radius R > 0. Choose N large enough so that at least three unbounded clusters interest B(0, N) with positive probability. Define a finite point process F exactly as in the proof of Lemma 19. The point process Π + F has at least three (N + R)-branches with positive probability and Theorem 2 (ii) implies that Π + F ≺ Π. Thus Π has at least three (N + R)-branches with positive probability.

Stable matching
Theorems 7 and 8 are consequences of the following lemmas. Let R be a point process with a unique one-colour stable matching scheme M. Define This is the set of R-points that would prefer some R-point in the ball B(0, 1), if one were present in the appropriate location, over their current partners. Also define H by (16) for the case of two-colour stable matching. A calculation given in [9, Proof of Theorem 5(i)] shows that, for one-colour and two-colour matchings, for some c = c(d) ∈ (0, ∞).

Lemma 22 (One-colour stable matching). Let R be a translationinvariant point process on R d with finite intensity that almost surely is non-equidistant and has no descending chains. If R is insertiontolerant, then P(#H
Lemma 23 (Two-colour stable matching). Let R and B be independent translation-invariant ergodic point processes on R d with equal finite intensity, such that the point process R+ B is non-equidistant and has no descending chains. If R is insertion-tolerant, then P(#H = ∞) = 1.

Remark 3.
Recall that in the case of two-colour stable matching we defined X in terms of the distance from an R-point to its partner. If we instead define X ′ by replacing R with B in (3), then X ′ d = X; see the discussion after [9,Proposition 7] for details. ♦ Proof of Theorem 7. Use Lemma 22 together with (17).
The following lemmas concerning stable matchings in a deterministic setting will be needed. A partial matching of a point measure µ ∈ M is the edge set m of simple graph ([µ], m) in which every vertex has degree at most one. A partial matching is a perfect matching if every vertex has degree exactly one. We write m(x) = y if and only if {x, y} ∈ m, and set m(x) = ∞ if x is unmatched. We say a partial matching is stable if there do not exist distinct points x, y ∈ [µ] satisfying Note that in any stable partial matching there can be at most one unmatched point. For each ε > 0, set For each y ∈ R d , set This is the set of µ-points that would prefer y ∈ R d over their partners.
Lemma 24. If µ ∈ M is non-equidistant and has no descending chains, then µ has an unique stable partial matching m. In addition, we have the following properties. Proof. The existence and uniqueness is given by [9,Lemma 15]. Thus for (i)-(iii) it suffices to check that the claimed matching is stable, which is immediate from the definition (18).
The next lemma is a simple consequence of Lemma 24.

Proof of Lemma 22: the case where R is insertion-tolerant.
Let R be insertion-tolerant. Note that H 1 (R) = H(R). First, we will show that Second, we will show that if P(0 < #H 1 (R) < ∞) > 0, then there exists a finite point process F such that F admits a nice conditional law given R, and Finally, note that by Corollary 14 and the insertion-tolerance of R that (21) and (20) are in contradiction. Thus P(#H 1 (R) = ∞) = 1. It remains to prove the first two assertions.
The following definition will be useful. Let M ′ be the set of point measures µ ∈ M such that µ has a unique stable perfect matching, has no descending chains, and is non-equidistant.
Let ε > 0. Let J be the set of point measures µ ∈ M ′ such that #H ε (µ) = 0. To show (20), it suffices to prove that P(R ∈ J ) = 0. Let µ ∈ J and let m be the unique stable perfect matching for µ. By Lemma 24 (ii), for Lebesgue-a.a. x ∈ B(0, ε) the unique stable partial matching for µ+δ x is m (and x is unmatched). If P(R ∈ J ) > 0, then it follows from the insertion-tolerance of R that with positive probability R does not have a perfect stable matching, a contradiction. Now let A be the set of point measures µ ∈ M ′ such that 0 < #H 1 (µ) < ∞ and 0 ∈ [µ]. If R ∈ A, then, by applying Lemma 25 repeatedly, there exists ρ = ρ(R) such that if a point is added within distance ρ of each point in H 1 (R) and each of their partners, then (for L-a.a. choices of such points) the resulting process R ′ satisfies H ρ (R ′ ) = 0. Let F be the finite point process whose conditional law given R is given as follows. Take independent uniformly random points in each of the appropriate balls of radius ρ provided R ∈ A; otherwise take F = 0. By the construction, lim ε→0 P #H ε (R + F ) = 0 | R ∈ A, ρ(R) > ε = 1, so (21) follows.
Proof of Lemma 22: the case where R is deletion-tolerant. Suppose R is deletion-tolerant. We will show that for any R-point Z #N(R, Z) = ∞ a.s.
From (22) it follows that if R(B(0, 1)) > 0, then #H = ∞. Since R is translation-invariant, P(R(B(0, 1)) > 0) > 0 and P(#H = ∞) > 0. It remains to show (22). Let Z be an R-point. Let F 1 be the point process with support N(R, Z), and let F 2 be the point process with support {M(y) : y ∈ N(R, Z)}. Consider the point process F defined by Let M ′ be given by We now turn to the two-colour case. Given two point measures µ, µ ′ ∈ M such that µ + µ ′ is a simple point measure, we say that m is a partial (respectively, perfect) matching of (µ, µ ′ ) if m is the edge set of a simple bipartite graph ([µ], [µ ′ ], m) in which every vertex has degree at most one (respectively, exactly one). We write m(x) = y if and only if {x, y} ∈ m and set m(x) = ∞ if x is unmatched. We say that m is stable if there do not exist x ∈ [µ] and y ∈ [µ ′ ] satisfying (18). If µ + µ ′ is non-equidistant and has no descending chains then there exists a unique stable partial matching of (µ, µ ′ ) [9, Lemma 15].

Remark 4. It is easy to verify that the two-colour analogues of Lemma 24 (i) and (iii) hold. ♦
We will need the following monotonicity facts about stable twocolour matchings. Similar results are proved in [8,Proposition 21], [5], and [15].
Lemma 26. Let µ, µ ∈ M and assume that µ + µ ′ is a simple point measure that is non-equidistant and has no descending chains. Let m be the stable partial matching of (µ, µ ′ ).
Proof of Lemma 23. The proof for the case when R is insertion-tolerant is given in [9, Theorem 6(i)]. In the case when R is deletion-tolerant we proceed similarly to the proof of Lemma 22. Recall that in the twocolour case, M denotes the two-colour stable matching scheme for R and B. Let Z be a B-point. Define N(R, Z) and F 1 as in the proof of Lemma 22, so that N(R, Z) is the set of R-points that would prefer Z over their partners and F 1 is the point process with support N(R, Z). Towards a contradiction assume that P(#N(R, Z) < ∞) > 0. There exists a unique stable partial matching for (R − F 1 , B) then Π is neither insertion-tolerant nor deletion-tolerant.
If there exists a deterministic sequence (n k ) with n k → ∞ and a discrete real-valued random variable N such that for all ℓ ∈ R, then Π is neither insertion-tolerant nor deletion-tolerant.
In our application of Proposition 27 (iii), N n will be integer-valued (see (32) below).
Proof of Proposition 27 (i). Let m n := EΠ(h n ). Since Π(h n ) − m n → 0 in probability, there exists a (deterministic) subsequence n k such that Proof of Proposition 27 (ii). Suppose that (26) holds for some deterministic sequence (n k ). Let m n k := EΠ(h n k ), and for each integer Thus S K (Π) → 0 in probability as K → ∞, and there exists a subsequence (K i ) so that S K i (Π) → 0 a.s. However, if U is uniformly distributed in B(0, 1), then S K i (Π + δ U ) → 1 a.s. Thus Π cannot be insertion-tolerant. Similarly, if Z is a Π-point, then S K i (Π − δ Z ) → −1 a.s. Thus Π cannot be deletion-tolerant.
Proof of Proposition 27 (iii). Suppose that (27) holds for some deterministic sequence (n k ) and some discrete random variable N. Let m n k := EΠ(h n k ), and let N n k (µ) := µ(h n ) − m n k for all µ ∈ M. For each integer K > 0, define Thus F K (Π, ℓ) → P(N ≤ ℓ) in probability as K → ∞ for all ℓ ∈ R.
(iii) There exists a deterministic sequence (n k ) with n k → ∞ such that (26) is satisfied with Λ in place of Π; that is, Lemma 28 parts (i) and (ii) will allow us to use a weak law of large numbers to prove (iii).

Proof of Lemma 28 (i). Note that
Thus by the independence of the Y z , we have we will split this sum into two parts. We write C 1 , C 2 for constants depending only on σ 2 and c.
Firstly, since h r has Lipschitz constant at most c/r, we have for all z ∈ Z 2 , Secondly, since h r has support in B(0, r), The result now follows by combining (29)-(31).
Proof of Lemma 28 (ii). Note that by Lemma 28 (i), for all r, R > 0, we have that Cov(Λ(h r ), Λ(h R )) < ∞. By (28) and independence of the Y z we have Since h R ↑ 1 as R → ∞, for each z ∈ Z 2 we have by the monotone convergence theorem that Eh R (z + Y z ) ↑ 1 as R → ∞. An additional application of the monotone convergence theorem shows that We will employ the following weak law of large numbers for dependent sequences to prove Lemma 28 (iii).
Lemma 29. Let Z 1 , Z 2 , . . . be real-valued random variables with finite second moments and zero means. If there exists a sequence b(k) with b(k) → 0 as k → ∞ such that E(Z n Z m ) ≤ b(n − m) for all n ≥ m, then (Z 1 + · · · + Z n )/n → 0 in probability as n → ∞. Corollary 30. Let Z 1 , Z 2 , . . . be real-valued random variables with finite second moments and zero means. Suppose that there exists C > 0, such that E|Z m | 2 ≤ C for all m ∈ Z + . If for all m ∈ Z + we have E(Z m Z n ) → 0 as n → ∞, then there exists an increasing sequence of positive of integers (r n ) such that (Z r 1 + · · · + Z rn )/n → 0 in probability as n → ∞. Furthermore, for any further subsequence (r n k ) we have (Z rn 1 + · · · + Z rn k )/k → 0 in probability as n → ∞.
Proof. Consider the sequence b(k) := 1/k, where we set b(0) = C. We will show that there exists a sequence r k so that E(Z rn Z rm ) ≤ 1/n for all n > m. Thus Z r k satisfies the conditions of Lemma 29 with b(k). We proceed by induction. Set r 1 = 1. Suppose that r 2 , . . . , r k−1 have already been defined and satisfy E(Z rn Z rm ) ≤ 1/n for all 1 ≤ m < n ≤ k − 1. It follows from Lemma 28 (ii) that there exists an integer R > 0 such that E(Z rm Z R ) ≤ 1/k for all 1 ≤ m ≤ k − 1; set r k := R. Furthermore, if (r n k ) is a subsequence of (r n ), we have that if m < k, then E(Z rn m Z rn k ) ≤ 1/n k ≤ 1/k. Thus Z rn k satisfies the conditions of Lemma 29 with b(k).  for all x ∈ R and set h n (x) := h(x/n) for x ∈ R and n ∈ Z + . For each n ∈ Z + , let N n := Λ(h n ) − EΛ(h n ). Assume that E|Y 0 | < ∞.
(i) The family of random variables (N n ) n∈Z + is tight and integervalued. (ii) For any k, ℓ ∈ R and a ∈ Z + , P(N a ≤ k, N n ≤ ℓ) − P(N a ≤ k) P(N n ≤ ℓ) → 0 as n → ∞.
(iii) There exists a deterministic sequence (n k ) with n k → ∞ and an integer-valued random variable N such that (27) is satisfied; that is, for all ℓ ∈ R, 1[N n k ≤ ℓ] P → P(N ≤ ℓ) as K → ∞.
As in the case d = 2, Lemma 31 parts (i) and (ii) will allow us to use a weak law of large numbers to prove (iii). Let us note that the assumption that E|Y 0 | < ∞ is not necessary for Lemma 31 part (ii).
For A, B ⊆ R, write T B A := # {z ∈ A ∩ Z : z + Y z ∈ B} ; that is, the number of Λ-points in B that originated from A. Observe that for n ∈ Z + , . (33) On the other hand, E|Y 0 | < ∞ implies easily that K + := ET [0,∞) < ∞. By translation-invariance, each term on the right side of (33) is bounded in expectation by one of these constants; for instance: ET Proof of Lemma 31 (ii). Let F n := σ({z + Y z ∈ [−n, n]} : z ∈ Z). We will show that for any event E ∈ σ(Y z : z ∈ Z), we have P(E | F n ) → P(E) a.s. as n → ∞.
From (34), the result follows, since {N n ≤ ℓ} ∈ F n . It suffices to check (34) for E in the generating algebra of events that depend on only finitely many of the Y z . But for such an event, say E ∈ σ(Y z : −m ≤ z ≤ m), we observe that P(E | F n ) equals the conditional probability of E given the finite σ-algebra σ({z+Y z ∈ [−n, n]} : −m ≤ z ≤ m), hence the required convergence follows from an elementary computation.
Proof of Lemma 31 (iii). By Lemma 31 (i) we may choose an integervalued N and a subsequence (c n ) so that N cn d → N as n → ∞. We will show that for all ℓ ∈ Z, there is a further subsequence c n k =: r k such that 1 n n k=1 1[N r k ≤ ℓ] − P(N r k ≤ ℓ) P → 0 as n → ∞.
Clearly, the result follows from (35) and the fact that N r k d → N as k → ∞.
We use Corollary 30 in conjunction with a diagonal argument to prove (35). Consider an enumeration of the integers given by ℓ 1 , ℓ 2 , . . . For each i ∈ Z + , let Z i k := 1[N c k ≤ ℓ i ] − P(N c k ≤ ℓ i ). By Lemma 31 (ii) and Corollary 30, there exists a subsequence c 1 n k := r 1 k such that (35) holds with r k replaced by r 1 k , and ℓ replaced by ℓ 1 . Similarly, we may choose (r 2 k ) to be a subsequence of (r 1 k ) so that (35) holds with r k replaced by r 2 k , and ℓ replaced by ℓ 2 ; moreover Corollary 30 assures us that (35) holds with r k replaced by r 2 k , and ℓ replaced by ℓ 1 . Similarly define the sequence (r i k ) for each i ∈ Z + . By taking the diagonal sequence r k := r k k , we see that (35) holds for all ℓ ∈ Z. 7.5. Gaussian zeros in the hyperbolic plane. The proof of Proposition 13 uses the following consequence of a result of Peres and Virág.
Proposition 32. If Υ D is the Gaussian zero process on the hyperbolic plane and Υ * D is its Palm version, then Υ * D ≺ Υ D +δ 0 and Υ D +δ 0 ≺ Υ * D . Proof. Let Υ D be the process of zeros of ∞ n=0 a n z n , where the a n 's are i.i.d. standard complex Gaussian random variables. Let E k be the event that Υ D (B(0, 1/k)) > 0. Peres and Virág [20, Lemma 18] prove that the conditional law of (a 0 , a 1 , . . .) given E k converges as k → ∞ to the law of (0, a 1 , a 2 , . . .), where a 1 is independent of the a n 's, and has a rotationally symmetric law with | a 1 | having probability density 2r 3 e −r 2 .
Let Υ D be the process of zeros of the power series with coefficients (0, a 1 , a 2 , . . .). Since the latter sequence is mutually absolutely continuous in law with (0, a 1 , a 2 . . .), we have that Υ D and Υ D +δ 0 are mutually absolutely continuous in law.
By Rouché's theorem from complex analysis [6, Ch. 8, p. 229], the above convergence implies that the conditional law of Υ D given E k converges to the law of Υ D (the convergence is in distribution with respect to the vague topology for point processes). By [12,Theorem 12.8] it follows that Υ D d = Υ * D . Proof of Proposition 13. It follows from Proposition 32 and Theorems 4 and 5 with Remark 1 that the Gaussian zero process on the hyperbolic plane is insertion-tolerant and deletion-tolerant.