m = m = m = m = Necessary condition

in N, and α, β, γ, δ in X such that α ≤ γ, β ≤ δ, we define I a := I K a

j, m =

K [ i=1 {k ∈ X : m i ≥ k δ − β + j i } 2.9 I b := I K b

j, m =

K [ i=1 {k ∈ X : γ − α + m i ≥ k j i } 2.10 I c := I K c

h, m =

K [ i=1 {k ∈ X : m i ≥ k γ − α + h i } 2.11 I d := I K d

h, m =

K [ i=1 {k ∈ X : δ − β + m i ≥ k h i } 2.12 An S particle system η t t ≥0 is stochastically larger than an f S particle system ξ t t ≥0 if and only if X k ∈X :kδ−β+ j 1 e Π 0,k α,β + X k ∈I a e Γ k α,β ≤ X l ∈X :l j 1 Π 0,l γ,δ + X l ∈I b Γ l γ,δ 2.13 X k ∈X :kh 1 e Π −k,0 α,β + X k ∈I d e Γ k α,β ≥ X l ∈X :lγ−α+h 1 Π −l,0 γ,δ + X l ∈I c Γ l γ,δ 2.14 for all choices of K ≤ e N α, β ∨ N γ, δ, h, j, m, α ≤ γ, β ≤ δ. Remark 2.5. The restriction K ≤ e N α, β ∨ N γ, δ avoids that an infinite number of K, I a , I b , I c , I d result in the same rate inequality. Since e Γ k α,β = 0 for each k e N α, β, if K e N α, β no terms are being added to the left hand side of 2.13, and adding more terms on the right hand side does not give any new restrictions. A similar statement holds for 2.14, with the corresponding condition K ≤ N γ, δ. Remark 2.6. If S = f S , Theorem 2.4 states necessary and sufficient conditions for attractiveness of S . We follow the approach in [8]: in order to characterize the stochastic ordering of two processes, first of all we find necessary conditions on the transition rates. Then we construct a Markovian increasing coupling, that is a coupled process ξ t , ζ t t ≥0 which has the property that ξ ≤ ζ implies P ξ , ζ {ξ t ≤ ζ t } = 1, for all t ≥ 0. Here P ξ , ζ denotes the distribution of ξ t , ζ t t ≥0 with initial state ξ , ζ .

2.2 Applications

We propose several applications to understand the meaning of Conditions 2.13–2.14. We think of α, β ≤ γ, δ as the configuration state of two processes η t ≤ ξ t on a fixed pair of sites x and y: namely η t x = α, η t y = β, ξ t x = γ, ξ t y = δ. 112

2.2.1 No multiple births, deaths or jumps

Let N = sup α,β∈X 2 N α, β: Proposition 2.7. If N = 1 then a change of at most one particle per time is allowed and Conditions 2.13 and 2.14 become e Π 0,1 α,β + e Γ 1 α,β ≤Π 0,1 γ,δ + Γ 1 γ,δ if β = δ and γ ≥ α, 2.15 e Π 0,1 α,β ≤Π 0,1 γ,δ if β = δ and γ = α, 2.16 e Π −1,0 α,β + e Γ 1 α,β ≥Π −1,0 γ,δ + Γ 1 γ,δ if γ = α and δ ≥ β, 2.17 e Π −1,0 α,β ≥Π −1,0 γ,δ if γ = α and δ = β. 2.18 Proof. . If β δ, then δ − β + j i ≥ δ − β + j 1 ≥ 1 for all K 0, 1 ≤ i ≤ K so that 1 ∈ I a by definition 2.9. Since N = 1 the left hand side of 2.13 is null; if β = δ the only case for which the left hand side of 2.13 is not null is j 1 = 0, which gives X k e Π 0,k α,β + X k ∈I a e Γ k α,β ≤ X l Π 0,l γ,β + X l ∈I b Γ l γ,β . Since N = 1, the value K = 1 covers all possible sets I a and I b , namely I a = {k : m 1 ≥ k 0} and I b = {γ − α + m 1 ≥ l 0}. If m 1 0, we get 2.15. If γ = α and m 1 = 0 we get 2.16. One can prove 2.17 in a similar way. If β = δ and γ ≥ α, Formula 2.15 expresses that the sum of the addition rates of the smaller process on y in state β must be smaller than the corresponding addition rates on y of the larger process on y in the same state. If β = δ and γ = α we also need that the birth rate of the smaller process on y is smaller than the one of the larger process, that is 2.16. Conditions 2.17–2.18 have a symmetric meaning with respect to subtraction of particles from x. Remark 2.8. If f S = S , when α = γ and β = δ conditions 2.16, 2.18 are trivially satisfied, and we only have to check 2.15 when α γ and 2.17 when β δ. Proposition 2.7 will be used in a companion paper for metapopulation models, see [3]. If R 0,k α,β = 0 for all α, β, k, the model is the reaction diffusion process studied by Chen see [4] and the attractiveness Conditions 2.15, 2.17 the only ones by Remark 2.8 reduce to Γ 1 α,β ≤ Γ 1 γ,β if γ α; Γ 1 α,δ ≥ Γ 1 α,β if δ β. In other words we need Γ 1 α,β to be non decreasing with respect to α for each fixed β, and non increasing with respect to β for each fixed α. In [4], the author introduces several couplings in order to find ergodicity conditions of reaction diffusion processes. All these couplings are identical to the coupling H introduced in Section 3.2 and detailed in Appendix A if N = 1, on configurations where an addition or a subtraction of particles may break the partial order, but differ from H on configurations where it cannot happen. 113

2.2.2 Multitype contact processes

If Γ k α,β = e Γ k α,β = 0, for all α, β ∈ X 2 , k ≥ 0, that is when no jumps of particles are present, all rates are contact-type interactions. Such a process is called multitype contact process. Conditions 2.13–2.14 reduce to: for all α, β ∈ X 2 , γ, δ ∈ X 2 , α, β ≤ γ, δ, h 1 ≥ 0, j 1 ≥ 0, i X k δ−β+ j 1 e Π 0,k α,β ≤ X l j 1 Π 0,l γ,δ ; ii X k h 1 e Π −k,0 α,β ≥ X l γ−α+h 1 e Π −l,0 γ,δ . 2.19 Many different multitype contact processes have been used to study biological models. We propose some examples with the corresponding conditions. Since the state space Ω = {0, 1, . . . , M} Z d where M ∞ is compact, we refer to the construction in [11]. Spread of tubercolosis model [17]. Here M represents the number of individuals in a population at a site x ∈ Z d . The transitions are: P 1 β = φβ1l {0≤β≤M−1} , R 0,1 α,β = 2dλα1l {β=0} , P −β β = 1l {1≤β≤M} , px, y = 1 2d 1l {x∼ y} . where y ∼ x is one of the 2d nearest neighbours of site x. Given two systems with parameters λ, φ, M and λ, φ, M , the proof of [17, Proposition 1] reduces to check Conditions 2.19: φβ1l {0≤β≤M−1} + 2dλα1l {β=0} ≤φδ1l {0≤δ≤M−1} + 2dλγ1l {δ=0} , if β = δ, j 1 = 0 1l {1≤α≤M,αh 1 } ≥1l {1≤γ≤M,γγ−α+h 1 } , if γ ≥ α, h 1 ≥ 0 which are satisfied if λ ≤ λ, φ ≤ φ and M ≤ M. In the following examples we suppose f S = S , that is we consider necessary and sufficient conditions for attractiveness. 2 -type contact process [14]. In this model M = 2. Since a value on a given site does not represent the number of particles on that site, we write the state space {A, B, C} Z d . The value B represents the presence of a type-B species, C the presence of a type-C species and A an empty site. If A = 0, B = 1, C = 2 then the transitions are R 0,1 α,β = 2dλ 1 1l {α=1,β=0} , R 0,2 α,β = 2dλ 2 1l {α=2,β=0} , P −β β = 1l {1≤β≤2} , px, y = 1 2d 1l {x∼ y} . By taking h 1 = 0, Condition 2.19 is X k δ−β 1l {k=1} 2d λ 1 1l {α=1,β=0} +1l {k=2} 2d λ 2 1l {α=2,β=0} ≤ 2dλ 1 1l {γ=1,δ=0} + 2dλ 2 1l {γ=2,δ=0} ; By taking β = 0, δ = 1, α = γ = 2 we get 2dλ 2 ≤ 0, which is not satisfied since λ 2 0. As already observed, see [20, Section 5.1], one can get an attractive process by changing the order between species: namely by taking A = 1, B = 0 and C = 2 the process is attractive. 114

2.2.3 Conservative dynamics

If Π 0,k α,β = 0, for all α, β ∈ X 2 , k ∈ N, we get a particular case of the model introduced in [8], for which neither particles births nor deaths are allowed, and the particle system is conservative. Suppose that in this model the rate Γ k α,β y − x has the form Γ k α,β p y − x for each k, α, β. Necessary and sufficient conditions for attractiveness are given by [8, Theorem 2.21]: X k δ−β+ j Γ k α,β p y − x ≤ X l j Γ l γ,δ p y − x for each j ≥ 0 2.20 X k h Γ k α,β p y − x ≥ X l γ−α+h Γ l γ,δ p y − x for each h ≥ 0 2.21 while 2.13 − 2.14 become X k ∈I a Γ k α,β p y − x ≤ X l ∈I b Γ l γ,δ p y − x 2.22 X k ∈I d Γ k α,β p y − x ≥ X l ∈I c Γ l γ,δ p y − x 2.23 for all I a , I b , I c and I d given by Theorem 2.4. Proposition 2.9. Conditions 2.20–2.21 and Conditions 2.22–2.23 are equivalent. Proof. . By choosing j i = j, h i = h and m i = m N α, β ∨ N γ, δ for each i in Theorem 2.4, Conditions 2.22–2.23 imply 2.20–2.21. For the opposite direction, by subtracting 2.21 with h = m i δ − β + j i to 2.20 with j = j i X m i ≥kδ−β+ j i Γ k α,β p y − x ≤ X γ−α+m i ≥l ′ j i Γ l ′ γ,δ p y − x 2.24 and we get 2.22 by summing K times 2.24 with different values of j i , m i , 1 ≤ i ≤ K. Condition 2.23 follows in a similar way.

2.2.4 Metapopulation model with Allee effect and mass migration

The third model investigated in [3] is a metapopulation dynamics model where migrations of many individuals per time are allowed to avoid the biological phenomenon of the Allee effect see [1], [19]. The state space is compact and on each site x ∈ Z d there is a local population of at most M individuals. Given M ≥ M A 0, M N 0, φ, φ A , λ positive real numbers, the transitions are P 1 β = β1l {β≤M−1} P −1 β = β φ A 1l {β≤M A } + φ1l {M A β} Γ k α,β = ¨ λ α − k ≥ M − N and β + k ≤ M, otherwise. for each α, β ∈ X , and px, y = 1 2d 1l { y∼x} . In other words each individual reproduces with rate 1, but dies with different rates: either φ A if the local population size is smaller than M A Allee effect or φ if it is larger. When a local population has more than M − N individuals a migration of more than one individual per time is allowed. Such a process is attractive by [3, Proposition 5.1, where N and M play opposite roles], which is an application of Theorem 2.4. 115

2.2.5 Individual recovery epidemic model

We apply Theorem 2.4 to get new ergodicity conditions for a model of spread of epidemics. The most investigated interacting particle system that models the spread of epidemics is the contact process, introduced by Harris [10]. It is a spin system η t t ≥0 on {0, 1} Z d ruled by the transitions η t x = 0 → 1 at rate X y ∼x η t y η t x = 1 → 0 at rate 1 See [11] and [12] for an exhaustive analysis of this model. In order to understand the role of social clusters in the spread of epidemics, Schinazi [17] intro- duced a generalization of the contact process. Then, Belhadji [2] investigated some generalizations of this model: on each site in Z d there is a cluster of M ≤ ∞ individuals, where each individual can be healthy or infected. A cluster is infected if there is at least one infected individual, otherwise it is healthy. The illness moves from an infected individual to a healthy one with rate φ if they are in the same cluster. The infection rate between different clusters is different: the epidemics moves from an infected individual in a cluster y to an individual in a neighboring cluster x with rate λ if x is healthy, and with rate β if x is infected. We focus on one of those models, the individual recovery epidemic model in a compact state space: each sick individual recovers after an exponential time and each cluster contains at most M individ- uals. The non-null transition rates are R 0,k ηx,η y = ¨ 2d ληx k = 1 and η y = 0 2d βηx k = 1 and 1 ≤ η y ≤ M − 1, 2.25 P k η y =    γ k = 1 and η y = 0 φη y k = 1 and 1 ≤ η y ≤ M − 1 η y k = −1 and 1 ≤ η y ≤ M, 2.26 px, y = 1 2d 1l { y∼x} for each x, y ∈ S 2 . 2.27 and Γ k ηx,η y = 0 for all k ∈ N. The rate γ represents a positive “pure birth” of the illness: by setting γ = 0, we get the epidemic model in [2], where the author analyses the system with M ∞ and M = ∞ and shows [2, Theorem 14] that different phase transitions occur with respect to λ and φ. Moreover [2, Theorem 15], if λ ∨ β 1 − φ 2d 2.28 the disease dies out for each cluster size M . By using the attractiveness of the model we improve this ergodicity condition. Notice that a depen- dence on the cluster size M appears. Theorem 2.10. Suppose λ ∨ β 1 − φ 2d1 − φ M , 2.29 with φ 1 and either i γ = 0, or ii γ 0 and β − λ ≤ γ2d. Then the system is ergodic. 116 Notice that if γ 0 and β ≤ λ hypothesis ii is trivially satisfied. If M = 1 and γ = 0 the process reduces to the contact process and the result is a well known and already improved ergodic result see for instance [11, Corollary 4.4, Chapter VI]; as a corollary we get the ergodicity result in the non compact case as M goes to infinity. In order to prove Theorem 2.10, we use a technique called u-criterion. It gives sufficient conditions on transition rates which yield ergodicity of an attractive translation invariant process. It has been used by several authors see [4], [5], [13] for reaction-diffusion processes. First of all we observe that Proposition 2.11. The process is attractive for all λ, β, γ, φ, M . Proof. . Since M = 1, then N = 1 and Conditions 2.13–2.14 reduce to 2.15, 2.17. Namely, given two configurations η ∈ Ω, ξ ∈ Ω with ξ ≤ η, necessary and sufficient conditions for attrac- tiveness are if ξx ηx and ξ y = η y, R 0,1 ξx,ξ y + P 1 ξ y ≤ R 0,1 ηx,η y + P 1 η y ; P −1 ξ y ≥ P −1 η y . If ξ y = η y ≥ 1, 2dβξx ≤ 2dβηx; if ξ y = η y = 0, 2dλξx ≤ 2dληx. In all cases the condition holds since ξ ≤ η. The key point for attractiveness is that R 0,1 ηx,η y is increasing in ηx. Given ε 0 and {u l ε} l ∈X such that u l ε 0 for all l ∈ X , let F ε : X × X → R + be defined by F ε x, y = 1l {x6= y} | y−x|−1 X j=0 u j ε 2.30 for all x, y ∈ X . When not necessary we omit the dependence on ε and we simply write F x, y = 1l {x6= y} | y−x|−1 X j=0 u j . Since u l ε 0, this is a metric on X and it induces in a natural way a metric on Ω. Namely, for each η and ξ in Ω we define ρ α η, ξ := X x ∈Z d F ηx, ξxαx 2.31 where {αx} x ∈Z d is a sequence such that αx ∈ R, αx 0 for each x ∈ Z d and X x ∈Z d αx ∞. 2.32 Denote by Ω m := {η ∈ Ω : ηx = m for each x ∈ Z d } and let η M ∈ Ω M and η ∈ Ω which is not an absorbing state if γ 0. The key idea consists in taking a “good sequence {u l } l ∈X and in looking for conditions on the rates under which the expected value e E · with respect to a coupled measure e P of the distance between η M t and η t converges to zero as t goes to infinity, uniformly 117 with respect to x ∈ S. We will use the generator properties and Gronwall’s Lemma to prove that if there exists ε 0 and a sequence {u l ε} l ∈X , u l ε 0 for all l ∈ X such that the metric F satisfies e E L F η t x, η M t x ≤ −εe E F η t x, η M t x 2.33 for each x ∈ Z d , then e E lim t →∞ F η t x, η M t x = 0 2.34 uniformly with respect to x ∈ S so that the distance ρ α ·, · between the larger and the smaller process converges to zero, and ergodicity follows. Condition 2.33 leads to Proposition 2.12 u-criterion. If there exists ε 0 and a sequence {u l ε} l ∈X such that for any l ∈ X    φlu l ε − lu l −1 ε ≤ −ε l −1 X j=0 u j ε − ¯ u ελ ∨ β2dl u l ε 0 2.35 where ¯ u ε := max l ∈X u l ε, u −1 = 0 by convention and u = U 0, then the system is ergodic. Hence we are left with checking the existence of ε 0 and positive {u l ε} l ∈X which satisfy 2.35. Such a choice is not unique. Remark 2.13. Given U = 1 0, if u l ε = 1 for each l ∈ X then Condition 2.35 reduces to the existence of ε 0 such that φl − l ≤ −εl − λ ∨ β2dl for all l ∈ X , that is ε ≤ 1 − φ − λ ∨ β2d, thus Condition 2.28. Notice that in this case F x, y = | y−x|−1 X j=0 1 = | y − x|. Another possible choice is Definition 2.14. Given ε 0 and U 0, we set u ε = U and we define u l ε l ∈X recursively through u l ε = 1 φl − ε l −1 X j=0 u j ε − Uλ ∨ β2dl + lu l −1 ε for each l ∈ X , l 6= 0. 2.36 Definition 2.14 gives a better choice of {u l ε} l ∈X , indeed the u-criterion is satisfied under the more general assumption 2.29. Proofs of Theorems 2.10 and 2.12 are detailed in Section 4. 3 Coupling construction and proof of Theorem 2.4 In this section we prove the main result: we begin with the necessary condition, based on [8, Proposition 2.24]. 118

3.1 Necessary condition

Proposition 3.1. If the particle system η t t ≥0 is stochastically larger than ξ t t ≥0 , then for all α, β, γ, δ ∈ X 2 × X 2 with α, β ≤ γ, δ, for all x, y ∈ S 2 , Conditions 2.13–2.14 hold. Proof. . Let ξ, η ∈ Ω × Ω be two configurations such that ξ ≤ η. Let V ⊂ Ω be an increasing cylinder set. If ξ ∈ V or η ∈ V , 1l V ξ = 1l V η. 3.1 Since η t is stochastically larger than ξ t , for all t ≥ 0, by Theorem 2.2 or [11, Theorem II.2.2] if we are interested in attractiveness e T t1l V ξ ≤ T t1l V η since ν ≤ ν ′ is equivalent to νV ≤ ν ′ V for all increasing sets. Combining this with 3.1, t −1 [ e T t1l V ξ − 1l V ξ] ≤ t −1 [T t1l V η − 1l V η], which gives, by Assumption 2.6, f L 1l V ξ ≤ L 1l V η. 3.2 We have, by using 2.1, f L 1l V ξ = X x, y ∈S px, y X α,β∈X χ x, y α,β ξ X k e Γ k α,β 1l V S −k,k x, y ξ − 1l V ξ + e Π 0,k α,β 1l V S 0,k x, y ξ − 1l V ξ + e Π −k,0 α,β 1l V S −k,0 x, y ξ − 1l V ξ = − 1l V ξ X x, y ∈S px, y X α,β∈X χ x, y α,β ξ X k ≥0 e Γ k α,β 1l Ω\V S −k,k x, y ξ + e Π 0,k α,β 1l Ω\V S 0,k x, y ξ + e Π −k,0 α,β 1l Ω\V S −k,0 x, y ξ + 1l Ω\V ξ X x, y ∈S px, y X α,β∈X χ x, y α,β ξ X k ≥0 e Γ k α,β 1l V S −k,k x, y ξ + e Π 0,k α,β 1l V S 0,k x, y ξ + e Π −k,0 α,β 1l V S −k,0 x, y ξ . 3.3 We write L 1l V η by using the corresponding rates of S . We fix y ∈ S, α z , γ z , β, δ ∈ X 4 with α z , β ≤ γ z , δ for all z ∈ S, z 6= y, and two configurations ξ, η ∈ Ω × Ω such that ξz = α z , ηz = γ z , for all z ∈ S, z 6= y, ξ y = β, η y = δ. Thus ξ ≤ η. We define the set C + y of sites which interact with y with an increase of the configuration on y, C + y = n z ∈ S : pz, y 0 and X k e Γ k α z , β + Γ k γ z , δ + e Π 0,k α z , β + Π 0,k γ z , δ o . 3.4 Denote by x = x 1 , . . . , x d the coordinates of each x ∈ S. We define, for each n ∈ N, C + y n = C + y ∩ {z ∈ S : P d i=1 |z i − y i | ≤ n}. We may suppose C + y 6= ;, since otherwise 2.13–2.14 would be 119 trivially satisfied. Given K ∈ N, we fix {p i z } i ≤K,z∈C + y such that for each i, z, p i z ∈ X and p i z ≤ ξz. Moreover we fix {p i y } i ≤K such that p i y δ, for each i. For n ∈ N, let I y n = K [ i=1 ζ ∈ Ω : ζ y ≥ p i y and ζz ≥ p i z , for all z ∈ C + y n . The union of increasing cylinder sets I y n is an increasing set, to which neither ξ nor η belong. We compute, using 3.3, f L 1l I y n ξ = X z ∈C + y n pz, y X k 1l S K i=1 {α z −k≥p i z , β+k≥p i y } e Γ k α z , β + 1l S K i=1 {β+k≥p i y } e Π 0,k α z , β + X z ∈C + y n pz, y X k 1l S K i=1 {β+k≥p i y } e Γ k α z , β + e Π 0,k α,β L 1l I y n η = X z ∈C + y n pz, y X l 1l S K i=1 {γ z −l≥p i z , δ+l≥p i y } Γ l γ z , δ + 1l S K i=1 {δ+l≥p i y } Π 0,l γ z , δ + X z ∈C + y n pz, y X l 1l S K i=1 {δ+l≥p i y } Γ l γ z , δ + Π 0,l γ,δ . So, by setting J p z , a, b := J {p i z } i ≤K , a, b = K [ i=1 {l : a − l ≥ p i z , b + l ≥ p i y }, and by 3.2, if p 1 y := min i {p i y }, we get X z ∈C + y n pz, y X k ∈Jp z , α z , β e Γ k α z , β + X k ≥p 1 y −β e Π 0,k α z , β + X z ∈C + y n pz, y X k ≥p 1 y −β e Π 0,k α z , β + e Γ k α z , β ≤ X z ∈C + y n pz, y X l ∈Jp z , γ z , δ Γ l γ z , δ + X l ≥p 1 y −δ Π 0,l γ z , δ + X z ∈C + y n pz, y X l ≥p 1 y −δ Π 0,l γ z , δ + Γ l γ z , δ Taking the monotone limit n → ∞ gives X z ∈S X k ∈Jp z , α z , β e Γ k α z , β + X k ≥p 1 y −β e Π 0,k α z , β pz, y ≤ X z ∈S X l ∈Jp z , γ z , β Γ l γ z , δ + X l ≥p 1 y −δ Π 0,l γ z , δ pz, y By choosing the values α z ≡ α, γ z ≡ γ, p i z ≡ p i α , since P z pz, y = 1, X k ∈Jp α , α,β e Γ k α,β + X k ≥p 1 y −β e Π 0,k α,β ≤ X l ∈Jp α , γ,δ Γ l γ,δ + X l ≥p 1 y −δ Π 0,l γ,δ . 3.5 A similar argument with C − x ={z ∈ S : pz, y 0 and X k e Γ k α,γz + Γ k γ,δ z + e Π −k,0 α,β z + Π −k,0 γ,δ z 0} 120 subsets C − x n, n ∈ N, {p i z } z ∈C − x n,i≤K ∈ X , {p i x } i ≤K ∈ X with p i x α, for each i, p i z ≥ δ z , for all z ∈ C − y n, i ≤ K, decreasing cylinder sets D x n = K [ i=1 ζ ∈ Ω : ζx ≤ p i x and ζz ≤ p i z , for all z ∈ C − y n to the complement of which ξ and η belong, and the application of inequality 3.2 to ξ, η, Ω\D x n which is an increasing set since it is the complement of an increasing one leads to X k ∈J − p γ , α,β e Γ k α,β + X α−k≤p 1 x e Π −k,0 α,β ≥ X l ∈J − p γ , γ,δ Γ l γ,δ + X γ−l≤p 1 x Π −l,0 γ,δ 3.6 where p 1 x = max i {p i x }, J − p γ , a, b := J − {p i γ } i ≤K , a, b = K [ i=1 {l : a − l ≤ p i x , b + l ≤ p i γ }. Finally, taking p i y = δ + j i + 1, p i α = α − m i in 3.5, p i x = α − h i − 1, p i γ = δ + m i in 3.6 gives 2.13–2.14.

3.2 Coupling construction