Hard-sphere disagreement percolation - Semantic Scholar

Disagreement percolation for the hard-sphere model

arXiv:1507.02521v2 [math.PR] 29 Feb 2016

Hofer-Temmel Christoph∗†

Abstract Disagreement percolation connects a Gibbs lattice gas and iid site percolation such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model, two point processes. The method implies uniqueness of the Gibbs measure at fugacity values at which the Boolean model with the same intensity value does not percolate. Lower bounds on the critical intensity for percolation of the Boolean model yield lower bounds on the critical fugacity for a phase transition. These lower bounds improve previous bounds obtained by cluster expansion techniques.

Keywords: hard-sphere model, disagreement percolation, unique Gibbs measure, stochastic domination, Janossy density MSC 2010: 82B21 (60E15 60K35 60G55 82B43 60D05)

Contents 1 Introduction 2 Setup 2.1 Space . . . . . . . . . . 2.2 Point processes . . . . . 2.3 The Boolean model . . . 2.4 The hard-sphere model . 2.5 Stochastic domination .

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3 Results and discussion 3.1 Disagreement percolation . . . . . . 3.2 Bounds . . . . . . . . . . . . . . . . 3.3 Comparison with expansion bounds . 3.4 Outlook . . . . . . . . . . . . . . . .

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4 Proof of Theorem 3.2

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∗ [email protected] † The author acknowledges the support of the VIDI project “Phase transitions, Euclidean fields and random fractals”, NWO 639.032.916.

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5 Disagreement coupling families 5.1 Prerequisites . . . . . . . . . . . . . . . . . . . 5.2 Joint Janossy measures and densities . . . . . . 5.2.1 Examples for couplings of two PP laws . 5.3 Dependently thinning Poisson to hard-sphere . 5.3.1 An integral equation . . . . . . . . . . . 5.3.2 Proof of Theorem 5.3 . . . . . . . . . . 5.4 The product disagreement coupling family . . . 5.5 The twisted disagreement coupling family . . .

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12 12 13 15 15 16 18 21 23

Introduction

Disagreement percolation by van den Berg and Maes [26] is a sufficient condition on the fugacity of a discrete Markov specification on a graph for uniqueness of the Gibbs measure. It implies the absence of phase transitions and the analyticity of the free energy in the high-temperature case. It has also been used to derive the Poincar´e inequality in the context of lattice Ising spin systems [4]. This paper generalises disagreement percolation to the hard-sphere model on Rd , the continuum equivalent of the well-studied hard-core model [27]. The core of disagreement percolation is a non-trivial coupling between three point processes on a bounded domain. Two are hard-sphere models with the same fugacity and possibly differing boundary conditions. The third one is a Boolean model stochastically dominating the symmetric difference between the two hard-sphere models. The connected components of the Gilbert graph of the Boolean model dominate the differing influence of the boundary conditions on the hard-sphere models. In the sub-critical phase of percolation, the almost-sure finite percolation clusters imply the equality of the two hard-sphere realisations with high probability inside a small domain inside a larger domain. Taking a limit along an exhaustive sequence of bounded domains implies the uniqueness of the Gibbs measure of the hard-sphere model. The disagreement coupling connects the fugacity of the hard-sphere models and the intensity of the Poisson point process. This allows the derivation of concrete bounds from the critical intensity of the Boolean model of percolation. In one dimension, the results replicate Tonk’s classic result of the complete absence of phase transitions [24]. In two dimensions and the high dimensional case, the new bounds improve upon the best known cluster expansion bounds [22, 8] by at least a factor of two. This work exclusively treats the hard-sphere model. One reason is its central importance in statistical mechanics and its easy and emblematic definition. More important though, the bounds in this paper stem from a simple product and a conjecturally optimal twisted approach. While a generalisation of the product approach to simple finite-range Markov point processes with bounded interaction-range seems possible, the twisted construction depends critically on the hard-sphere constraint. The twisted approach takes inspiration from an improvement of disagreement percolation for the hard-core model [27]. The disagreement couplings in this paper use conditional couplings between a hard-sphere model and its dominating Poisson point process. The measurability of these conditional couplings with respect to its boundary conditions has not been a topic in the relevant literature on couplings [9] yet. One solution to

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the measurability problem is the joint Janossy density of a coupling of several point processes. These allow to model non-trivial couplings with almost-sure common points. They permit to describe the disagreement couplings by a recursive family of regular conditional densities of couplings of point processes. The explicit form of the densities shows their measurability directly. This permits an interpretation of the disagreement couplings in this paper as conditional dependent thinnings of a base Poisson point process. The thinning approach is the key to ignore the uncountable non-graph structure of Rd and restrict to the finite set of points where interesting things happens. Section 2 introduces notation and basic terms. The main theorems, resulting bounds and discussion are in Section 3. Proofs follow in Section 4 and Section 5. Joint Janossy densities are in Section 5.2 and only appear in the proofs, but not the main results.

2

Setup

This section introduces the notation and models used in this work.

2.1

Space

Consider the Euclidean space Rd with the Euclidean metric ||.|| and the Lebesgue measure L. The bounded and all Borel sets of Rd are Bb and B respectively. Fix a positive finite radius R. For x ∈ Rd , let S(x) := {y ∈ Rd | ||x − y|| < R} be the open sphere S of radius R around x. The volume of S(x) is vd Rd . For B ∈ B, let S(B) := x∈B S(x) and R(B) := S(B) \ B be the sphere and ring of radius R around B respectively. A van Hove sequence [22, Def 2.1.1] is a monotone increasing sequence (Bn )n∈N of bounded Borel sets converging to Rd and eventually containing every bounded Borel set. The increasing hypercubes ([−n, n]d )n∈N are a van Hove sequence. For B ∈ B, let CB be the locally finite point configurations on B, i.e., for each C ∈ CB and A ∈ Bb , |C ∩ A| < ∞. Let FB be the σ-algebra on CB generated by n , the standard product σ-algebra is {{C ∈ CB | C ∩ A = ∅} | B ⊇ A ∈ B}. On CB ⊗n FB . For J : CB → R measurable and non-negative, consider the integral Z J(C)dC := CB

Z n ∞ Y X 1 J({x1 , . . . , xn }) dxi . n! B n n=0 i=1

(1)

This notation ignores the incidental ordering of points introduced by the integrals in the rhs of (1) in generating functions over CB . The measure on CB with infinitesimal dC on the lhs of (1) is L?B defined by L?B ({C ∈ CB | C ∩ A = k}) :=

∞ X 1 L(B \ A)n−k L(A)k . n!

n=k

R−con

Two points x and y are R-connected by a configuration C, written x ←−−−→ y, in C

if there is a finite path of jumps of less than R distance between x and y using only points in C as intermediate nodes. Two Borel sets are R-connected by a configuration C, if there is a R-connected pair of points, with one point from 3

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each set. A configuration C is R-connected, if all its members are R-connected. R−con In other words, for all {x, y} ⊆ C, x ←−−−→ y. It is also equivalent to the conin C

nectedness of the graph on C with edges {{x, y} ⊆ C | ||x − y|| < R}. In this case, call C a R-cluster,

2.2

Point processes

A simple point process (short PP) on a Borel set B ∈ B is a random variable taking values in CB . This work treats a PP as a locally finite random subset of points of Rd , instead of as a random measure or as a collection of marginal counting rvs. Let P be a PP law and denote by ξ the canonical variable on CRd . A Borel measure M on (CB , FB ) is the local Janossy measure [5, after (5.3.2)] of P on B ∈ Bb , if Z ∀E ∈ FB : P(ξ ∩ B ∈ E) = M (dC) . (2) E

A measurable function J : CB → [0, ∞[ is the local Janossy density [5, after (5.3.2)] of P on B ∈ Bb , if Z ∀E ∈ FB : P(ξ ∩ B ∈ E) = J(C)dC . (3) E

The Janossy density J is the is the Radon-Nikodyn derivative of M with respect to L?B . These definitions of local Janossy measure and density are portmanteau versions of the traditional definitions on generating cylinder sets. If ξ has finite moments measures of all orders under P, then all local Janossy measures exist [5, Theorem 5.4.I]. For B ∈ Bb and C ∈ CB , write the infinitesimal of the local Janossy measure of P on B at C as P(ξ ∩ B = dC). If P admits a Janossy density on B, then denote it at C ∈ CB by the expression P(ξ ∩B = C). The expression P(ξ ∩B = ∅) is both the void probability and the Janossy density at ∅, which always coincide. The view of the Janossy density as Radon-Nikodyn derivative [5, Lemma 5.4.III] gives L?B -a.e. P(ξ ∩ B = dC) = P(ξ ∩ B = C)dC .

(4)

Because the local Janossy measure (density) on B ⊇ A ∈ Bb of a PP law P on B equals the Janossy measure of the restriction of the law to A, the remainder of this paper drops the quantifier “local”. Let [.] be Iverson brackets. For disjoint A, B ∈ Bb and E ∈ FB , the following identities (stated only in the density version) hold. Z P(ξ ∩ A = Y ) = P(ξ ∩ (A ∪ B) = Y ∪ Z)dZ , (5a) C Z B P(ξ ∩ A = Y, ξ ∩ B ∈ E) = [Z ∈ E]P(ξ ∩ (A ∪ B) = Y ∪ Z)dZ , (5b) C Z B [Z ∈ E]P(ξ ∩ (A ∪ B) = Y ∪ Z) dZ . (5c) P(ξ ∩ A = Y |ξ ∩ B ∈ E) = P(ξ ∩ B ∈ E) CB The intensity measure of the PP law P is the average R number of points on bounded Borel sets. For B ∈ Bb , this is the quantity CB |C|P(ξ ∩ B = dC). 4

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If the intensity measure is absolutely continuous with respect to L, then the Radon-Nikodyn derivative of the intensity measure is the intensity. Poi The classic example of a PP law is the Poisson PP law PB,α of intensity α on B ∈ Bb . Its Janossy density at C ∈ CB fulfils Poi PB,α (ξ = C) = exp(−αL(B))α|C| .

2.3

(6)

The Boolean model

A C ∈ CRd R-percolates, if it contains an infinite R-connected component. The bounded finiteness of C renders this equivalent to the existence of an unbounded R-connected component. The Boolean model of intensity α is a PRPoi d ,α -distributed PP ϕ, with discs of radius R/2 centred at the points of ϕ. If discs overlap, then the corresponding centre points are connected. This is just R-connectivity from Section 2.1. The Boolean model percolates, if ϕ contains an infinite R-connected component. Adding more points improves R-connectivity. Hence, the probability of percolation is monotone increasing in α. The Poissonian nature of the Boolean model makes percolation a tail event, i.e., it holds with either probability 0 or 1. Thus, a critical intensity separating the non-percolating and percolating regimes exists. Theorem 2.1 ([17, Theorem 3.3]). For d ≥ 2, a λb (d) ∈]0, ∞[ separates the sub-critical (almost-never percolating) from the super-critical (almost-surely percolating) intensities. If α < λb (d) and (Bn )n∈N is van Hove, then R−con

Poi PB (A ←−−−→ R(Bn )) −−−−→ 0 . n ,α n→∞

in ξ

(7)

In dimension one, percolation almost-never happens at finite intensities. Whence, λb (1) = ∞ [17, Theorem 3.1]. In the subcritical regime, the size of the cluster containing the origin decays exponentially [17, Section 3.7]. Section 3.2 discusses bounds on λb (d).

2.4

The hard-sphere model

For disjoint Y, C ∈ CRd , the indicator of the conditional hard-core constraint of Y under condition C is Y Y H(Y |C) := [||x − y|| ≥ R] [||x − y|| ≥ R] . (8) y∈Y,x∈C

{x,y}⊆Y

For a bounded domain B ∈ Bb , a boundary condition C ∈ CB c and a fugacity hs λ ∈ [0, ∞[, consider the hard-sphere model with law PB,C,λ . It has the partition function Z λ|Y | H(Y |C)dY .

(9)

λ|Y | H(Y |C) Poi = PB,λ (ξ = Y |H(ξ|C) = 1) . Z(B, C, λ)

(10)

Z(B, C, λ) := CB

Its Janossy density is hs PB,C,λ (ξ = Y ) =

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The rhs expression in (10) refers to the fact that the hard-sphere model is a conditioned Poisson PP of intensity λ. Because of the bounded range interaction in H(Y |C) in (8), one may restrict the boundary condition C ∈ CB c to C ∩ R(B) ∈ CR(B) . hs A Gibbs measure is a weak limit of a sequence (PB ) along a van Hove n ,Cn ,λ n∈N sequence (Bn )n∈N and a sequence (Cn )n∈N of boundary conditions with Cn ∈ CBnc [20, Sections 2 and 3]. The Gibbs measures Gλ of the specification Pλhs := hs (PB,C,λ )B∈Bb ,C∈CBc form a simplex. Unlike in the lattice case [23], in the continuum case of dimension greater than one, the existence of a finite critical fugacity at which a phase transition happens is widely believed, but not yet proven. See the solution in one dimension [24] and the discussion of the state of the problem in higher dimensions [16, Section 3.3].

2.5

Stochastic domination

Let [n] := {1, . . . , n}. A coupling P of n PP laws P1 , . . . , Pn on B ∈ B is ⊗n n , FB ) such that, for all i ∈ [n] and E ∈ FB , a probability measure on (CB n are ξ1 , . . . , ξn . P(ξi ∈ E) = Pi (ξ ∈ E). The canonical variables on CB A PP law P2 stochastically dominates a PP P1 , if there exists a coupling P of them with P(ξ1 ⊆ ξ2 ) = 1. A Poisson PP stochastically dominates a hard-sphere model of the same fugacity as the intensity of the Poisson PP. Lemma 2.2 ([9, Example 2.2]). Let B ∈ Bb , C ∈ CB c and α, λ ∈ [0, ∞[. If Poi hs dom α ≥ λ, then PB,α stochastically dominates PB,C,λ . Let PB,C,λ be the dominating coupling. The Papangelou intensity [6, (15.6.13)] ρP (x, Y ) of a PP law P at x ∈ B ∈ Bb with condition Y ∈ CB\{x} denotes the infinitesimal cost under P of adding a new point at x to the configuration Y . The Papangelou intensity of a hardsphere PP law with fugacity λ is bounded from above by the one of a Poisson PP law of intensity λ. Poi (x, Y ) . ρPB,C,λ (x, Y ) = H({x}|Y ∪ C)λdx ≤ λdx = ρPB,λ hs

(11)

This is the key point behind Lemma 2.2.

3 3.1

Results and discussion Disagreement percolation

At the core of disagreement percolation is a coupling of two instances of the hardsphere model on the same finite volume, but with differing boundary conditions, such that the set of points differing between the two instances (the disagreement cluster ) is stochastically dominated by a Poisson point process. Therefore, one may control the disagreement clusters and the influence of the differing boundary conditions by the percolation clusters of the boolean disc model. If the intensity of the dominating Poisson point process is below the critical value for percolation in the boolean disc model, then the finiteness of percolation clusters controls the influence of the differing boundary conditions. The influence vanishes as the finite volume tends to the whole space. This implies the uniqueness of the Gibbs measure of the hard-sphere model. 6

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The remainder of this section formalises the above outline. The symmetric difference S1 4 S2 between sets S1 and S2 equals (S1 \ S2 ) ∪ (S2 \ S1 ). A disagreement coupling couples three PPs in the fashion described above. Definition 3.1. For α, λ ∈ [0, ∞[. A disagreement coupling on B ∈ Bb and C1 , C2 ∈ CB c at intensity α and fugacity λ is a law P := PB,C1 ,C2 ,λ,α on ⊗3 3 (CB , FB ) with ∀i ∈ [2], E ∈ FB : ∀E ∈ FB :

hs P(ξi ∈ E) = PB,C (ξ ∈ E) , i ,λ

(12a)

Poi PB,α (ξ

(12b)

P(ξ3 ∈ E) =

∈ E) ,

P(ξ1 4 ξ2 ⊆ ξ3 ) = 1 , R−con

P(∀x ∈ ξ1 4 ξ2 : x ←−−−−→ C1 4 C2 ) = 1 . in ξ1 4ξ2

(12c) (12d)

A disagreement coupling family at intensity α and fugacity λ is a family of disagreement couplings (PB,C1 ,C2 ,λ,α )B∈Bb ,C1 ,C2 ∈CBc . A disagreement coupling in the sub-critical phase of the Boolean model implies uniqueness of the Gibbs measure. Theorem 3.2. If there exists a disagreement coupling family of intensity α < λb (d) at fugacity λ, then Gλ consists of a single Gibbs measure. The proof of Theorem 3.2 is in Section 4. A simple attempt to construct a disagreement coupling is via the following product construction. Algorithm 3.3. Fix λ ∈ [0, ∞[, B ∈ Bb and C1 , C2 ∈ CB c . Sample (η1 , ϕ1 ) dom dom according to PB,C and (η2 , ϕ2 ) according to PB,C independently of each other. 1 2 Regard the CS(B) -valued random (γn )n∈N given by ( ∅ if n = 1 , γn := (13a) {x ∈ η1 4 η2 | x ∈ S(γn−1 ∪ (C1 4 C2 ))} if n > 1 . The sequence γ is monotone increasing and stabilizes almost-surely after a finite number of steps. Let χ := S( lim γn ) ∩ B. Independent of before and each other, n→∞

dom Poi sample (η3 , ϕ3 ) according to PB\χ,∅ and ϕ4 according to PB\χ,λ . Regard the PP triple

((η1 ∩ χ) ∪ η3 , (η2 ∩ χ) ∪ η3 , (ϕ1 ∩ χ) ∪ (ϕ2 ∩ χ) ∪ ϕ3 ∪ ϕ4 ) .

(13b)

The above sampling procedure is well-defined (see Proposition 5.11). It is easy to see that it fulfils properties (12c) and (12d). Determining if the coupling has the correct marginals, i.e., properties (12a) and (12b), poses a problem, though. For example, what is the law of (η1 ∩ χ)|χ? Our knowledge of the dom coupling PB,C is not sufficient to answer this. We remedy this problem by describing an explicit coupling between a hard-sphere PP law and its dominating Poisson PP law obtained via a dependent thinning in Section 5.3. The new coupling has an explicit density measurable in the boundary condition. Using this as a building block, we derive an explicit disagreement coupling in the spirit of Algorithm 3.3 in Section 5.4. As a consequence, we obtain a first sufficient condition for uniqueness of the Gibbs measure. 7

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Theorem 3.4. There exists a disagreement coupling family of intensity 2λ for Pλhs . If λ < 12 λb (d), then Gλ is a singleton. The proof of Theorem 3.4 is in Section 5.4. Another possible approach to Theorem 3.4 might be a generalisation of the first disagreement approach by hs hs van den Berg [25]. Take the product law PB,C × PB,C and apply the glueing 1 2 lemma to bound the symmetric difference ξ1 4 ξ2 , which is just ξ1 ∪ ξ2 under Poi the preceding product law, by a PB,2λ . Then, control a disagreement cluster growing from B ⊇ A ∈ Bb by the percolation clusters. Again, the problem, as in algorithm 3.3, is that we do not know enough about the properties of the conditional laws under the coupling. An optimisation of the coupling behind Theorem 3.4 yields a twisted version halving the Poisson intensity. The “twist” refers to the fact two hard-sphere models of fugacity λ fit under a single dominating Poisson(λ) PP. The result is better sufficient condition for uniqueness of the Gibbs measure. Theorem 3.5. There exists a disagreement coupling family of intensity λ for Pλhs . If λ < λb (d), then Gλ is a singleton. The proof of Theorem 3.5 is in Section 5.5.

3.2

Bounds

This section combines Theorem 3.5 with knowledge about the critical intensity of the Boolean model. Bounds on λb (d) translate directly into sufficient conditions for the uniqueness of the Gibbs measure. In one dimension the Boolean model never percolates [17, Theorem 3.1]. λb (1) = ∞ .

(14a)

0.843 In dimension two, rigorous bounds on λb (2) are [ 0.174 R2 , R2 ] [17, Theorem 3.10]. More recent high confidence bounds in [1], taken from [18, equation (2)], are

λb (2) >

1.127 0.358 > . 2 πR R2

(14b)

See also very similar simulation bounds in [18, table 1] and [21]. In high dimensions, the asymptotic behaviour of the critical intensity [19], taken from [17, Section 3.10], is lim λb (d)vd Rd = 1 . (14c) d→∞

3.3

Comparison with expansion bounds

Popular methods to study the absence of phase transitions, in particular to guarantee the uniqueness of the Gibbs measure, are virial and cluster expansion methods [22]. Both deliver analyticity of the free energy, too. Let λce (d) be the radius of the cluster expansion in d dimensions. In one dimension, disagreement percolation (14a) replicates Tonks’ classic result of the complete absence of phase transitions via virial expansion methods [3, 12, 15, 24]. In terms of the fugacity, it is known that the radius of the cluster expansion is exactly [3, 10, 15] λce (1) = 8

1 . eR

(15a)

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General bounds on the cluster expansion radius are [22, Section 4] 1 1 ≤ λce (d) ≤ . evd Rd vd R d

(15b)

In two dimensions, the best currently known cluster expansion bound [8] is λce (2) ≥

0.5107 0.1625 ∼ . 2 πR R2

(15c)

This is about 2.2 times less than the disagreement percolation bound (14b). The upper bound (15e) equals λce (2) ≤

0.2342 2 = . eπR2 R2

(15d)

This shows that cluster expansion bounds are always worse than disagreement percolation in two dimensions. A general upper bound on the cluster expansion radius [11]: λce (d) ≤

2 . evd Rd

(15e)

The upper bound is tight in one dimension, as v1 = 2 and (15a) holds. We conjecture the upper bound to not be tight in other dimensions. We base this on the following conjecture about the asymptotic behaviour in high dimensions lim λce (d)vd Rd =

d→∞

1 . e

(15f)

Comparing (15f) with (14c), we see that the asymptotic improvement should be by a factor e. This is not completely surprising. On the infinite k-regular 1 tree Tk , the critical percolation probability is k−1 and the radius of the cluster k−2

1 . Both Zd and Rd behave for large d as T2d , for expansion is (k−2) ∼ e(k−1) (k−1)k−1 both percolation and cluster expansion. Finally, I conjecture that disagreement percolation is always (i.e., on every reasonable space) better than cluster expansion. I also believe that the improvement on Rd decreases as the dimension d increases, and that different lower order terms are hidden behind the limits (14c) and (15f).

3.4

Outlook

I think that an extension of the results to the case of general simple finite-range Markov point processes as well as the physically more interesting case of marked Markov point processes with bounded interaction-range, and maybe even finite but unbounded interaction-range is possible, but demands a notational and definitional base exceeding the limits of a single paper. Beyond yielding a sufficient condition for uniqueness of the Gibbs measure, one should be able to derive complete analyticity of the free energy (as pointed out by Schonmann [26, Note added in proof]) and the Poincare inequality for Markov point processes [4]. Both results should build on the exponential decay of the size of subcritical percolation clusters in the Boolean model [17, Section 3.7]. 9

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For dimension 3, it would be nice to have the generalisation `a la [2] of the high confidence critical percolation probability [1] not only for site percolation but also for continuum percolation. Another sufficient condition for uniqueness Gibbs measure, and even complete analyticity of the free energy, is Dobrushin’s uniqueness condition [7]. There have been some generalisations to the continuous case [13, 14], but I have not been able to work out the particular form for the hard-sphere model, yet. hs-poi dom Both the couplings PB,C from [9] and PB,C from Section 5.3 show that a Poi hs PB,λ law stochastically dominates every PB,C,λ law. It is, to my knowledge, not yet proven, that this is the smallest intensity Poisson law one can use. Even if it would be the smallest, the piecewise recursive construction of the disagreement coupling in Section 5.5 demands only stochastic domination on some restricted class of domains with free boundary conditions. I do not know, if this allows to lower the intensity of the dominating Poisson PP.

4

Proof of Theorem 3.2

The proof of Theorem 3.2 follows closely the one in the discrete case [26, proof of corollaries 1 and 2]. Proposition 4.1 applies a disagreement coupling to bound the difference between the two hard-sphere models by a percolation connection probability. Lemma 4.2 is a generic bound on differences of integrals of bounded functions. Theorem 3.2 uses a disagreement coupling family to exploit these bounds on increasing scales. First, it restricts to a small domain, then it applies the bounds from disagreement coupling and Lemma 4.2 and finally, it uses the sub-criticality of the Boolean model to tighten the bound to zero as the domain increases. Proposition 4.1. Let A, B ∈ Bb with A ⊆ B, C1 , C2 ∈ CB c and α, λ ∈ [0, ∞[.S Let P := PB,C1 ,C2 ,λ,α be a disagreement coupling. Let E ∈ FA and F := Y ∈E {Y } × CB\A be its embedding into FB . The disagreement bound is R−con

hs hs Poi |PB,C (F ) − PB,C (F )| ≤ PB,α (A ←−−−→ C1 4 C2 ) . 1 2 in ξ

(16)

Proof. First, reduce the difference by cancelling symmetric parts. (12a)

hs hs |PB,C (F ) − PB,C (F )| = |P(ξ1 ∈ F ) − P(ξ2 ∈ F )| 1 2

= |P(ξ1 ∈ F, ξ2 6∈ F ) − P(ξ1 6∈ F, ξ2 ∈ F )| ≤ max{P(ξ1 ∈ F, ξ2 6∈ F ), P(ξ1 6∈ F, ξ2 ∈ F )} . Second, relax the asymmetric event to disagreement and use the properties of disagreement percolation. relax

P(ξ1 ∈ F, ξ2 6∈ F ) ≤ P((ξ1 4 ξ2 ) ∩ A 6= ∅) (12d)

R−con

= P(A ←−−−−→ C1 4 C2 ) in ξ1 4ξ2

(12c)

R−con

≤ P(A ←−−−→ C1 4 C2 ) in ξ3

(12b)

=

R−con Poi PB,α (A ←−−−→ C1 in ξ

10

4 C2 ) .

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Lemma 4.2. Let µ1 and µ2 be probability measures on the measurable space (Ω, A). For all f : Ω → [0, 1] measurable, Z Z f dµ1 − f dµ2 ≤ sup{|f (ω1 ) − f (ω2 )| | ω1 , ω2 ∈ Ω} . (17) Proof. Let f + := sup{f (ω) | ω ∈ Ω} ≤ 1 and f − := inf{f (ω) | ω ∈ Ω} ≥ 0. It follows that Z Z Z Z f dµ1 − f dµ2 ≤ f + dµ1 − f − dµ2 Z Z = f + dµ1 − f − dµ2 = f+ − f− = sup{f (ω1 ) | ω1 ∈ Ω} − inf{f (ω2 ) | ω2 ∈ Ω} = sup{f (ω1 ) | ω1 ∈ Ω} + sup{−f (ω2 ) | ω2 ∈ Ω} ≤ sup{f (ω1 ) − f (ω2 ) | ω1 , ω2 ∈ Ω} ≤ sup{|f (ω1 ) − f (ω2 )| | ω1 , ω2 ∈ Ω} . Proof of Theorem 3.2. Let ν1 , ν2 ∈ Gλ . The aim is to show that ν1 = ν2 . This is equivalent to ∀A ∈ Bb , E ∈ FA :

ν1 (ξ ∩ A ∈ E) = ν2 (ξ ∩ A ∈ E) .

(18)

The hard-sphere property ensures that a Gibbs measure in Gλ has moment measures of all orders [5, (5.4.9)]. Thus, its local Janossy measures exist. The local Janossy densities might not exist, because, for λ going to infinity, the Gibbs measure might be concentrated on a single configuration. Let S (Bn )n∈N be a van Hove sequence with A ⊆ B1 . For every n ∈ N, let En := Y ∈E {Y } × CBn \A be the embedding of E into FBn . For each Gibbs measure ν ∈ Gλ and n ∈ N, the Markov property restricts the discussion to the bounded Borel set Bn . Z ν(ξ ∩ A ∈ E) = ν(ξ ∩ A ∈ E, ξ ∩ Bnc = dC) c CBn

(5)

Z

=

ν(ξ ∩ A ∈ E|ξ ∩ Bnc = C)ν(ξ ∩ Bnc = dC)

c CBn

Z = c CBn

Z = c CBn

hs PB (ξ ∩ A ∈ E)ν(ξ ∩ Bnc = dC) n ,C hs PB (En )ν(ξ ∩ Bnc = dC) . n ,C

Second, the existence of a disagreement coupling family at intensity α and Lemma 4.2 controls the difference between different Gibbs measures by the connection probability of the Boolean model. Taking the limit along the van Hove sequence shows that the difference is zero. |ν1 (ξ ∩ A ∈ E) − ν2 (ξ ∩ A ∈ E)|

11

Hard-sphere disagreement percolation

Hofer-Temmel

Z = P hs (E )ν (ξ ∩ Bnc = dC1 ) CBc Bn ,C1 n 1 n Z hs c − PBn ,C2 (En )ν1 (ξ ∩ Bn = dC2 ) CB c n

(17)

 hs hs (En ) − PB (En ) C1 , C2 ∈ CBnc ≤ sup PB n ,C1 n ,C2   (16) R−con Poi c C , C ∈ C ≤ sup PB (A ← − − − → C 4 C ) 2 Bn 1 2 1 n ,α in ξ

relax

R−con

Poi ≤ PB (A ←−−−→ Bnc ) n ,α in ξ

(7)

−−−−→ 0 . n→∞

5

Disagreement coupling families

Section 5.1 introduces helper quantities and notation. The proof of Theorem 3.2, i.e., that disagreement percolation guarantees uniqueness of the Gibbs measure, is in Section 4. Section 5.2 introduces the joint Janossy measure and density of a PP coupling. Section 5.3 states the joint Janossy density of a new coupling between the hard-sphere and Poisson PP laws explicitly. The coupling is a key building block of the disagreement coupling families in sections 5.4 and 5.5. The product and twisted disagreement coupling families are in sections 5.4 and 5.5 respectively. If λ is clear from context, it may be omitted in subscripts.

5.1

Prerequisites

A notion of size adapted to the hard-sphere setting is h(B) := sup{|C| | C ∈ CB , H(C) = 1} .

(19)

In particular, h(B) ≤ (diam(B) + 1)d . The basis for several inductive proofs is the strict monotonicity of h(B) under removing spheres around its member points. ∀x ∈ B : h(B \ S(x)) < h(B) . (20) The conditional hard-sphere constraint chains. ∀X, Y, Z ∈ CRd :

H(X ∪ Y |Z) = H(X|Y ∪ Z)H(Y |Z) .

(21)

So do the Janossy densities of the hard-sphere model. For Y ∈ CB and B ⊇ A ∈ Bb , one has hs hs hs PB,C (ξ ∩ A = Y ∩ A)PB\A,C∪(Y ∩A) (ξ = Y \ A) = PB,C (ξ = Y ) .

(22)

Removing a point from the boundary condition restricts the partition function to a smaller volume. In general, this leads to Z(B, C ] {x}) = Z(B \ S(x), C)

and

Z(B, C) = Z(B \ S(C), ∅) .

(23)

Fix λ ∈ [0, ∞[ and B ∈ Bb . The function CB c × CB → {0, 1}

(C, Y ) 7→ H(Y |C) 12

(24)

Hard-sphere disagreement percolation

Hofer-Temmel

is measurable on (CB c ×CB , FB c ⊗FB ) as a product of measurable functions (8). This implies that the function Z CB c → [0, ∞[ C 7→ Z(B, C, λ) = H(Y |C)λ|Y | dY (25) CB

is measurable on (CB c , FB c ). This function is also monotone increasing in B and monotone decreasing in C. The partition function has the uniform bounds Z 1 ≤ Z(B, C, λ) ≤ λ|Y | dY = exp(λL(B \ S(C))) . (26) CB

The density has the uniform bounds hs 0 ≤ PB,C,λ (ξ = Y ) ≤ λh(B) .

5.2

(27)

Joint Janossy measures and densities

This section extends the concept of Janossy measure and density from a PP law to a coupling of PP laws. A closely related problem appears around Janossy measures of non-simple PPs [5, Proposition 5.4.V]. Let n ≥ 2 and P be a coupling of n PP laws. ⊗n n , FB ) is the (local) joint Janossy Definition 5.1. A Borel measure M on (CB measure of P on B ∈ Bb , if, for all E1 , . . . , En ∈ FB , Z Y P(∀i ∈ [n] : ξi ∩ B ∈ Ei ) = [Yi ∈ Ei ]M (dY ) . (28) n CB i∈[n]

Because the local joint Janossy measure on A ⊆ B of a coupling P on B equals the joint Janossy measure of the restriction of the coupling to A, the remainder of this paper drops the quantifier “local”. This definition of a joint Janossy measure is between the portmanteau style of the classicQcase (2) and the n explicit style on generating sets in [5, Section 5.3]. As the sets i=1 Ei generate ⊗n FB , there is no loss of generality. If P admits a joint Janossy measure on B, n . then P(∀i ∈ [n] : ξi ∩ B = dYi ) denotes it at Y ∈ CB Even if all the marginal Janossy measures are diffuse, there may be positive n M -mass on lower-dimensional subspaces of CB . These describes dependencies of the PPs under the coupling. For stochastic domination, positive mass lies on the diagonal of the space according to the dominated PP law. One would like to define a joint Janossy density by an analogue of the identity (4). The possibly non-trivial mass on diagonals is an impediment, because the product measure L?B ⊗n does not assign mass to these. A solution is to track the PPs a point belongs to explicitly and adapt the integration appropriately. n The space CB has the canonical variables (ξi )i∈[n] . Let In := {I ⊆ [n] | I 6= In ∅}. The space CB has the canonical variables (ξI )I∈In . The identities  !  \ [ [ ξi = ξI and ξI = ξi \  ξj  (29) i∈I

i∈I⊆[n]

j∈[n]\I

In In n n describe a surjection from CB to CB and an injection from CB to CB . The In ? ⊗In product measure LB assigns no mass to diagonal subspaces of CB . Thus,

13

Hard-sphere disagreement percolation

Hofer-Temmel

the identities (29) describe a L-a.e. (short for L?B ⊗n -a.e. on the lhs and L?B ⊗In a.e. on the rhs) bijection. Definition 5.2. Let n ≥ 2 and P be a coupling of n PP laws. A measurable In function J : CB → [0, ∞[ is the (local) joint Janossy density of P on B ∈ Bb , if, for all E1 , . . . , En ∈ FB , 



Z P(∀i ∈ [n] : ξi ∩ B ∈ Ei ) =

In CB



Y

J(X) 

[ 

i∈[n]

XI ∈ Ei  dX .

(30)

i∈I⊆[n]

Because the local joint Janossy density on A ⊆ B of a coupling P on B equals the joint Janossy density of the restriction of the coupling to A, the remainder of this paper drops the quantifier “local”. The definition uses the same semi-portmanteau style on generating sets as the definition of the joint Janossy measure (28). The rhs of (30) uses the L-a.e. bijection (29) to model intersections between [ the PPs explicitly. For example, the points common to ξi and ξj are in XI . If P admits a joint Janossy density on B ∈ Bb , then {i,j}⊆I⊆[n] In . Using the L-a.e. bijection (29), P(∀I ∈ In : ξi ∩ B = XI ) denotes it at X ∈ CB n a shorthand of the joint Janossy density is P(∀i ∈ [n] : ξi ∩ B = Yi ) at Y ∈ CB . In Integration over this shorthand remains over CB , though. For a product coupling, the joint Janossy density is the product of the n gives marginal Janossy densities. Using the L-a.e. bijection (29) at Y ∈ CB 
T S XI := i∈I Yi \ j∈[n]\I Yj and

P(∀i ∈ [n] : ξi ∩ B = Yi ) n Y Y = [Yi ∩ Yj = ∅] Pi (ξ ∩ B = Yi )  = 

(31)

i=1

{i,j}⊆[n]

 Y

[XI = ∅]

! Y

Pi (ξ ∩ B = X{i} )

.

i∈n

I∈In ,|I|≥2

The joint Janossy measure and the joint Janossy density do not live on the same space. Nevertheless, because the lhs of (28) and (30) agree, a L-a.e. n identification of the rhs is possible. For Y ∈ CB , the L-a.e. bijection (29) implies that P(∀i ∈ [n] : ξi ∩ B = dYi )  = P ∀I ∈ In : ξI ∩ B = XI :=



!  \ i∈I

Yi

\

[

j∈[n]\I

Yj 

Y

dXI . (32)

I∈In

In n This identity holds with respect to integration over CB and CB on the left- and right-hand sides respectively. The identities (5) generalise directly from the classic to the joint case. Joint Janossy measures (densities) of marginals of a coupling P result from integrating out the joint Janossy measure (density) over the complementary variables.

14

Hard-sphere disagreement percolation

Hofer-Temmel

5.2.1

Examples for couplings of two PP laws

For a coupling P of two PP laws, condition (30) reduces to ∀E1 , E2 ∈ FB : P(ξ1 ∩ B ∈ E1 , ξ2 ∩ B ∈ E2 ) Z = [Y1 ∪ X ∈ E1 ][Y2 ∪ X ∈ E2 ]J(Y1 ∪ X, Y2 ∪ X)dY1 dY2 dX . 3 CB

Here, X models the points common to both PPs. The Janossy density of the first marginal on B ∈ Bb at Y ∈ CB follows by integrating out over the additional points of ξ2 . XZ P(ξ1 ∩ B = Y ) = P(ξ1 ∩ B = Y, ξ2 ∩ B = X ∪ Z)dX . Z⊆Y

CB

For a product coupling, the joint Janossy density is the product of the marginal Janossy densities (31). (P1 × P2 )(ξ1 ∩ B = Y1 , ξ2 ∩ B = Y2 ) = P1 (ξ ∩ B = Y1 )P2 (ξ ∩ B = Y2 ) .

5.3

Dependently thinning Poisson to hard-sphere

This section presents a coupling between a hard-sphere PP law and a dominating Poisson PP law in Theorem 5.3. The joint Janossy density of the coupling is explicit, in contrast to the opaque coupling from Lemma 2.2. This explicit form shows its measurability with respect to the boundary condition. For x ∈ Rd , let dn,i (x) be the nth binary digit of the ith coordinate of x. Define the order ≺ on Rd , by ordering (dn,i (.))n∈Z,i∈[d] lexicographically first over n ∈ Z and then over i ∈ [d]. The order ≺ is total and measurable, i.e., the sets of the form {x | x ≺ y}, for y ∈ Rd , are L-measurable. The symbols ±∞ extend ≺ with elements being bigger and smaller than each element of Rd . For a, b ∈ Rd ∪ {±∞} with a ≺ b, there is the interval ]a, b] := {x | a ≺ x  b}, as well as all standard variations thereof. Each bounded Borel set is contained in a finite interval. Fix λ ∈ [0, ∞[, B ∈ Bb and C ∈ CB c . Implicitly restrict intervals ]a, b] to B (i.e., ]a, b] ∩ B). For x ∈ B and Y ∈ CB , let Yx := Y ∩] − ∞, x[, Z([x, ∞[\S(x), C ∪ Yx , λ) Z([x, ∞[, C ∪ Yx , λ)

(33a)

qB,C (x, Y ) := [x ∈ Y ]pB,C (x, Y ) + [x 6∈ Y ](1 − pB,C (x, Y )) .

(33b)

pB,C (x, Y ) := H({x}|C ∪ Yx ) and

The monotonicity of Z(.) in the domain implies that pB,C (x, Y ) ∈ [0, 1]. hs-poi hs Theorem 5.3. There is a coupling PB,C,λ between a hard-sphere law PB,C,λ I2 Poi and a dominating Poisson law PB,λ , whose joint Janossy density at X ∈ CB is hs-poi PB,C,λ (∀I ∈ I2 : ξI = XI ) Poi = [X{1} = ∅]PB,λ (ξ = X{1,2} ∪ X{2} )

Y x∈X{1,2} ∪X{2}

15

qB,C (x, X{1,2} ) . (33c)

Hard-sphere disagreement percolation

Hofer-Temmel

I2 ⊗I2 The joint Janossy density is measurable in (C, X) on (CB c × CB , FB c ⊗ F B ) and integrable in C for finite Borel measures on (CB c , FB c ). Its joint Janossy 2 measure at Y ∈ CB is Y hs-poi Poi PB,C,λ (ξ1 = dY1 , ξ2 = dY2 ) = [Y1 ⊆ Y2 ]PB,λ qB,C (x, Y1 ) . (33d) (ξ = dY2 ) x∈Y2

The boundary condition C may be restricted to CR(B) . Theorem 5.3 follows from the propositions in Section 5.3.2. The coupling in Theorem 5.3 is conceptually just a dependent thinning from the dominating Poisson PP. Take a realisation of a Poisson PP with iid [0, 1] marks, order the points by ≺ and finally thin dependently with the keeping probabilities (33a). The thinning probability rewrites into   1 ∂ . (33e) log Z(]y, ∞[, C ∪ Yy , λ) pB,C (x, Y ) = − λ ∂y y=x In this way, pB,C (x, Y ) may be seen as a dependent disintegration of the free energy of the not yet considered part of B with respect to the Poisson intensity. The proof of (33e) is in Section 5.3.1. 5.3.1

An integral equation

Proposition 5.4 gives the probability of deleting all points of the Poisson PP within an interval. Proposition 5.4. For all a, b ∈ B with a ≺ b and Y ∈ CB with Y ∩]a, b[= ∅, Z C]a,b[

λ|Z|

Y

(1 − pB,C (z, Y ))dZ

z∈Z

= exp(λL(]a, b[))

Z(]b, ∞[, C ∪ (Y ∩] − ∞, a])) . (34) Z(]a, ∞[, C ∪ (Y ∩] − ∞, a]))

Proof. Let Y 0 := Y ∩] − ∞, a]. For each x ∈]a, b[, Yx = Y ∩] − ∞, x[= Y 0 and let p(x) := pB,C (x, Y 0 ) = pB,C (x, Y ). Let Z Y f: ]a, b[→ R x 7→ λ|Z| (1 − p(z))dZ . C]x,b[

z∈Z

The aim is to calculate f (a). If ]x, b[ contains a point, then splitting the smallest point off yields an integral equation for f . Z Y f (x) = λ|Z| (1 − p(z))dZ C]x,b[

z∈Z

Z

Z λ(1 − p(y))

=1+

λ|Z|

C]y,b[

]x,b[

Z (1 − p(y))f (y)dy .

=1+λ ]x,b[

16

Y z∈Z

(1 − p(z))dZdy

(35a)

Hard-sphere disagreement percolation

Hofer-Temmel

Because f is continuous, there is the boundary condition Z Y f (b) = lim f (x) = 1 + lim [|Z| > 0]λ|Z| (1 − p(z))dZ = 1 . x→b

x→b

C]x,b[

(35b)

z∈Z

The problem of calculating (34) transforms into solving the integral equation with boundary condition described by (35). Let Bb → [1, ∞[

z:

A 7→ Z(A∩]a, ∞[, C ∪ Y 0 , λ) .

Regard the functions Z ]a, b[→ [0, ∞[

l:

x 7→

dy = L(]x, b[) , ]x,b[

e:

]a, b[→ [1, ∞[

x 7→ exp(λL(]x, b[)) = exp(λl(x)) ,

i:

]a, b[→ [1, ∞[

x 7→ z(]x, ∞[) ,

f:

]a, b[→ [0, ∞[

x 7→

e(x) . i(x)

− + For ε > 0 and x ∈ B, let x− ε and xε be the points with L(]xε , x[) = ε and + L(]x, xε [) = ε respectively. For ε small enough, this is well-defined for L-a.e. x ∈ B. The derivative of a function g :]a, b[→ R is − ∂g g(x+ ε ) − g(xε ) (x) := lim . ε→0 ∂x 2ε

This is just the usual one-dimensional derivative applied to the measurable total ordering of Rd by ≺. The Lebesgue differentiation theorem asserts that for a measurable g : Rd → [0, ∞[, L-a.e. the limit Z 1 g(y)dy = g(x) (36) lim ε→0 2ε ]x− ,x+ [ ε ε holds. Investigate the derivatives of l, e, i and f. The easy cases are ∂l (x) = −1 ∂x

∂e (x) = −λe(x) . ∂x

and

More advanced is
∂i 1 − (x) = lim z(]x+ ε , ∞[) − z(]xε , ∞[) ε→0 2ε ∂x Z −1 = lim ])λ|Z| H(Z|C ∪ Y 0 )dZ (1 − [Z ∈ C]x+ ε ,∞[ ε→0 2ε C − ]xε ,∞[

− Because Z∩]x− ε , ∞[6= ∅, the strategy is to split off the smallest point in ]xε , ∞[ and factorise the hard-sphere constraints with (21).

−1 = lim ε→0 2ε

Z

−1 ε→0 2ε

Z

0

Z

λH({y}|C ∪ Y ) + ]x− ε ,xε ]

= lim

+ ]x− ε ,xε ]

λ|Z| H(Z|C ∪ Y 0 ∪ {y})dZdy

C]y,∞[ 0 λH({y}|C ∪ Y 0 )Z(]x− ε , ∞[, C ∪ Y ∪ {y}, λ)dy

17

Hard-sphere disagreement percolation

Hofer-Temmel

1 ε→0 2ε

(23)

Z

= − λ lim

+ ]x− ε ,xε ]

H({y}|C ∪ Y 0 )z((]x− ε , ∞[) \ S(y))dy

(36)

= − λH({x}|C ∪ Y 0 )z(]x, ∞[\S(x)) .

Finally, ∂i ∂e (x) − e(x) ∂x (x) i(x) ∂x ∂f (x) = 2 ∂x i(x) i(x)(−λe(x)) − e(x)(−λH({x}|C ∪ Y 0 )z(]x, ∞[\S(x))) = i(x)2   λe(x) H({x}|C ∪ Y 0 )z(]x, ∞[\S(x))) = − 1− i(x) z(]x, ∞[) (33a)

=

− λf(x)(1 − p(x)) .

Because f(b) =

e(b) exp(λL(]b, b[)) 1 = = , i(b) z(]b, ∞[) z(]b, ∞[)

integration yields Z f(x) = f(b) − ]x,b[

∂f 1 (y)dy = +λ ∂x z(]b, ∞[)

Z f(y)(1 − p(y))dy . ]x,b[

The function z(]b, ∞[)f is L-a.e. a solution of (35a) and z(]b, ∞[)f(a) yields (34).

5.3.2

Proof of Theorem 5.3

This section contains the propositions proving the various parts of Theorem 5.3. The freedom to restrict the boundary condition C to C ∩ R(B) ∈ CR(B) follows from the same freedom for H and Z (23). The first step is a summation over the dependent thinning probabilities. Proposition 5.5. For all disjoint Y, X ∈ CB , one has X Y qB,C (x, X ∪ Z) = 1 .

(37)

Z⊆Y x∈Y

Proof. Ascertain (37) by induction on the size of Y . The induction base for Y = ∅ is obviously true. In all other cases, let y := min Y (with respect to ≺) and Y 0 := Y \ {y}. Thus, X Y qB,C (x, X ∪ Z) Z⊆Y x∈Y

=

X Y Z 0 ⊆Y 0

qB,C (x, X ∪ {y} ∪ Z 0 ) +

x∈Y

= pB,C (x, X)

Y

qB,C (x, X ∪ Z 0 )

x∈Y

X Y

qB,C (x, X ∪ {y} ∪ Z 0 )

Z 0 ⊆Y 0 x∈Y 0

+ (1 − pB,C (x, X))

X Y Z 0 ⊆Y 0 x∈Y 0

18

qB,C (x, X ∪ Z 0 )

Hard-sphere disagreement percolation

Hofer-Temmel

= pB,C (x, X) + (1 − pB,C (x, X)) = 1. Proposition 5.6. The rhs of (33c) is the joint Janossy density of a coupling of two PPs and the rhs of (33d) is the infinitesimal of the corresponding joint Janossy measure. Proof. The properties of a σ-finite measure follow from integration over (33c) by choosing suitable integration domains. The total mass equals one, because Z hs-poi PB,C,λ (∀I ∈ I2 : ξI = XI )dX I

CB2

Z =

I CB2

Z = 3 CB

Z = 2 CB

Z CB (37)

Z

qB,C (x, X{1,2} )dX

x∈X{1,2} ∪X{2}

Poi [X = ∅]PB,λ (ξ = Y ∪ Z)

Y

qB,C (x, Y )dXdY dZ

x∈Y ∪Z Poi PB,λ (ξ = Y ∪ Z)

Y

qB,C (x, Y )dY dZ

x∈Y ∪Z Poi PB,λ (ξ = Z)

=

Y

Poi [X{1} = ∅]PB,λ (ξ = X{1,2} ∪ X{2} )

X Y

qB,C (x, Y )dZ

Y ⊆Z x∈Z Poi PB,λ (ξ = Z)dZ

=

CB

= 1. 2 . Check the equivalence of infinitesimals at Y ∈ CB hs-poi PB,C,λ (ξ1 = dY1 , ξ2 = dY2 ) Y Poi = [Y1 ⊆ Y2 ]PB,λ (ξ = dY2 ) qB,C (x, Y1 ) x∈Y2 Poi = [Y1 \ Y2 = ∅]d(Y1 \ Y2 )PB,λ (ξ = Y2 )dY2

Y

qB,C (x, Y1 ∩ Y2 )

x∈Y2 hs-poi = PB,C,λ (ξ{1} = Y1 \ Y2 , ξ{1,2} = Y1 ∩ Y2 , ξ{2} = Y2 \ Y1 )

× d(Y1 \ Y2 )d(Y1 ∩ Y2 )d(Y2 \ Y1 ) . hs-poi Proposition 5.7. The joint Janossy density of PB,C is measurable in (C, X) I2 ⊗I2 on (CB c × CB , FB c ⊗ F B ) and integrable in C for finite Borel measures on (CB c , FB c ).

Proof. The joint Janossy density consists of the measurable expressions from (25), (33a) and (33b) combined by measurability-preserving operations, whence it is I2 measurable. For fixed X ∈ CB , there is the bound hs-poi PB,C (∀I ∈ I2 : ξI = XI ) ≤ λ|X{1} ∪X{1,2} | .

Boundedness and measurability imply integrability.

19

Hard-sphere disagreement percolation

Hofer-Temmel

hs-poi Proposition 5.8. The coupling PB,C,λ describes a stochastic domination. hs-poi PB,C,λ (ξ1 ⊆ ξ2 ) = 1 .

(38)

Proof. hs-poi PB,C,λ (ξ1 ⊆ ξ2 ) =

Z

hs-poi [Y1 ⊆ Y2 ]PB,C,λ (ξ1 = dY1 , ξ2 = dY2 )

2 CB

(33d)

Z

=

2 CB

hs-poi PB,C,λ (ξ1 = dY1 , ξ2 = dY2 ) = 1 .

hs-poi Proposition 5.9. The second marginal of PB,C,λ is Poissonian. hs-poi Poi PB,C,λ (ξ2 = Y ) = PB,λ (ξ = Y ) .

∀Y ∈ CB :

(39)

Proof. The Janossy density is hs-poi PB,C,λ (ξ2 = Y ) X Z hs-poi PB,C,λ = (ξ1 = X ∪ Z, ξ2 = Y )dZ X⊆Y (33c)

=

CB

X Z X⊆Y

Poi [Z = ∅]PB,λ (ξ = Y )

CB

Y

qB,C (x, X)dZ

x∈Y

Z=∅

Poi = PB,λ (ξ = Y )

X Y

qB,C (x, X ∪ Z)

X⊆Y x∈Y (37)

Poi = PB,λ (ξ = Y ) .

hs-poi Proposition 5.10. The first marginal of PB,C,λ is hard-sphere. hs-poi hs PB,C,λ (ξ1 = Y ) = PB,C,λ (ξ = Y ) .

∀Y ∈ CB :

(40)

Proof. The Janossy density is hs-poi PB,C,λ (ξ1 = Y ) Z Y (33c) X Poi = [Y \ X = ∅]PB,λ (ξ = X ∪ Z) qB,C (x, Y )dZ X⊆Y X=Y

Z

CB

x∈X∪Z

Y

Poi PB,λ (ξ = Y ∪ Z)

=

CB

qB,C (x, Y )dZ

x∈Y ∪Z

Let n := |Y |. Order Y =: {y1 , . . . , yn } increasingly by ≺ and let y0 := −∞ and yn+1 := ∞. For i ∈ {0, . . . , n}, let Bi :=]yi , yi+1 [. =

n Y i=1

(33a)

=

! pB,C (yi , Y )λ

n Z Y i=0

CBi

Poi PB (ξ = Z) i ,λ

Y

(1 − pB,C (z, Y ))dZ

z∈Z

n Y

Z([yi , ∞[\S(yi ), C ∪ Yyi ) λH({yi }|C ∪ Yyi ) Z([yi , ∞[, C ∪ Yyi ) i=1 20

!

Hard-sphere disagreement percolation

Hofer-Temmel

×

n Y

Z CBi

i=0 (34)

= λ

n

×

λ|Z|

exp(−λL(Bi ))

n Y

! H({yi }|C ∪ Yyi )

i=1 n Y

Y

(1 − pB,C (z, Y ))dZ

z∈Z n Y Z([yi , ∞[\S(yi ), C ∪ Yyi ) Z([yi , ∞[, C ∪ Yyi ) i=1

exp(−λL(Bi )) exp(λL(Bi ))

i=0

!

Z(]yi+1 , ∞[, C ∪ (Y ∩] − ∞, yi ])) Z(]yi , ∞[, C ∪ (Y ∩] − ∞, yi ]))

For i ∈ [n], let Yi := {y1 , . . . , yi } and Y0 = ∅. Thus, Yyi = Yi−1 and Y ∩] − ∞, yi ] = Yi . Also, combine the hard-sphere constraints by (21). = λn H(Y |C)

n Y Z([yi , ∞[\S(yi ), C ∪ Yi−1 ) i=1

×

!

Z([yi , ∞[, C ∪ Yi−1 )

n Y Z(]yi+1 , ∞[, C ∪ Yi ) i=0

Z(]yi , ∞[, C ∪ Yi ))

Because Z([yi , ∞[, C ∪ Yi ) = Z([yi , ∞[\S(yi ), C ∪ Yi−1 ) and S(∞) := ∅, one has 1 ≤ Z([yn+1 , ∞[\S(yn+1 ), C ∪ Yn ) ≤ Z([yn+1 , ∞[, C ∪ Yn ) ≤ Z(∅, C ∪ Yn ) = 1 . Join the two products and reduce the resulting telescoping product. = λn H(Y |C) n Y Z([yi+1 , ∞[\S(yi+1 ), C ∪ Yi ) Z(]yi+1 , ∞[, C ∪ Yi ) × Z([y , ∞[, C ∪ Y ) Z(]y i+1 i i , ∞[\S(yi ), C ∪ Yi−1 )) i=0 Z([yn+1 , ∞[\S(yn+1 ), C ∪ Yn ) Z([y0 , ∞[, C ∪ Y0 ) Z(∅, C ∪ Y ) = λn H(Y |C) Z(]a, b[, C) λn H(Y |C) = . Z(]a, b[, C) = λn H(Y |C)

5.4

The product disagreement coupling family

First, Proposition 5.11 checks that algorithm 3.3 yields a well-defined probability measure. Definition 5.12 gives a recursive explicit description of a joint Janossy density of a family of couplings inspired by algorithm 3.3. The name “product disagreement coupling” stems from this construction. Proposition 5.13 shows that Definition 5.12 defines a disagreement coupling family at intensity λ. Theorem 3.4 follows from combining Theorem 3.2 and Proposition 5.13. Proposition 5.11. The construction in algorithm 3.3 is well-defined. Proof. Every step in (γn )n∈N is measurable. Therefore, γ and χ are well-defined rvs. The conditional sampling step is well-defined, because of the following sampling procedure. With ϑ := R(B \ χ) ∩ B, let

21

Hard-sphere disagreement percolation

Hofer-Temmel

hs dom p := PB,∅ (ξ ∩ ϑ = ∅) = PB,∅ (ξ1 ∩ ϑ = ∅) dom ≥ PB,∅ (ξ2 ∩ ϑ = ∅) = exp(−λL(ϑ)) ≥ exp(−λL(B)) > 0 . dom Sample (η, ϕ) independently and repeatedly from PB,∅ -distributed until ϕ∩ϑ = ∅ happens. As p > 0, the number of resamples has a geometric distribution. As ϕ ∩ ϑ = ∅ happens, let (η3 , ϕ3 ) be (η \ χ, ϕ \ χ). For ϕ4 , take an independent Poi PB,λ -distributed PP and project it to B \ χ.

Another problem in the proof of Proposition 5.11 is that the law of (η \ χ, ϕ \ hs-poi dom χ)|ϕ ∩ ϑ = ∅ is not better known. A solution is to use PB,C,λ instead of PB,C,λ . The explicit description of the joint Janossy density permits to describe such conditional laws. The result is a recursive description of the joint Janossy densities of a disagreement coupling family. The recursive part of the construction mimics algorithm 3.3. Definition 5.12 (Product coupling). For B ∈ Bb and C1 , C2 ∈ CB c , define ⊗3 3  , FB ) as follows. Let the disagreement on (CB a probability measure PB,C 1 ,C2 zone be D := S(C1 4 C2 ) ∩ B . (41a)  restricted to D and describe it by an auxiliary First, construct the law PB,C 1 ,C2 ⊗3  3 law PB,C1 ,C2 on (CD , FD ). Second, apply the construction recursively to B \ D,  with boundary conditions depending on the realisation of (ξ1 , ξ2 ) under PB,C . 1 ,C2 Case D = ∅: By construction, (C1 4 C2 ) ∩ R(B) = ∅ and C := C1 ∩ R(B) = hs-poi C2 ∩ R(B). Take a PB,C coupling between a hard-sphere PP and a dominating Poisson PP. Add a second independent Poisson PP of intensity λ. Treat the points of the hard-sphere PP as belonging to both target hard-sphere PPs. The measurable map

f:

3 3 CB → CB

(Y1 , Y2 , Y3 ) 7→ (Y1 , Y1 , Y2 ∪ Y3 )

describes these multiplexing and merging operations. Thus,   hs-poi  Poi [D = ∅]PB,C = f ◦ PB,C × PB,λ . 1 ,C2

(41b)

 3 is The joint Janossy density of [D = ∅]PB,C at Y ∈ CB 1 ,C2  [D = ∅]PB,C (∀i ∈ [3] : ξi = Yi ) = [Y1 = Y3 ] 1 ,C2 X hs-poi Poi × PB,C (ξ1 ∩ D = Y1 , ξ2 ∩ D = Z) × PB,λ (ξ = Y3 \ Z) . (41c) Y1 ⊆Z⊆Y3

Case D 6= ∅:  The helper measure PB,C is the projection of a product of two independent 1 ,C2 hard-sphere-Poisson couplings with merged Poisson components. The measurable map g:

4 3 CB → CB

(Y1 , Y2 , Y3 , Y4 ) 7→ (Y1 ∩ D, Y3 ∩ D, (Y2 ∪ Y4 ) ∩ D)

describes these projections and the merging. Thus,   hs-poi hs-poi  . PB,C = g ◦ P × P B,C2 B,C1 1 ,C2 22

(41d)

Hard-sphere disagreement percolation

Hofer-Temmel

 3 The joint Janossy density of PB,C at Y ∈ CD is 1 ,C2  PB,C (∀i ∈ [3] : ξi = Yi ) = 1 ,C2

X

hs-poi PB,C (ξ1 ∩ D = Y1 , ξ2 ∩ D = Y3 \ Z) 1

Z⊆Y3 hs-poi × PB,C (ξ1 ∩ D = Y2 , ξ2 ∩ D = Z) . (41e) 2  Define the density of PB,C recursively by 1 ,C2   [D 6= ∅]PB,C (∀i ∈ [3] : ξi = Yi ) := PB,C (∀i ∈ [3] : ξi = Yi ∩ D) 1 ,C2 1 ,C2  (∀i ∈ [3] : ξi = Yi \ D) . (41f) × PB\D,C 1 ∪(Y1 ∩D),C2 ∪(Y2 ∩D)

Proposition 5.13. Definition 5.12 describes the joint Janossy densities of  a disagreement coupling family (PB,C ) c at intensity 2λ. The 1 ,C2 B∈Bb ,C1 ,C2 ∈CB I3 ⊗I3 ⊗2 2 joint Janossy densities are measurable in (C, Y ) on (CB ) c × C B , FB c ⊗ F B ⊗2 2 and integrable in (C1 , C2 ) for finite Borel measures on (CB , F ). The boundary c Bc conditions C1 and C2 may be restricted to CR(B) . Proof. The complicated twisted model in Section 5.5 generalises the product model. Thus, the proof is omitted.

5.5

The twisted disagreement coupling family

Definition 5.12 describes a product disagreement coupling family in a recursive way amenable to inductive proofs. The obvious point for improvement is the def inition of the helper measure PB,C on D (41d). It is just the product of two 1 ,C2 hard-sphere Poisson couplings with merged Poisson components. A better analysis of the constraints on the hard-sphere PPs in the disagreement zone D (41a) shows that one may partition D into subsets on which at most one of the two hard-sphere PPs may have points. This key insight leads to a coupling twisting two hard-sphere PPs dominated by a single Poisson PP of intensity λ on D. Recursion leads to the joint Janossy density of the twisted coupling described in Definition 5.14. Proposition 5.16 shows that it is well-defined and has nice measurability and integrability properties with respect to the boundary conditions. Theorem 3.5 follows from combining Theorem 3.2 an Proposition 5.17. Definition 5.14 (Twisted coupling). For B ∈ Bb and C1 , C2 ∈ CB c , define ⊗3 M 3 a probability measure PB,C on (CB , FB ) as follows. For i ∈ [2], let the 1 ,C2 forbidden zone Fi be [ Fi := B ∩ S(x) . (42a) x∈Ci

Let D := F1 ∪ F2 be the disagreement zone. It is the same as in (41a). First, M construct the law PB,C restricted to D and describe it by an auxiliary law 1 ,C2 ⊗3 N 3 PB,C1 ,C2 on (CD , FD ). Second, apply the construction recursively to B \D, with N boundary conditions depending on the realisation of (ξ1 , ξ2 ) under PB,C . 1 ,C2 Case D = ∅: By construction, (C1 4 C2 ) ∩ R(B) = ∅ and C := C1 ∩ R(B) = hs-poi C2 ∩ R(B). Take a PB,C coupling between a hard-sphere PP and a dominating Poisson PP. Treat the points of the hard-sphere PP as belonging to both target hard-sphere PPs. The measurable map f:

2 3 CB → CB

(Y1 , Y2 ) 7→ (Y1 , Y1 , Y2 ) 23

Hard-sphere disagreement percolation

Hofer-Temmel

describes this multiplexing operation. Thus, hs-poi M [D = ∅]PB,C = f ◦ PB,C . 1 ,C2

(42b)

Case D 6= ∅: Partition D into D1 := F2 \F1 , D2 := F1 \F2 and D0 := F1 ∩F2 . N The construction of PB,C is independent on the elements of the partition 1 ,C2 {D0 , D1 , D2 }. On D1 , only the first hard-sphere PP may have points. So, project a coupling of the first hard-sphere PP with a dominating Poisson PP and set the second hard-sphere PP to empty. On D2 , the roles of the first and second hard-sphere PP switch. On D0 , neither hard-sphere PP may have points. So, let the two hard-sphere PPs be empty and take a Poisson PP of intensity λ. The combination of these independent constructions is the helper coupling. Regard the measurable maps f1 :

3 3 CB → CD 1

(Y1 , Y2 , Y3 ) 7→ (Y2 ∩ D1 , Y1 ∩ D1 , Y3 ∩ D1 ) ,

f2 :

3 CB

(Y1 , Y2 , Y3 ) 7→ (Y1 ∩ D2 , Y2 ∩ D2 , Y3 ∩ D2 ) ,

g:

→

3 CD 0

×

3 CD 2 3 CD 1

×

3 CD 2

→

3 CD

(Yij )i,j∈{1,2,3} 7→

3 [ i=1

Yi1 ,

3 [

Yi2 ,

i=1

3 [

! Yi3

.

i=1

The name-giving twist in f1 and f2 maps the hard-sphere points to the appropriate marginal. The function g combines the different elements of the partition. ∅ For A ∈ Bb , let PA be the Dirac mass on the empty configuration ∅ ∈ CA . Define ⊗3 N 3 the twisted helper measure PB,C on (CD , FD ) by 1 ,C2 g◦



     hs-poi hs-poi ∅ ∅ Poi ∅ ∅ PD ×P ×P × f ◦ (P ×P ) × f ◦ (P ×P ) . (42c) 1 2 D0 ,λ D0 B B B,C1 B,C2 0

3 M recursively by at Y ∈ CB Define the density of PB,C 1 ,C2 M N [D 6= ∅]PB,C (∀i ∈ [3] : ξi = Yi ) := PB,C (∀i ∈ [3] : ξi = Yi ∩ D) 1 ,C2 1 ,C2 M × PB\D,C (∀i ∈ [3] : ξi = Yi \ D) . (42d) 1 ∪(Y1 ∩D),C2 ∪(Y2 ∩D) N 3 Proposition 5.15. The joint Janossy density of PB,C at Y ∈ CD is 1 ,C2 N Poi PB,C (∀i ∈ [3] : ξi = Yi ) = PD (ξ = Y3 ∩ D0 ) 1 ,C2 0 ,λ hs-poi × [Y1 ⊆ D1 ]PB,C (ξ1 ∩ D1 = Y1 , ξ2 ∩ D1 = Y3 ∩ D1 ) 1 hs-poi × [Y2 ⊆ D2 ]PB,C (ξ1 ∩ D2 = Y2 , ξ2 ∩ D2 = Y3 ∩ D2 ) . (43a) 2 I3 ⊗I3 ⊗2 2 It is measurable in (C, Y ) on (CB ) and integrable in (C1 , C2 ) c × C D , FB c ⊗ F D ⊗2 2 for finite Borel measures on (CB c , FB c ). The boundary conditions C1 and C2 may be restricted to CR(B) . The marginal Janossy densities at Y ∈ CD are hardsphere, as

∀i ∈ [2] :

N hs PB,C (ξi = Y ) = PB,C (ξ ∩ D = Y ) , 1 ,C2 i

(43b)

and Poissonian, as N Poi PB,C (ξ3 = Y ) = PD,λ (ξ = Y ) . 1 ,C2

24

(43c)

Hard-sphere disagreement percolation

Hofer-Temmel

Proof. The disjoint construction on the partition {D0 , D1 , D2 } of D in (42c) translates directly into the joint Janossy density. The freedom to restrict the boundary conditions to CR(B) , the measurability and integrability all follow from hs-poi the same properties of the dominating hard-sphere Poisson couplings PB,. in Theorem 5.3 and the product construction (42c). The Janossy density of the third marginal at Y ∈ CD looks as follows.

=

N PB,C ,C (ξ3 = Y ) Z 1 2 X 3 CD X1 ]X2 ]X12 ⊆Y

N PB,C (ξ1 = X1 ∪ X12 ∪ Z1 ∪ Z12 , ξ2 = X2 ∪ X12 ∪ Z2 ∪ Z12 , ξ3 = Y )dZ 1 ,C2

Substitute (43a), then derive Xi ⊆ Di and Zi ⊆ Di , for i ∈ [2], and X12 ⊆ D1 ∩ D2 = ∅ and Z12 ⊆ D1 ∩ D2 = ∅. Poi = PD (ξ = Y ∩ D0 ) 0  X Z Y  ×

=

CDi

X⊆Y ∩Di

i∈[2] Poi PD (ξ 0

= Y ∩ D0 )



Y

hs-poi PB,C (ξ1 i

∩ Di = X ∪ Z, ξ2 ∩ Di = Y ∩ Di )dZ 

hs-poi PB,C (ξ2 ∩ Di = Y ∩ Di ) i

i∈[2] (39)

Poi = PD (ξ = Y ∩ D0 ) 0

Y

Poi PD (ξ = Y ∩ Di ) i

i∈[2]

=

Poi PB (ξ

= Y ).

N By the symmetry in the definition of PB,C (42c), it suffices to consider 1 ,C2 the Janossy density of the first marginal at Y ∈ CD .

=

N PB,C ,C (ξ1 = Y ) Z 1 2 X 3 CD X2 ]X3 ]X23 ⊆Y

N PB,C (ξ1 = Y, ξ2 = X2 ∪ X23 ∪ Z2 ∪ Z23 , ξ3 = X3 ∪ X23 ∪ Z3 ∪ Z23 )dZ 1 ,C2

Substitution of (43a), resolution of the Iverson indicators and consideration of the other constraints together imply that X2 ⊆ D1 ∩D2 = ∅, X23 ⊆ D1 ∩D2 = ∅ and X3 ⊆ D1 , as well as Z2 ⊆ D2 and Z23 ⊆ D2 . Split Z3 over the partition {D0 , D1 , D2 } of D. Z = CD0

Poi PD (ξ = Z)dZ 0





× [Y ⊆ D1 ]

X Z X⊆Y

3 CD 1

hs-poi PB,C (ξ1 ∩ D1 = Y, ξ2 ∩ Di = X ∪ Z)dZ  1

!

Z × 3 CD

hs-poi (ξ1 PB,C 2

∩ D2 = Z2 ∪ Z23 , ξ2 ∩ D2 = Z3 ∪ Z23 )dZ

2

25

Hard-sphere disagreement percolation

Hofer-Temmel

hs-poi = 1 × [Y ⊆ D1 ]PB,C (ξ1 ∩ D1 = Y ) × 1 1 hs (ξ ∩ D1 = Y ) = [Y ⊆ D1 ]PB,C 1 hs = PB,C (ξ ∩ D = Y ) . 1

The following proofs use case analysis on D and induction on h(B). The induction base case is D = ∅. If D 6= ∅, then let B 0 := B \ D, apply the recursion step (42d) and check whether h(B 0 ) < h(B) holds. If it does, then recursion proceeds controlled. If it does not, then apply the recursion step (42d) a second time. Let φ := (ξ1 4 ξ2 ) ∩ D, D0 := B 0 ∩ S(φ) and B 00 := B 0 \ D0 . If φ = ∅, then D0 = ∅ and the recursion steps into the induction base case. If φ 6= ∅, then D0 6= ∅ and h(B 00 ) < h(B) by (20). If h(B) = 0, then D ⊆ B = ∅ and the induction base case applies. This two-step induction/recursion reduces the proofs to simple checks on the cases induction base case D = ∅ and recursive case D 6= ∅. Proposition 5.16. Definition 5.14 describes the joint Janossy densities of a M disagreement coupling family (PB,C ) c at intensity λ. The joint 1 ,C2 B∈Bb ,C1 ,C2 ∈CB I3 ⊗I3 ⊗2 2 , FB ) and Janossy densities are measurable in (C, Y ) on (CB c × CB c ⊗ FB ⊗2 2 integrable in (C1 , C2 ) for finite Borel measures on (CB c , FB c ). The boundary conditions C1 and C2 may be restricted to CR(B) . In the recursion step (42d), the boundary conditions may be restricted to CD . Proof. This proof uses two-step induction. 3 is Case D = ∅: The joint Janossy density at Y ∈ CB M [D = ∅]PB,C (ξ1 = Y1 , ξ2 = Y2 , ξ3 = Y3 ) 1 ,C2 hs-poi = [D = ∅][Y1 = Y2 ]PB,C (ξ1 = Y2 , ξ2 = Y3 ) . (44) M a probability Its definition as push-forward measure makes [D = ∅]PB,C 1 ,C2 ⊗3 3 measure on (CB , FB ). Measurability, integrability and the freedom to restrict hs-poi the boundary conditions follow from the same properties of PB,C in Theorem 5.3. Case D 6= ∅: The definition of D implies that S(B \ D) ∩ (C1 4 C2 ) = ∅. Thus, for i ∈ [2], the boundary conditions (Yi ∩ D) instead of Ci ∪ (Yi ∩ D) suffices in the recursive Definition (42d). The measurability with respect to the boundary conditions is crucial to integrate out over the boundary condition and see the density as a right con⊗3 3 3 ditional joint Janossy density. The isomorphism between (CB , FB ) and (CD × ⊗3 ⊗3 3 3 3 CB\D , FD ⊗ FB\D ) establishes probabilities for E1 × E2 ∈ CD × CB\D . M PB,C ,C (E1 × E2 ) Z 1 2 M = [Y ∈ E1 × E2 ]PB,C (∀i ∈ [3] : ξi = dYi ) 1 ,C2 3 CB

Z [Z ∈ E1 ]

= 3 CD

!

Z [Z ∈ 3 CB\D

M E2 ]PB,Y (∀i 1 ∩D,Y2 ∩D

∈ [3] : ξi = dZi )

N × PB,C (∀i ∈ [3] : ξi = dYi ) 1 ,C2 Z M N = [Z ∈ E1 ]PB,Y (E2 )PB,C (∀i ∈ [3] : ξi = dYi ) . 1 ∩D,Y2 ∩D 1 ,C2 3 CD

26

Hard-sphere disagreement percolation

Hofer-Temmel

Extend from these sets to the full σ-algebra by completion. The σ-finite properties of the resulting measure follow by adopting the domains of integration appropriately. The total mass is one, because Z M PB,C (∀i ∈ [3] : ξi = dYi ) 1 ,C2 3 CB

Z

!

Z

= 3 CD

Z = 3 CD

3 CB\D

M N PB,Y (∀i ∈ [3] : ξi = dZi ) PB,C (∀i ∈ [3] : ξi = dYi ) 1 ∩D,Y2 ∩D 1 ,C2

N 1 × PB,C (∀i ∈ [3] : ξi = dYi ) 1 ,C2

= 1. Measurability, integrability and the freedom to restrict the boundary conditions N follow from the same properties of PB,C in Proposition 5.15 and the recursive 1 ,C2 M properties of PB,Y1 ∩D,Y2 ∩D . Proposition 5.17. Definition 5.14 describes the joint Janossy densities of a M disagreement coupling family (PB,C ) c at intensity λ. 1 ,C2 B∈Bb ,C1 ,C2 ∈CB Proof. Check the four properties (12) of a disagreement coupling. Properties (12a), (12b), (12b) and (12d) are a consequence of propositions 5.18, 5.19, 5.20 and 5.21 respectively. 3 3 , let /Z ∈ CB For Z ∈ CD N/M

N/M

JB,C1 ,C2 (Z1 , Z2 , Z3 ) := PB,C1 ,C2 (∀i ∈ [3] : ξi = Zi ) .

(45)

M hs Proposition 5.18. For i ∈ [2], the marginal of ξi under PB,C is PB,C . 1 ,C2 i

∀i ∈ [2], Y ∈ CB :

M hs PB,C (ξi = Y ) = PB,C (ξ = Y ) . 1 ,C2 i

(46)

Proof. This proof uses two-step induction. (42b)

(40)

hs-poi hs M (ξ1 = Y ) = PB,C (Y ). Case D = ∅: PB,C (ξi = Y ) = PB,C i 1 ,C2 i Case D = 6 ∅: Let J := {{2}, {3}, {2, 3}}. By the symmetry in Definition 5.14, only the case i = 1 is to consider.

=

M PB,C (ξ1 = Y ) 1 ,C2 X

Z

X{1,2} ]X{1,3} ]X{1,2,3} ⊆Y split

J CB

J CB\D

XI )

3∈I

Y

dXJ

J∈J

X Z{1,2} ]Z{1,3} ]Z{1,2,3} ⊆Y \D

\ D,

[

ZI ,

2∈I

[

XI ,

2∈I

=

[

! M S JB\D,Y ∩D, 2∈I XI (Y

N × JB,C (Y ∩ D, 1 ,C2 recurse

XI ,

2∈I

J CD

X{1,2} ]X{1,3} ]X{1,2,3} ⊆Y ∩D

Z

[

Z

X

=

M JB,C (Y, 1 ,C2

X X{1,2} ]X{1,3} ]X{1,2,3} ⊆Y ∩D

[ 3∈I

Z J CD

27

XI )

[

ZI )

3∈I

Y

Y

dZJ

J∈J

dXJ

J∈J M S PB\D,Y ∩D, 2∈I XI (ξ1 = Y \ D)

Hard-sphere disagreement percolation

Hofer-Temmel

N × JB,C (Y ∩ D, 1 ,C2

[

XI ,

2∈I

[

XI )

3∈I

Y

dXJ

J∈J

(46)

hs N = PB\D,Y ∩D (ξ = Y \ D)PB,C1 ,C2 (ξ1 ∩ D = Y ∩ D)

(43b)

hs hs = PB\D,Y ∩D (ξ = Y \ D)PB,C1 (ξ ∩ D = Y ∩ D)

(22)

hs = PB,C (ξ = Y ) . 1

M Proposition 5.19. The marginal of ξ3 under PB,C is Poissonian. 1 ,C2

∀Y ∈ CB :

M Poi PB,C (ξ3 = dY ) = PB,λ (dY ) . 1 ,C2

(47)

Proof. This proof uses two-step induction. (42b)

(39)

hs-poi M Poi (ξ3 = dY ) = PB,C (ξ2 = dY ) = PB,λ (dY ). Case D = ∅: PB,C 1 ,C2 1 ∩R(B) Case D = 6 ∅: Using the shorthand from (45) and letting J := {{1}, {2}, {1, 2}}, one has

=

M PB,C (ξ3 = Y ) 1 ,C2 X

Z J CB

X{1,3} ]X{2,3} ]X{1,2,3} ⊆Y split

=

M JB,C ( 1 ,C2

[

XI ,

1∈I

Z

X X{1,3} ]X{2,3} ]X{1,2,3} ⊆Y ∩D

[

XI , Y )

2∈I

Y

Z

X

J CD

dXJ

J∈J

Z{1,3} ]Z{2,3} ]Z{1,2,3} ⊆Y \D

J CB\D

! M S S ×JB\D, ( 1∈I XI , 2∈I XI

[

ZI ,

1∈I N × JB,C ( 1 ,C2

[

XI ,

1∈I recurse

=

[ 2∈I

Z

1∈I

XI ,

[

Y

dZJ

J∈J

Y

dXJ

J∈J

X{1,3} ]X{2,3} ]X{1,2,3} ⊆Y ∩D

[

ZI , Y \ D)

2∈I

XI , Y ∩ D)

X N × JB,C ( 1 ,C2

[

J CD

M S S (ξ3 = Y \ D) PB\D, 1∈I XI , 2∈I XI

XI , Y ∩ D)

2∈I

Y

dXJ

J∈J

(47)

Poi N = PB\D,λ (ξ = Y \ D)PB,C (ξ3 ∩ D = Y ∩ D) 1 ,C2

(43c)

Poi Poi = PB\D,λ (ξ = Y \ D)PD,λ (ξ ∩ D = Y ∩ D)

(22)

Poi = PB,λ (ξ = Y ) .

M Proposition 5.20. The union of the first two margionals is PB,C -a.s. con1 ,C2 tained in the third one. M PB,C (ξ1 ∪ ξ2 ⊆ ξ3 ) = 1 . 1 ,C2

(48)

Proof. This proof uses two-step induction. (42b)

(38)

hs-poi M Case D = ∅: PB,C (ξ1 ∪ ξ2 ⊆ ξ3 ) = PB,C (ξ1 ⊆ ξ2 ) = 1. 1 ,C2 1 ∩R(B) Case D = 6 ∅: There is

28

Hard-sphere disagreement percolation

Hofer-Temmel

N PB,C (ξ1 ∪ ξ2 ⊆ ξ3 ) 1 ,C2 (42c)

(38)

hs-poi hs-poi = PD (ξ1 ∩ D1 ⊆ ξ2 ∩ D1 )PD (ξ1 ∩ D2 ⊆ ξ2 ∩ D2 ) = 1 . (49) 1 2

Therefore, M PB,C ,C (ξ1 ∪ ξ2 ⊆ ξ3 ) Z 1 2 M = [Y1 ∪ Y2 ⊆ Y3 ]PB,C (∀i ∈ [3] : ξi = dYi ) 1 ,C2 3 CB

(42d)

Z

=

3 CB

split

Z

=

3 CD

recurse

=

N [Y1 ∪ Y2 ⊆ Y3 ]PB,C (∀i ∈ [3] : ξi = d(Yi ∩ D)) 1 ,C2 M × PB\D,Y (∀i ∈ [3] : ξi = d(Yi \ D)) 1 ∩D,Y2 ∩D ! Z M [Z1 ∪ Z2 ⊆ Z3 ]PB,Y1 ,Y2 (∀i ∈ [3] : ξi = dZi ) 3 CB\D

N × [Y1 ∪ Y2 ⊆ Y3 ]PB,C (∀i ∈ [3] : ξi = dYi ) 1 ,C2 Z M N PB,Y (ξ1 ∪ ξ2 ⊆ ξ3 )[Y1 ∪ Y2 ⊆ Y3 ]PB,C (∀i ∈ [3] : ξi = dYi ) 1 ,Y2 1 ,C2 3 CD

(48)

N = PB,C (ξ1 ∪ ξ2 ⊆ ξ3 ) 1 ,C2

(49)

= 1.

Proposition 5.21. The disagreeing points of the hard-sphere marginals are M PB,C -a.s. R-connected to the disagreeing points of the boundary conditions. 1 ,C2 R−con

M PB,C (∀x ∈ ξ1 4 ξ2 : x ←−−−−→ C1 4 C2 ) = 1 . 1 ,C2 in ξ1 4ξ2

(50)

Proof. This proof uses two-step induction. Case D = ∅: There is R−con

M PB,C (∀x ∈ ξ1 4 ξ2 : x ←−−−−→ C1 4 C2 ) 1 ,C2 in ξ1 4ξ2

(42b)

R−con

hs-poi = PB,C (∀x ∈ ξ1 4 ξ1 = ∅ : x ←−−−−−−→ C1 4 C2 ) = 1 . 1 ∩R(B) in ξ1 4ξ1 =∅

Case D 6= ∅: As Di ⊆ S(Cj \ Ci ), for {i, j} := [2], hs-poi PB,C (∀x ∈ ξ1 ∩ Di : x ∈ S(Cj \ Ci )) i (40)

hs = PB,C (∀x ∈ ξ ∩ Di : x ∈ S(Cj \ Ci )) = 1 . (51) i

Also (42c)

hs-poi N PB,C (ξi ⊆ Di ) = PB,C (ξi ∩ D ⊆ Di ) 1 ,C2 i (40)

hs = PB,C (ξi ∩ D ⊆ Di ) i

rewrite

hs = PB,C (ξi ∩ Fi = ∅) i

(42a)

= 1. 29

(52)

Hard-sphere disagreement percolation

Hofer-Temmel

This leads to R−con

N PB,C (∀x ∈ ξ1 4 ξ2 : x ←−−−−→ C1 4 C2 ) 1 ,C2 in ξ1 4ξ2

relax

N ≥ PB,C (∀x ∈ ξ1 4 ξ2 : x ∈ S(C1 4 C2 )) 1 ,C2

(52)

N = PB,C (∀x ∈ ξ1 ∩ D1 : x ∈ S(C2 \ C1 ), ∀y ∈ ξ1 ∩ D2 : y ∈ S(C1 \ C2 )) 1 ,C2

(42c)

hs-poi = PB,C (∀x ∈ ξ1 ∩ D1 : x ∈ S(C2 \ C1 )) 1 hs-poi × PB,C (∀x ∈ ξ1 ∩ D2 : x ∈ S(C1 \ C2 )) 2

(51)

= 1.

Finally, R−con

M PB,C (∀x ∈ ξ1 4 ξ2 : x ←−−−−→ C1 4 C2 ) 1 ,C2 in ξ1 4ξ2

relax

R−con

M ≥ PB,C (∀x ∈ (ξ1 4 ξ2 ) \ D : x ←−−−−−−−→ (ξ1 4 ξ2 ) ∩ D 1 ,C2 in (ξ1 4ξ2 )\D | {z } =:E

R−con

, ∀y ∈ (ξ1 4 ξ2 ) ∩ D : y ←−−−−−−−→ C1 4 C2 ) in (ξ1 4ξ2 )∩D | {z } ⊗2 -measurable =:E 0 , is FD

expand

Z

=

2 Y ∈CD

(42d)

Z

=

2 Y ∈CD

recurse

=

M PB,C ((ξ1 , ξ2 ) ∈ E ∩ E 0 , ξ1 ∩ D = dY1 , ξ2 ∩ D = dY2 ) 1 ,C2 R−con

M PB\D,Y (∀x ∈ ξ1 4 ξ2 : x ←−−−−→ Y1 4 Y2 ) 1 ,Y2 in ξ1 4ξ2

N × PB,C ((ξ1 , ξ2 ) ∈ E 0 , ξ1 ∩ D = dY1 , ξ2 ∩ D = dY2 ) 1 ,C2 Z N 1 × PB,C ((ξ1 , ξ2 ) ∈ E 0 , ξ1 ∩ D = dY1 , ξ2 ∩ D = dY2 ) 1 ,C2 2 Y ∈CD

N = PB,C ((ξ1 , ξ2 ) ∈ E 0 ) 1 ,C2

=1

by the previous calculation.

Acknowledgements He thanks Marie-Colette van Lieshout, Jacob van den Berg, Ronald Meester and Erik Broman for helpful discussions surrounding this topic.

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