Disagreement percolation for the hard-sphere model

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. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds improve upon known bounds obtained by cluster expansion techniques. The proof uses a novel dependent thinning from a Poisson point process to the hard-sphere model, with the thinning probability related to a derivative of the free energy.


Introduction
Disagreement percolation by van den Berg and Maes [30] is a sufficient condition on the activity of a discrete Gibbs 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 Poincaré inequality in the context of lattice Ising spin systems [4]. This paper generalises disagreement percolation to the hard-sphere model on R d , the continuum equivalent of the well-studied hard-core model [31].
The core of disagreement percolation is a coupling between three point processes on a bounded domain. Two are hard-sphere models with the same activity and differing boundary conditions. The third one is a Boolean model stochastically dominating the points of disagreement between the two hardsphere models. The connected components of the Gilbert graph of the Boolean model connected to the boundary control the extent of 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 van Hove sequence [23, Def 2.1.1] is a monotone increasing sequence (B n ) n∈N of bounded Borel sets converging to R d and eventually containing every bounded Borel set. The increasing hypercubes ([−n, n] d ) n∈N are a van Hove sequence.
The Gilbert graph of a configuration C has vertices C and edges {{x, y} ⊆ C | ||x − y|| ≤ R}, i.e., connects points at distance at most R. The configuration C is a R-cluster, if it is R-connected, i.e., connected in its Gilbert graph. Two points x and y are R-connected by a configuration C, written x R−con there is a finite path of jumps of at most distance R between x and y using only points in C as intermediate points. Two Borel sets are R-connected by a configuration C, if there is a R-connected pair of points, with one point from each set.

Point processes
For B ∈ B, let C B be the locally finite point configurations on B, i.e., for each C ∈ C B and A ∈ B b , |C ∩ A| < ∞. Let F B be the σ-algebra on C B generated by {{C ∈ C B | C ∩ A = ∅} | B ⊇ A ∈ B}, i.e., compatible with the Fell topology.
A simple point process (short PP) on a Borel set B ∈ B is a random variable taking values in C B . This work treats a PP as a locally finite random subset of points of R d , 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 C R d .
A Borel measure M on (C B , F B ) is the local Janossy measure [5, after (5.3. This definition of local Janossy measure is a portmanteau version of the traditional definitions on generating cylinder sets. Because the local Janossy measure of a PP law P on B ∈ B on B ⊇ A ∈ B b equals the Janossy measure of the restriction of the law to A, the remainder of this paper drops the quantifier "local". If ξ has finite moment measures of all orders under P, then the Janossy measure in (1) exists [5,Theorem 5.4.I]. For B ∈ B b and C ∈ C B , write the infinitesimal of the Janossy measure of P on B at C as P(ξ ∩ B = dC).
The intensity measure of the PP law P is the average number of points on bounded Borel sets. For B ∈ B b , it equals C B |C|P(ξ ∩ B = dC).

The Boolean model
The classic PP is the Poisson PP law P poi B,α of intensity α on B ∈ B, i.e., with intensity measure αL.
A configuration C ∈ C R d R-percolates, if it contains an infinite R-cluster. The bounded finiteness of C renders this equivalent to the existence of an unbounded R-cluster. The Boolean model of intensity α is a P poi R d ,α -distributed PP, with closed spheres of radius R/2 centred at the points. If spheres overlap, then the corresponding points are connected. This is just R-connectivity from Section 2.1. The Boolean model percolates, if it contains an infinite R-cluster.
Adding more points improves R-connectivity. Hence, the probability of percolation is monotone increasing in α. The Poissonian nature of the Boolean disagreement percolation model makes percolation a tail event, i.e., it holds with either probability 0 or 1. Thus, a critical intensity separates the non-percolating and percolating regimes.

2.4
The hard-sphere model Let [.] be Iverson brackets 1 . For disjoint Y, C ∈ C R d , the indicator function H of the conditional hard-core constraint of Y under condition C is given by For a bounded domain B ∈ B b , a boundary condition C ∈ C B c and an activity λ ∈ [0, ∞[, consider the hard-sphere model with law P hs B,C,λ . As it is the Poisson PP of intensity λ conditioned to be hard-core, its Janossy infinitesimal is The alternative definition in statistical mechanics uses the pair potential The Hamiltonian of n ordered points in B is The density of x ∈ B n is where the partition function Z is The convention e −∞ = 0 encodes (3) by (5b). The remainder of this paper uses the PP notation as in (4), except for the partition function. Because of disagreement percolation the bounded range interaction in H(Y |C) in (3), one may restrict the boundary condition to C R(B) . A Gibbs measure is a weak limit of a sequence (P hs Bn,Cn,λ ) n∈N along a van Hove sequence (B n ) n∈N and a sequence (C n ) n∈N of boundary conditions with C n ∈ C B c n [21, Sections 2 and 3]. The Gibbs measures G λ of the specification P hs λ := (P hs B,C,λ ) B∈B b ,C∈C B c form a simplex. Unlike in the lattice case [24], in the continuum case of dimension greater than one, the existence of a finite critical activity at which a phase transition happens is widely believed, but not yet proven. See the solution in one dimension [26], the absence of positional phase transition in two dimensions [22], which does not exclude a conjectured orientational phase transition, and the discussion of the state of the problem in higher dimensions [17,Section 3.3]. If R = 0, then there is no interaction, the hardsphere model reduces to a Poisson PP and P poi R d ,λ is the unique Gibbs measure.

Stochastic domination
On C n B , the standard product σ-algebra is F ⊗n B . The canonical variables on C n B are ξ := (ξ 1 , . . . , ξ n ). A coupling P of n PP laws P 1 , . . . , P n on B ∈ B is a probability measure on (C n B , F ⊗n B ) such that, for all 1 ≤ i ≤ n and E ∈ F B , P(ξ i ∈ E) = P i (ξ ∈ E).
A PP law P 2 stochastically dominates a PP P 1 , if there exists a coupling P of them with P(ξ 1 ⊆ ξ 2 ) = 1. A Poisson PP stochastically dominates a hardsphere model with the same activity as the intensity of the Poisson PP [10, Example 2.2].

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 model.
If the intensity of the dominating Poisson point process is below the critical value for percolation in the Boolean 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. Furthermore, as the cluster size of the Boolean model decays exponentially in the subcritical phase, controls of the Gibbs measure such as the influence of boundary conditions or the reduced pair correlation function decay exponentially, too.
The remainder of this section formalises the preceding outline. The proofs are in Section 4. The symmetric difference S 1 S 2 between sets S 1 and S 2 equals (S 1 \ S 2 ) ∪ (S 2 \ S 1 ).
A disagreement coupling family of intensity α and activity λ is a family of disagreement couplings (P B,C1,C2,λ,α ) B∈B b ,C1,C2∈C B c .
A disagreement coupling family in the sub-critical phase of the Boolean model implies uniqueness of the Gibbs measure. Disagreement percolation also implies that the sensitivity to boundary conditions (8a), the finite volume error (8b), reduced second moment measure (second factorial cumulant measure) (8c) and reduced pair correlation function (8d) decay exponentially. The rate of exponential decay is the same as the one of the Boolean model (7), which holds in the whole subcritical regime of the Boolean model [18,Section 3.7]. Theorem 3.3. Suppose that there exists a disagreement coupling family of intensity α < λ b (d) at activity λ, and there exist K ≥ 1, κ > 0 such that, for all For all A, For every x, y ∈ R d , the reduced pair correlation function ρ decays as

Bounds from disagreement percolation
The hard-sphere model admits a disagreement coupling family of the same intensity as its activity.
Theorem 3.4. There exists a disagreement coupling family of intensity λ for P hs λ . If λ < λ b (d), then G λ is a singleton and exponential decay as in (8) holds. disagreement percolation Theorem 3.4 follows from the disagreement coupling family in Section 6 and theorems 3.2 and 3.3. A motivation of this coupling is in Section 3.4 and discussion of generalisations and other approaches in Section 3.5.
Bounds on λ b (d) translate directly into sufficient conditions for the uniqueness of the Gibbs measure. In one dimension the Boolean model never percolates [18,Theorem 3.1].
In dimension two, rigorous bounds on More recent high confidence bounds in [1], taken from [19, equation (2)], are For dimensions 2 to 10, simulation bounds are in [27,28]. Another set of high confidence and rigorous bounds via an Ornstein-Zernike approach are in [32, Table 4]. The asymptotic behaviour of the critical intensity [20], taken from [18, Section 3.10], is lim

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 [23]. Both deliver analyticity of the free energy, too. Let λ ce (d) be the radius of the cluster expansion in d dimensions.
In one dimension, disagreement percolation (9a) replicates Tonks' classic result of the complete absence of phase transitions via virial expansion methods [3,12,15,26]. In terms of the activity, it is known that the radius of the cluster expansion is exactly [3,11,15] In two dimensions, the best currently known lower [9] and upper bounds [23, The bounds (10b) are between 0.45% and 0.65% of the disagreement percolation bound (9b). General bounds on the cluster expansion radius from [23, Section As v 1 = 2, equation (10a) shows that the upper bound is tight. I conjecture that the asymptotic behaviour in high dimensions is Comparing (9c) and (10d), the asymptotic improvement should be by a factor of e. This is not completely surprising, because on the infinite k-regular tree T k , the critical percolation probability is 1 k−1 and the radius of the cluster expan- . Both Z d and R d behave for large d as T 2d , for both percolation and cluster expansion. Extrapolating arguments of [25,Section 8] gives a heuristic for the upper bound in (10c), too. Finally, I conjecture that disagreement percolation is always better than cluster expansion. A possible approach is recent work connecting the Ornstein-Zernike equation for the Boolean model with Ruelle-like sufficient conditions for cluster expansion [16].

Motivation behind the dependent thinning and twisted coupling
This section assumes familiarity with the dependent thinning in Definition 5.2 and the twisted disagreement coupling family in Definition 6.1.
The approach to disagreement percolation in [30] is a vertex-wise conditional coupling of two Markov fields on a finite graph. A uniform control of those couplings allows stochastic domination by a Bernoulli product field. This poses a problem on R d . The key insight is to flip the picture around. Start with the Bernoulli random field and reinterpret the conditional couplings as simultaneous dependent thinnings to the two dominated Markov fields. Transferring this to the PP case is non-trivial, but helpfully [31] introduced an optimisation for the hard-core model. This reduces the thinning probability onto two hard-core models on a single vertex to a thinning probability of a single hard-core model. In the PP case, this enables the independent construction on the disjoint domains in (26a). It allows to "twist" two hard-sphere models of activity λ under a single P poi λ PP, i.e., have joint stochastic domination in (28e). The overall recursive approach from the dependent vertex-wise couplings stays and translates into the recursive definition (26b). The recursive definition of P tw-rec B,C1,C2 demands that it is jointly measurable in the boundary conditions C 1 and C 2 . By the above outline, this comes back to the measurability of P thin B,C in the boundary condition C. The classic dominating couplings between a Poisson PP and a hard-sphere model in [10] or following [21] are implicit. But the D = ∅ case in (26b) suggests to use the dependent thinning approach for a single dominated hard-sphere model, too. In this case, the calculations are doable and lead to the dependent thinning in Definition 5.2.
The Papangelou intensity [6, (15.6.13)] is the infinitesimal cost of adding another point to a given configuration. It is H({x}|Y ∪ C)λ for the hard-sphere model. The Poisson PP has constant Papangelou intensity λ. Thus, one can control the hard-sphere model pointwise incrementally by a Poisson PP. All three stochastic dominations of a hard-sphere model by a Poisson PP (the dependent thinning in Definition 5.2, [10] and [21]) build upon this fact. In the D = ∅ case, P tw-rec reduces to the same setting, too. It is yet unknown if this is the smallest Poisson intensity needed to dominate the hard-sphere model.
Another natural question is whether the depending thinning factorises over clusters of the dominating Poisson PP. Although it looks likely to be true, because the answer is not relevant here, it is not investigated.

Outlook
In the lattice case, disagreement percolation implies the complete analyticity of the free energy, pointed out by Schonmann [30, Note added in proof]), and the Poincare inequality for the usual spin-flip dynamics [4]. In principle, both results should be generalisable to the hard-sphere model, too. The exponential control in (8) looks exactly like what is needed in the discrete case for complete analyticity [8], but a theory for PPs is still missing.
The proof of theorems 3.2 and 3.3 is not dependent on the hard-sphere model and applies to arbitrary Gibbs PPs with bounded range interaction. A generalisation to the physically interesting case of marked Gibbs PP models with finite, but unbounded, range should be possible. This demands a notational and definitional base exceeding the limits of a single paper, though.
Beyond the hard-sphere model, one could do a product construction in (26a) and compensate by adding an additional P poi B,λ in the D = ∅ case in (26a). This would lead to a disagreement coupling family of intensity 2λ, for a repulsive potential. The recursive construction still demands the dominating coupling to be measurable in the boundary conditions.
The more simple product approach from [29] with a swapping argument yields only a lower bound of λ b (d)/2. Thus, it is not strong enough for the comparison in Section 3.3. Also, the same measurability concerns as in the twisted approach surface, too.
Another sufficient condition for uniqueness Gibbs measure, and even complete analyticity of the free energy, is Dobrushin's uniqueness condition [7]. There have been generalisations to the PP case [13,14], but I make no explicit comparison here.

Proof of theorems in Section 3.1
The proof of Theorem 3.2 follows closely the one in the discrete case [30, 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. This proposition is the key control of the influence of the differing boundary conditions. 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 finally, it uses the sub-criticality of the Boolean model to tighten the bound to zero as the domain increases. Theorem 3.3 uses Proposition 4.1 to control the influence of the differing boundary conditions.
To lighten the notation, this section drops the λ parameter in P hs B,C,λ .
Proof. Let F be the embedding of E into F B . First, reduce the difference by cancelling symmetric parts.
Second, relax the asymmetric event to disagreement and use the properties of disagreement percolation.
For disjoint A, B ∈ B b and E ∈ F B , the following identities hold for the Janossy infinitesimals.
Proof of Theorem 3.2. Let ν 1 , ν 2 ∈ G λ . The aim is to show that ν 1 = ν 2 . This is equivalent to The hard-sphere property ensures that a Gibbs measure in G λ has moment measures of all orders [5, (5.4.9)]. Thus, its local Janossy measures exist. The following result controls the difference between two measures. Let µ 1 and µ 2 be probability measures on the measurable space (Ω, A). For all f : Ω → [0, 1] measurable, Let (B n ) n∈N be a van Hove sequence with A ⊆ B 1 . For each Gibbs measure ν ∈ G λ and n ∈ N, the Gibbs property restricts the discussion to the bounded Borel set B n . Second, the existence of a disagreement coupling family of intensity α and (13) 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.
Statement (8d) follows from (8c) by disintegration with respect to the product of the intensity measure of ν, which has a density with respect to L.

Dependently thinning Poisson to hard-sphere
Sections 5.1 and 5.2 contain additional facts about joint Janossy measures and the hard-sphere model respectively. The dependent thinning from a P poi B,λ to a P hs B,C,λ is in Section 5.3. This section fixes λ ∈ [0, ∞[. Hence, it drops the λ parameter in P hs B,C,λ and Z(B, C, λ). Also, P poi B stands for P poi B,λ .

Joint Janossy measure
Let n ≥ 2 and P be a coupling of n PP laws. A Borel measure M on (C n B , F ⊗n B ) is the (local) joint Janossy measure of P on B ∈ B b , if, for all E 1 , . . . , E n ∈ F B , 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 classic case (1) and the explicit style on generating sets in [5,Section 5.3]. As the sets n i=1 E i generate F ⊗n B , there is no loss of generality. If P admits a joint Janossy measure on B, then P(ξ ∩ B = dY ) := P(∀1 ≤ i ≤ n : ξ i ∩ B = dY i ) denotes its infinitesimal at Y ∈ C n B . The identities (12) generalise directly from the classic to the joint case. Joint Janossy measures of marginals of a coupling P result from integrating out the joint Janossy measure over the complement.

More about the hard-sphere model
The conditional hard-sphere constraint H chains.
The function H is C B c × C B → {0, 1} and measurable on ( as a product of measurable functions (3). It is monotone decreasing in both arguments. For B ∈ B b , the function is measurable on (C B c , F B c ) and monotone decreasing. As a consequence, P hs B,C is measurable in the boundary condition C, too. For C ∈ C R d , the function is monotone increasing. Finally, the relation between (4) and (5d) is The hard-sphere model fulfils the DLR conditions [21, (2.2)-(2.4)]). That is, for A, B ∈ B b , X ∈ C A and Y ∈ C B , the Janossy infinitesimal chains.

The thinning
This section presents a coupling between a hard-sphere PP law and a dominating Poisson PP law. The coupling is an explicit dependent thinning from the dominating Poisson PP. The thinning probability is related to the logarithm of the free energy. Its explicit form implies the measurability of the coupling with respect to the boundary condition. For x ∈ R d , let d n,i (x) be the n th binary digit of the i th coordinate of x. Define the order ≺ on R d , by ordering (d n,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 ∈ R d , are L-measurable. The symbols ±∞ extend ≺ with elements being bigger and smaller than each element of R d . For as well as all standard variations thereof. Each bounded Borel set is contained in a finite interval.
For the remainder of this section, fix B ∈ B b and C ∈ C B c . Implicitly restrict intervals ]a, b] to B, i.e., ]a, b] ∩ B.
The proof of Proposition 5.1 is in Section 5.4. Proposition 5.1 calculates the dependent disintegration of the free energy of a right-open interval in B with respect to the Poisson intensity. The monotonicity of Z in the domain applied to the lhs of (20) implies that it grows monotone in x outside of S(C ∪ Y ). The monotonicity applied to the rhs of (20) implies that its value lies in [0, 1].
and the helper function The Janossy infinitesimal of the thinning is Theorem 5.3. The dependent thinning P thin B,C is a dominating coupling between a Poisson PP and a hard-sphere PP, as and The boundary condition may be restricted to R(B). The law P thin B,C is measurable in C.
The proof of Theorem 5.3 is in Section 5.6.

Proof of Proposition 5.1
For L-a.e. x ∈ B, there exists ε > 0 and and exists L-a.s. [2, Lemma 5.1.3]. This is the usual one-dimensional derivative applied along the measurable total ordering ≺ of R d .
. Because of the monotonicity of Z in the domain (17), p(x, Y ) ∈ [0, 1]. The remainder of this section shows that z = −λhs.
Because L-a.e. point of B is a density point of B [2, 5.8(ii)], for L-a.e. a, b ∈ B with a ≺ b, L(]a, b[) > 0. Therefore, for given Y , x ∈ Y , there exists ε > 0 and points For ε small enough, X ⊆ A ε consists of a single point y. Thus, Restate everything inside the outer integral as a function of y. Expand domain of the inner integration paying only an exponential penalty. Rewrite z (x) as Apply (18) and the Lebesgue differentiation theorem [2, Thm 5.6.2] to get Aε h(y)s(y)λdy = −λh(x)s(x) .

Thinning all points and an integral equation
This section calculates the probability of thinning all points within an interval and it holds that The solution comes from an integral equation. For each The aim is to calculate f (a) e(a) . If ]x, b[ contains a point, then splitting the smallest point off yields an integral equation for f .

Proof of Theorem 5.3
The thinning P thin B,C is well-defined, as The proof of (25) is easy by induction over the cardinality of Z and by splitting of the point min Z with respect to ≺. The freedom to restrict the boundary condition C to C ∩ R(B) ∈ C R(B) follows from the same freedom for H and Z (5d). The measurability in the boundary condition follows from the measurability preserving operations in (21) and the measurability of the rhs of (20).
The fact that P thin B,C is a stochastic domination is evident from the construction. The construction as thinning implies that the second marginal is Poisson (22b). The remainder of this section shows that the first marginal is hardsphere (22a).
Let Y ∈ C B with n := |Y |. Order Y =: {y 1 , . . . , y n } increasingly by ≺. Let y 0 := −∞ and y n+1 : For 0 ≤ i ≤ n, Combine these rewritings to get Combine the hard-sphere constraints by (15). Join the two products and cancel the factors except the denominator at index 0 and the numerator at index n in the left product.
6 The twisted coupling family Definition 6.1 defines a family of couplings recursively. Proposition 6.3 shows that it is a disagreement coupling family of intensity λ for the hard-sphere model. The notational conventions outlined at the beginning of Section 5 apply.
Let D := F 1 ∪ F 2 be the zone of disagreement and partition it into Define the joint Janossy intensity of the law P tw-zone ) .

(26a)
Define the joint Janossy intensity of law P tw-rec B,C1,C2 on (C 3 B , F ⊗3 B ) recursively by The idea behind the recursive construction of P tw-rec is simple. The sets F 1 and F 2 describe the parts of the domain forbidden by the respective boundary conditions. If one can construct the disagreement coupling on D, then recursion takes care of the rest.
If D = ∅, a dominating coupling with an identification of the two hardsphere PPs is already a disagreement coupling. Since there are no disagreeing points, no connection to the disagreeing boundary is needed. For 1 ≤ i ≤ 2, let If D = ∅, then the partition {D 0 , D 1 , D 2 } comes into play. Points of ξ 1 and ξ 2 can only lie in D 1 and D 2 respectively. Independent projections of a dominating coupling take care of that. This also connects the disagreeing points to the boundary for free. On D 0 , an independent Poisson PP of intensity λ ensures that there is a Poisson PP on all of D. This is the "twist".
The event of connecting disagreement with the boundary in F ⊗2 B is Proposition 6.2. The boundary conditions of P tw-zone B,C1,C2 may be restricted to C R(B) . The law P tw-zone B,C1,C2 is jointly measurable in (C 1 , C 2 ) and has the right marginals It also has the useful properties Proof. The freedom to restrict the boundary conditions to C R(B) follows from the same property of P thin B,. in Theorem 5.3. The measurability in the boundary conditions follow from the same properties of the law P poi B and P thin B,. in Theorem 5.3 and the other measurable indicator terms in construction (26a).
The Poisson marginal (28b) is a straightforward integration over (26a). The hard-sphere marginals (28a) use the hard-core exclusion together with the properties of the partition {D 0 , D 1 , D 2 } in addition to integration.
Properties (28c) and (28d) follows directly from the [Y i ⊆ D i ] terms, for each 1 ≤ i ≤ 2. Property (28e) follows from the fact that P thin D1,C1 and P thin D2,C2 are dominating couplings (22c). Property (28f) follows trivially from the definition of D 1 and D 2 and (28c). Proposition 6.3. The boundary conditions of P tw-rec B,C1,C2 may be restricted to C R(B) . The coupling P tw-rec B,C1,C2 is well-defined and jointly measurable in (C 1 , C 2 ). Its marginals are ∀1 ≤ i ≤ 2 : P tw-rec B,C1,C2 (ξ i = dY ) = P hs B,Ci (dY ) , P tw-rec B,C1,C2 (ξ 3 = dY ) = P poi B (dY ) .  (31). An induction step in a proof of a recursive property may have to recurse at most twice before being able to apply the inductive hypothesis. The recursion terminates after at most 2s(B) steps.
The freedom to restrict the boundary conditions to C R(B) follows from the same property of P thin B,. in Theorem 5.3, P tw-zone B\D,.,. in Proposition 6.2 and itself. The measurability in the boundary conditions follow from the same properties of the law P thin B,. in Theorem 5.3, P tw-zone B\D,.,. in Proposition 6.2, the other measurable indicator terms in (26a) and itself.
The property (29c) follows directly from the [Y 1 = Y 2 ] identification and the dominating property of P thin B,∅ (22c) in the D = ∅ case. In the D = ∅ case, it follows from (28c), (28d) and (28e) and itself recursively.
Property (29d) is trivial in the D = ∅ case and follows from (28f) and itself recursively in the D = ∅ case.
Step ( ) of the following proof of (29d) in the nontrivial D = ∅ case demonstrates the need for the measurability of the coupling in the boundary conditions.