Geometry of Lipschitz percolation

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www.imstat.org/aihp 2012, Vol. 48, No. 2, 309–326

DOI:10.1214/10-AIHP403

Geometry of Lipschitz percolation

G. R. Grimmett

^a

and A. E. Holroyd

^b

aStatistical Laboratory, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 0WB, UK.

E-mail:[email protected]; url:http://www.statslab.cam.ac.uk/~grg/

bMicrosoft Research, 1 Microsoft Way, Redmond, WA 98052, USA and Department of Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada. E-mail:[email protected]; url:http://research.microsoft.com/~holroyd/

Received 22 July 2010; revised 3 November 2010; accepted 10 November 2010

Abstract. We prove several facts concerning Lipschitz percolation, including the following. The critical probabilityp_Lfor the existence of an open Lipschitz surface in site percolation onZ^dwithd≥2 satisfies the improved boundp_L≤1−1/[8(d−1)]. Wheneverp > p_L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. Forpsufficiently close to 1, the connected regions ofZ^d⁻¹above which the surface has height 2 or more exhibit stretched-exponential tail behaviour.

The last statement is proved via a stochastic inequality stating that the lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets ofZ^d.

Résumé. Nous démontrons plusieurs résultats concernant la percolation Lipschitzienne. La probabilité critiquep_Lpour l’existence d’une surface Lipschitzienne ouverte dans la percolation par site surZ^d (lorsqued≥2) satisfait l’estimation amélioréep_L≤ 1−1/[8(d−1)]. Pour toutp > p_L, la hauteur de la plus basse surface Lipschitzienne au-dessus de l’origine a une queue qui décroît exponentiellement vite. Lorsquepest suffisamment proche de 1, la taille des régions connexes deZ^d⁻¹au-dessus desquelles cette surface a une hauteur supérieure ou égale à 2 possède un comportement exponentiel étiré. Ce dernier résultat provient d’une inégalité stochastique qui montre que la plus basse surface est dominée stochastiquement par la frontière de l’union de certains ensembles aléatoires deZ^dindépendants et identiquement distribués.

MSC:60K35; 82B20

Keywords:Percolation; Lipschitz embedding; Random surface; Branching process; Total progeny

1. Lipschitz percolation

We consider site percolation with parameterpon the latticeZ^d withd≥2, with law denotedPp. The existence of open Lipschitz surfaces was investigated in [6], the main theorem of which may be summarized as follows. Let · denote the¹-norm, and writeZ⁺= {1,2, . . .}andZ⁺₀ = {0} ∪Z⁺. A functionF :Z^d⁻¹→Z⁺is calledLipschitzif:

for anyx, y∈Z^d⁻¹withx−y =1,we have|F (x)−F (y)| ≤1. (1) LetLIPbe the event that there exists a Lipschitz functionF:Z^d⁻¹→Z⁺such that,

for eachx∈Z^d⁻¹, the site(x, F (x))∈Z^dis open. (2)

The eventLIPis clearly increasing. Since it is invariant under translation ofZ^d by the vector(1,0, . . . ,0), its probability equals either 0 or 1. Therefore, there existspL∈ [0,1]such that:

Pp(LIP)=

0 ifp < pL, 1 ifp > pL.

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It was proved in [6] that 0< p_L<1, and more concretely that 0< p_L≤1−(2d)⁻². As noted in [6], whenp > p_L, there exists a Lipschitz functionF satisfying (1) with the property that the random field(F (x): x∈Z^d⁻¹)is stationary and ergodic. Applications of these and related statements may be found in [7,14,15].

2. Main results

Our first result is an improvement of the upper bound forp_Lof [6].

Theorem 1. Ford≥2we havep_L≤1− [8(d−1)]⁻¹. This is proved in Section4. The complementary inequality

pL≥1−1+o(1)

2d asd→ ∞

is proved in Section5, yielding that 1/dis the correct order of magnitude for 1−p_L(d)in the limit asd→ ∞.

A Lipschitz functionF satisfying (2) is calledopen. For any familyF of Lipschitz functions, the minimum (or

‘lowest’) function F (x):=min

F (x): F ∈F

is Lipschitz also. If there exists an open Lipschitz function, there exists necessarily a lowest such function, and we refer to it as the ‘lowest open Lipschitz function.’ We shall sometimes use the term ‘Lipschitz surface’ to describe the subset{(x, F (x)): x∈Z^d⁻¹}ofZ^d, for some LipschitzF. See Fig.1. We emphasize that Lipschitz functions are always required to take values in thepositiveintegers.

For reasons of exposition, if there exists no open Lipschitz function, we define the lowest open Lipschitz function byF (x)= ∞for allx∈Z^d⁻¹. Our second main result is the following.

Fig. 1. Part of the lowest open Lipschitz surface (on a finite torus) whend=3 andp=0.82. Each cube represents an open site in the surface.

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Theorem 2. Letd≥2and letFbe the lowest open Lipschitz function.There existsα=α(d, p)satisfyingα(d, p) >0 forp > p_Lsuch that

Pp

F (0) > n

≤e⁻^αn, n≥0. (3)

This is proved in Section6by an adaptation of Menshikov’s proof of exponential decay for subcritical percolation.

Since the law ofF (x)is the same for allx∈Z^d⁻¹, the choice of the origin 0 in (3) is arbitrary. Theorem2extends the exponential-decay theorem of [6] by removing the condition onpthat is present in that work.

Our third result is a bound of a different type on the lowest open Lipschitz functionF. Recall that, by definition, F ≥1. LetSbe the set of allx∈Z^d⁻¹for whichF (x) >1. LetS₀be the vertex-set of the component containing 0 in the subgraph of the nearest-neighbour lattice ofZ^d⁻¹induced byS(and takeS₀:=∅if 0∈/S).

Theorem 3. Letd≥2.There existsp_M<1such that,forp > p_Mandε >0, exp

−λn^1/(d⁻¹⁾

≤Pp

|S₀|> n

≤exp

−γ n^1/(d⁻¹⁾⁻^ε

, n≥1, (4)

whereλ=λ(d, p)andγ=γ (d, p, ε)are positive and finite.Ifd =3then(4)holds even withε=0.

The above statement is similar in spirit to Dobrushin’s theorem [8] concerning the existence of ‘flat’ interfaces in the three-dimensional Ising model with mixed boundary conditions (see also [9] and [12], Chapter 7). The proof utilizes a bound for the tail of the total progeny in a subcritical branching processes with ‘stretched-exponential’ family-sizes.

(The term ‘subexponential’ is sometimes used in the literature.) The ε of (4) arises from a certain instance in the large-deviation theory of heavy-tailed distributions; see the discussion of Section8. A mild extension of Theorem3is proved in Section8.

The principal ingredient in the proof of Theorem3is the following stochastic inequality, which has other potential applications also. In preparation for this, we introduce the concept of a local cover. Fory∈Z^d⁻¹, letFybe the set of functionsf:Z^d⁻¹→Z⁺₀ that are Lipschitz in the sense of (1), and that satisfy:

(a) f (y) >0, and

(b) for allx∈Z^d⁻¹, eitherf (x)=0 or the site(x, f (x))is open.

Thelocal coveratyis the Lipschitz functionf_ygiven by f_y(z):=min

f (z): f∈Fy

, z∈Z^d⁻¹.

It is shown in [6] that the lowest open Lipschitz functionF is given by F (x)=sup

f_y(x): y∈Z^d⁻¹

. (5)

Now let(g_y: y∈Z^d⁻¹)beindependentrandom functions fromZ^d⁻¹toZ⁺₀ such that, for eachy∈Z^d⁻¹,g_y has the same law asf_y. Let

G(x):=sup

g_y(x): y∈Z^d⁻¹

. (6)

Theorem 4. The lowest open Lipschitz functionF is dominated stochastically byGin that,for any increasing subset A⊆ [0,∞]^Z^d⁻¹,

Pp

F (x): x∈Z^d⁻¹

∈A

≤P

G(x): x∈Z^d⁻¹

∈A .

Section3contains the basic estimate that leads in Section4to the proof of Theorem1. Lower bounds for the critical valuep_Lare found in Section5. The exponential-decay Theorem2is proved in Section6, followed in Section7by the proof of Theorem4. The final Section8contains the proof of Theorem3.

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3. A basic estimate

This section contains a basic estimate (Proposition5) similar to the principal Lemma 3 of [6], together with a lemma (Lemma6) that will be useful in the proof of Theorem1. We begin by introducing some terminology.

Letd≥2 andp=1−q∈ [0,1]. The site percolation model onZ^dis defined as usual by letting each sitex∈Z^dbe openwith probabilityp, or elseclosed, with the states of different sites independent. The sample space isΩ= {0,1}^Z^d where 1 represents ‘open,’ and 0 represents ‘closed.’ The appropriate product probability measure is writtenPp, and expectation asEp. See [11] for a general account of percolation.

As explained in [6], the lowest open Lipschitz functionF may be constructed as a blocking surface to certain paths.

Lete₁, e₂, . . . , e_d∈Z^dbe the standard basis vectors ofZ^d. We define aΛ-pathfromutovto be any finite sequence of distinct sitesu=x₀, x₁, . . . , x_k=vofZ^dsuch that, for eachi=1,2, . . . , k,

xi−xi−1∈ {±ed} ∪ {−ed±ej: j=1, . . . , d−1}. (7) The directed stepw:=x_i−x_i₋₁is called:upward(U) ifw=e_d;downward vertical(DV) ifw= −e_d;downward diagonal(DD) otherwise. The notation ‘Λ-path’ was introduced in [6] to convey an impression of the DD steps available to such a path.

A Λ-path is calledadmissible if every upward step terminates at a closed site, which is to say that, for each i=1,2, . . . , k,

ifxi−xi−1=edthenxiis closed. (8)

Foru=(u₁, u₂, . . . , u_d)∈Z^d, we writeh(u)=u_dfor itsheight. Let L:=Z^d⁻¹× {0} ⊂Z^d

be the hyperplane of height zero.

AΛ-path is called a λ-pathif it has no downward vertical steps. Denote byuv (respectively, uλv) the event that there exists an admissibleΛ-path (respectively,λ-path) fromutov. We writeu⁺ vandu⁺λvfor the corresponding events given in terms of paths using no vertexw∈Z^dwithh(w) <0. More generally, forA, B⊆Z^d we writeAB (and similarly for the other relations) if ab for some a∈Aandb∈B. Similarly, we write

‘ABinC’ if such a path exists using only sites inC⊆Z^d.

Proposition 5. Letd≥2and letq=1−psatisfyρ:=8(d−1)q <1.Then

u∈L:u≥r

Pp(0⁺λu)≤

∞ n=r

2n n

2⁻²ⁿρⁿ, r≥1. (9)

Proof. Leta=2d−2. Any path contributing to the event 0⁺λuwithu∈Luses some numberUof upward steps and some numberDof downward diagonal steps. Furthermore,U=D≥ u. Therefore,

u∈L:u≥r

Pp(0⁺λu)≤

U=D≥r

U+D U

q^Ua^D

=

∞ n=r

2n n

(qa)ⁿ

as required.

Since_2n

n

≤2²ⁿ, it follows from Proposition5whenρ <1 that

u∈L:u≥r

Pp(0⁺λu)≤ ρ^r

1−ρ, r≥0. (10)

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A marginally improved upper bound may be derived by using either Stirling’s formula or the local central limit theorem.

Here is a lemma concerning the relationship betweenΛ-paths andλ-paths. ForS⊆Z^d⁻¹×Z⁺₀, write

↓S=

x∈Z^d⁻¹×Z⁺₀: x=s−kedfor somes∈Sandk≥0 . Lemma 6. Forω∈Ω,we have that

x∈Z^d⁻¹×Z⁺₀: 0⁺ x

= ↓

x∈Z^d⁻¹×Z⁺₀: 0⁺λx .

Proof. Since everyλ-path is aΛ-path, andΛ-paths may end with any number of downward vertical steps without restriction, the right side is a subset of the left side. It remains to show that the left side is a subset of the right side.

Letx∈Z^d⁻¹×Z⁺₀ be such that 0⁺ x, and letπ be an admissibleΛ-path from 0 tox of shortest length. Ifπ contains no DV step, then it is aλ-path, and we are done. Suppose there is a DV step inπ, and consider the last one, denotedV, in the natural order of the path. Then V is an ordered pair(x, y)of sites ofZ^d withy=x−e_d. Since the sites of the path are distinct,V is not followed by a U step. Therefore, V is either at the end of the pathπ, or it is followed by a DD step, which we write as the ordered pair(y, z)withz=y−e_d+αe_j for someα∈ {−1,1}

andj∈ {1,2, . . . , d−1}. In the latter case, we may interchangeV with this DD step, which is to say that the ordered triple(x, y, z)inπis replaced by(x, y, z)withy=y+αej. This change does not alter the admissibility of the path or its endpoints.

A small complication would arise if y∈π. If this were to hold, the site sequence thus obtained would contain a loop, and we may erase this loop to obtain an admissible path from 0 tox with fewer steps thanπ. This would contradict the minimality of the length ofπ, whencey∈/π.

Proceeding iteratively, the position in the path of the last DV step may be delayed until: either it is the last step of the path, or it precedes a U step, denotedU. In the latter case, we may removeUtogether with the previous DV step to obtain a new admissibleΛ-path from 0 tox of shorter length thanπ, a contradiction. Proceeding thus with every DV step ofπ, we arrive at an admissibleΛ-path from 0 tox comprising an admissibleλ-path followed by a number

of DV steps. The claim follows.

4. Hills, mountains, and the proof of Theorem1

We describe next the use of Proposition5in the proof of Theorem1. Fory∈L, thehillH_yis given by H_y:=

z∈Z^d: y⁺ z

. (11)

Hills combine as follows to form ‘mountains.’ Forx∈L, themountainM_xofxis given by M_x=

{H_y: y∈Lsuch thatx∈H_y}. (12)

Letx∈LandS⊆Z^d. TheheightofSatxis defined as l_x(S)=sup{k: x+ke_d∈S},

where the supremum of the empty set is interpreted as 0. Thelocal heightof the mountainM_xis defined as its height l_x(M_x)above x. By the definition ofΛ-paths, we have that: eitherl_x(M_x) <∞for allx∈L, orl_x(M_x)= ∞for allx.

Define the eventI =

x∈L{l_x(M_x) <∞}. The eventI is invariant under the action of translation ofZ^d by any vector inL, whence it has probability either 0 or 1. By the above,Pp(I )=1 if and only ifPp(l₀(M₀) <∞)=1.

Let the functionF:Z^d⁻¹→Z⁺∪ {∞}be defined by

F (x)=1+lx(Mx), x∈L, (13)

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as in (5). By the above discussion,F is finite-valued if and only if I occurs. It is explained in [6] that F is the lowest open Lipschitz function. In conclusion, the lowest open Lipschitz surface is finite if and only ifPp(I )=1. In particular,

p_L=inf

p: Pp(I )=1

. (14)

The random field(F (x): x∈L)is stationary and ergodic under the action of translation of Lby any ej withj ∈ {1,2, . . . , d−1}.

We show in the remainder of this section thatPp(I )=1 under the condition of the following lemma. ForS⊆Z^d, theradiusofSwith respect to 0∈Z^dis given by

rad(S)=sup

s: s∈S .

Lemma 7. Letd≥2,and letρ:=8(d−1)q <1.Then Pp

rad(H0)≥r

≤ ρ^r

1−ρ, r≥1, (15)

and there exists an absolute constantc=c(d)such that Pp

rad(M0)≥r

≤cr^d⁻¹ ρ^r/2

1−ρ, r≥1. (16)

Proof of Theorem1. Sincel₀(M₀)≤rad(M0), we have by (16) thatl₀(M₀) <∞a.s. whenρ <1. By (14),p_L≤

1−1/[8(d−1)].

The following corollary of Lemma 7 will be used in Section 8. The footprint L(S) of a subset S⊆Z^d is its projection ontoL:

L(S):=

(s₁, s₂, . . . , s_d₋₁,0): (s₁, s₂, . . . , s_d)∈S

. (17)

Corollary 8. Letd≥2,p=1−q∈(0,1)andρ:=8(d−1)q.

(a) There existsα=α(d, p) <∞such that PpL(H₀)≥n

≥exp

−αn^1/(d⁻¹⁾

, n≥2.

(b) There existsβ=β(d, p)satisfyingβ(d, p) >0whenρ <1,such that PpL(M₀)≥n

≤exp

−βn^1/(d⁻¹⁾

, n≥2.

Proof of Lemma7. Letρ <1 andr≥1. Suppose 0⁺λuwhereu∈Z^d⁻¹×Z⁺₀ andu ≥r. By considering all sitesvsuch that there is aλ-path fromutovusing downward diagonal steps only, there must existv∈Lwith 0⁺λv andv ≥r. Therefore, by (10),

Pp

rad H₀^λ

≥r

≤

u∈L:u≥r

Pp(0⁺λu)≤ ρ^r 1−ρ, where

H_y^λ:=

z∈Z^d: y⁺λz

, y∈L.

By Lemma6,H0= ↓H₀^λ, and in particular rad(H0)=rad(H₀^λ). Inequality (15) follows.

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By the definition ofM₀and the triangle inequality, Pp

rad(M0)≥r

≤

y∈L

Pp

0∈H_y,rad(Hy)≥r− y .

The last sum equals

y∈L

Pp

y∈H₀,rad(H0)≥r− y ,

which we split into two sums depending on whether or not y ≤r/2. The first such sum is no larger than cr^d⁻¹Pp(rad(H0)≥r/2)for some constantc. By Lemma6and (10), the second satisfies

y∈L:y>r/2

Pp(y∈H₀)=

y∈L:y>r/2

Pp(0⁺λy)

≤ ρ^r/2 1−ρ

as required.

Proof of Corollary8. (a) There existsc >0 such that, if every site in{ke_d: 1≤k≤m}is closed, then|L(H₀)| ≥ cm^d⁻¹. Therefore,

PpL(H₀)≥cm^d⁻¹

≥q^m, (18)

and the first claim follows.

(b) There existsc >0 such that|L(M₀)| ≤crad(M0)^d⁻¹, and the second claim follows by (16).

5. Inequalities for Lipschitz critical points

By Theorem1, we have 1−p_L(d)≥1/[8(d−1)]. In this section, we derive further results concerning the values p_L(d). In particular we prove a lower bound forp_L that implies that the correct order of magnitude of 1−p_L(d)is indeed 1/din the limit of larged.

Consider the hypercubic latticeL^d with vertex-setZ^d. Letπ be a (finite or infinite) directed self-avoiding path ofL^dwith verticesx₁, x₂, . . . .We callπ acceptableif it contains no upward steps, i.e., ifx_i₊₁−x_i =e_dfor alli.

Consider site percolation onZ^d. An acceptable path is calledopen(respectively,closed) if all of its sites are open (respectively, closed). Letp_c^↓(d)be the critical probability for the existence of an infinite open acceptable path from the origin.

Proposition 9. The sequence(p_L(d): d≥2)is non-decreasing and satisfiesp_L(d)≥1−p^↓_c(d).

Letpc(d)be the critical probability of site percolation onL^d, and letpc(d)be the critical probability of the oriented site percolation process onL^din which every edge is oriented in the direction of increasing coordinate direction. It is elementary by graph inclusion that

p_c(d)≤p^↓_c(d)≤min

p_c(d−1),p_c(d) .

Several proofs are known that 2dpc(d)→1 asd→ ∞(see [11], p. 30; indeed the lace expansion permits an expansion ofp_c(d)in inverse powers of 2d). Hence, by Proposition9,

1−p_L(d)≤1+o(1)

2d asd→ ∞.

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Fig. 2. Part of the latticeL²_altof [13], obtained by adding oriented edges toZ².

The value ofpL(2)may be expressed as the critical value of a certain percolation model. Consider the oriented graphL²_altobtained fromZ²by placing an oriented bond fromutovif and only ifv−u∈ {e1, e1±e2}. This graph was used in [13], and is illustrated in Fig.2. Letp_alt be the critical probability of oriented site percolation onL²_alt. It is shown in [19], Theorem 5.1, that p_alt ≥ ¹₂. It is elementary thatp_alt≤ p_c(2), and it was proved in [1] that

pc(2)≤0.7491 (see also [17]). In summary, 1

2 ≤p_alt≤ p_c(2)≤0.7491.

Simulations in [19] indicate thatp_alt≈0.535.

Proposition 10. We have thatp_L(2)=p_alt.

The equationpL(2)=palt≥¹₂has been noted in independent work of Berenguer (personal communication).

Proof of Proposition9. LetF:Z^d⁻¹→Z⁺be Lipschitz. The restrictionG:Z^d⁻²→Z⁺given by G(x1, x2, . . . , xd−2):=F (x1, x2, . . . , xd−2,0)

is Lipschitz also, and the monotonicity ofpL(d)follows.

Letq=1−psatisfyq > p^↓_c(d). Suppose there exists an acceptable closed path fromnedto some site inL. Fix such a path, and letxbe its earliest site lying inL. Then we see thatx⁺ ned, and hence 0∈Hxandl0(M0)≥n. By (13),F (0) > n.

On the other hand, by a standard argument of percolation theory, on the event that there exists an infinite acceptable closed path starting fromned, there exists a.s. an acceptable closed path fromnedto some site inL. Hence

Pp

F (0) > n

≥θ^↓(q) >0,

whereθ^↓(q)is the probability that the origin lies in an infinite acceptable closed path. Thus p≤pL, and hence

1−p^↓c ≤pL.

Proof of Proposition10. This is similar to part of the proof of [15], Theorem 6; see also [15], Lemma 7. Letd=2.

Ifp > pL, there exists a.s. a sitezon the 2-coordinate axis ofZ²such thatzis the starting point of some infinite open oriented path ofL²_alt. Hencep≥palt, so thatpL≥palt.

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Conversely, letp > p_alt. By the block construction of [13] or otherwise (see also [15], Lemma 7), there is a strictly positive probability θ⁺(p)that the site (0,1)of Z² is the starting point of an infinite open oriented path of L²_alt using no site with 2-coordinate lying in(−∞,0]. By considering a reflection ofL²_altin the 2-coordinate axis, there is probability at leastp⁻¹θ⁺(p)²that there exists an open Lipschitz functionF:Z²→Z⁺. Therefore,p≥p_L, so that

p_alt≥p_L.

6. Exponential decay

In this section, we prove exponential decay of the tail ofF (0)whenp > p_L, as in Theorem2. Letd ≥2 andL_n= Z^d⁻¹× {n}, so thatL=L0. Forx∈L, let

Kx:=sup{n:xLn}.

Recall the lowest open Lipschitz functionF of (13).

Lemma 11. Letp∈(0,1)andx∈L.The random variablesKxandF (x)−1=lx(Mx)have the same distribution.

Proof. This holds by a process of path-reversal. It is convenient in this proof to work with bond percolation rather than site percolation. Each bond of the hypercubic lattice Z^d is designated ‘open’ with probability p and ‘closed’

otherwise, different bonds receiving independent states. We call aΛ-pathadmissibleif any directed step along a bond from some y toy+e_d is closed. It is clear that the set of admissible paths has the same law as in the formulation using site percolation.

AΛ⁻-path fromutovis defined as any finite sequence of distinct sitesu=x₀, x₁, . . . , x_k=vofZ^dsuch that, for eachi=1,2, . . . , k,

x_i−x_i₋₁∈ {±e_d} ∪ {e_d±e_j: j =1, . . . , d−1}. (19) Any step in the direction −e_d is calleddownward. A Λ⁻-path is called (−)-admissible if every downward step traverses a closed bond.

Letπ be aΛ-path ofZ^d, and letρπ be theΛ⁻-path obtained by reversing each step. Note thatπ is admissible if and only ifρπis(−)-admissible.

It suffices to assumex=0. LetΠ_nbe the union overy∈L₀of the set ofΛ-paths fromytone_d. ThenρΠ_nis the set ofΛ⁻-paths fromne_dtoL, so that

Pp(Lne_d)=Pp(ne_d⁻ L),

where⁻ denotes connection by an admissibleΛ⁻-path. By a reflection of the lattice, the last probability equals Pp(0L_n), so that

Pp

l₀(M₀)≥n

=Pp(0L_n),

and the claim follows by (13).

Theorem 12. There existsα=α(d, p)satisfyingα(d, p) >0forp > p_Lsuch that Pp(0L_n)≤e⁻^αn, n≥0.

Proof of Theorem2. This is immediate by Theorem12and Lemma11.

Proof of Theorem12. The proof is an adaptation of Menshikov’s proof of a corresponding fact for percolation, as presented in [11], Section 5.2. We shall use the BK inequality of [3] (see also [11], Section 2.3). The application of the BK inequality (but not the inequality itself) differs slightly in the current context from that of regular percolation,

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and we illustrate this as follows. Leta, b, u, v∈Z^d. The ‘disjoint occurrence’ of the events{ab}and{uv} is written as usual{ab} ◦ {uv}. In this setting, it comprises the set of configurations such that: there exist admissibleΛ-pathsπ_a,bfroma tob, andπ_u,vfromutov, such that each directed edge from somex tox+e_dlies in no more than one ofπa,b,πu,v. That is, the paths must have no upward step in common; they are permitted to have downward steps in common.

Letm≥1 andR_m= [−m, m]^d⁻¹×Z. Forx∈R_m∩Landr≥0, let g_p^x,m(r)=Pp(xL_r inR_m), g_p(r)=Pp(0L_r)

and

h^m_p(r)=max

g_p^x,m(r): x∈Rm∩L

. (20)

On recalling the definition of an admissibleΛ-path, we see that the event{x Lr inRm}is a finite-dimensional cylinder event, and so eachg_p^x,m(r)is a polynomial inp. Note that

h^m_p(r)≥g_p^0,m(r)→gp(r) asm→ ∞. Sinceg^x,m_p (r)≤g_p^x,^∞(r)=gp(r), we have that

h^m_p(r)→gp(r) asm→ ∞. (21)

By (20),h^m_p(r)is the maximum of a finite set of polynomials inp. Therefore,h^m_p(r)is a continuous function ofp, and there exists a finite setD^m(r)⊆(0,1)such that: forp /∈D^m(r), there existsx=x_p^m(r)∈Rm∩Lwith

d

dph^m_p(r)= d

dpg^x,m_p (r). (22)

Let x∈R_m∩L and A^x,m(n)= {xL_ninR_m}, so that g^x,m_p (n)=Pp(A^x,m(n)). By Russo’s formula, [11], Eq. (5.10),

d dplogPp

A^x,m(n)

= − 1 1−pEp

N

A^x,m(n)

|A^x,m(n)

, (23)

whereN (A)is the number of pivotal sites for a decreasing eventA. We claim, as in [11], Eq. (5.18), that Ep

N

A^x,m(n)

|A^x,m(n)

≥ n

n

r=0h^m_p(r)−1. (24)

Once this is proved, it follows by (23) that d

dplogg_p^x,m(n)≤ − 1 1−p

n _n

r=0h^m_p(r)−1

.

Therefore, by (22), forp /∈D^m(n), d

dplogh^m_p(n)≤ − 1 1−p

n _n

r=0h^m_p(r)−1

. (25)

We integrate (25) to obtain, for 0< α < β <1, h^m_β(n)≤h^m_α(n)exp

−(β−α)

n n

r=0h^m_α(r)−1

,

(11)

as in [11], Eq. (5.22). Letm→ ∞, and deduce by (21) that g_β(n)≤g_α(n)exp

−(β−α) n

n

r=0gα(r)−1

. (26)

This last inequality may be analyzed just as in [11], Section 5.2, to obtain the claim of the theorem.

It remains to prove (24), and the proof is essentially that of [11], Lemma 5.17. Fixx∈R_m∩L, and letV be a random variable taking values in the non-negative integers with

P(V ≥k)=h^m_p(k), k≥0.

SupposeA^x,m(n)occurs, and letz1, z2, . . . , zN be the pivotal sites forA^x,m(n), listed in the order in which they are encountered beginning atx. Letρ₁:=max{0, h(z1)−1}and

ρi:=max

0, h(zi)−h(zi−1)−1

, i=2,3, . . . , N. (27)

Thus,ρ_i measures the positive part of the ‘vertical’ displacement between the(i−1)th andith pivotal sites.

We claim as in [11], Eq. (5.19), that, fork≥1, Pp

ρ₁+ · · · +ρ_k≤n−k|A^x,m(n)

≥P(V₁+ · · · +V_k≤n−k), (28) where theV_i are independent, identically distributed copies ofV. Equation (24) follows from this as in [11], Sec- tion 5.2, and, as in that reference, it suffices for (28) to show the equivalent in the current setting of [11], Lemma 5.12,

namely the next lemma.

Lemma 13. Letkbe a positive integer,and letr₁, r₂, . . . , r_k be non-negative integers with sum not exceedingn−k.

With the above notation,forx∈Rm∩L, Pp

ρ_k≤r_k, ρ_i=r_ifor 1≤i < k|A^x,m(n)

≥P(V ≤r_k)Pp

ρ_i=r_i for 1≤i < k|A^x,m(n)

. (29)

Proof. Suppose first thatk=1 and 0≤r₁≤n−1. By the BK inequality, Pp

{ρ₁> r₁} ∩A^x,m(n)

≤Pp

A^x,m(r₁+1)◦A^x,m(n)

≤Pp

A^x,m(r₁+1) Pp

A^x,m(n)

≤P(V > r1)Pp

A^x,m(n) , so that (29) holds withk=1.

Turning to the general case, letk≥1 and let ther_i satisfy the given conditions. For a sitez, letD_z be the set of sites attainable fromx along admissibleΛ-paths ofRmnot containing the upward step fromz−ed toz. LetBz be the event that the following statements hold:

(a) z−e_d∈D_z, andz /∈D_z, (b) zis closed,

(c) D_zcontains no vertex ofL_n,

(d) the pivotal sites for the event{0z}are, taken in order,z1, z2, . . . , zk−1=z, with theρi of (27) satisfying ρ_i=r_i for 1≤i < k.

LetB=

zB_z, and note that

B∩A^x,m(n)= {ρi=ri for 1≤i < k} ∩A^x,m(n). (30)

Forω∈A^x,m(n)∩B, there exists a unique siteζ=ζ (ω)such thatB_ζ occurs.

LetΔdenote the ordered pair(D_ζ, ζ ). Now, Pp

A^x,m(n)∩B

=

δ

Pp

B∩ {Δ=δ} Pp

A^x,m(n)|B∩ {Δ=δ}

, (31)

(12)

where the sum is over all possible valuesδ=(δ_z, z)of the random pairΔ. Note that Pp

A^x,m(n)|B∩ {Δ=δ}

=Pp(zL_ninR_m\δ_z).

By a similar argument and the BK inequality, Pp

{ρ_k> r_k} ∩A^x,m(n)∩B

=

δ

Pp

B∩ {Δ=δ} Pp

{ρ_k> r_k} ∩A^x,m(n)|B∩ {Δ=δ}

(32) and

Pp

{ρ_k> r_k} ∩A^x,m(n)|B∩ {Δ=δ}

≤P(V > r_k)Pp

A^x,m(n)|B∩ {Δ=δ} . On dividing (32) by (31), we deduce that

Pp

ρ_k> r_k|A^x,m(n)∩B

≤P(V > r_k).

The lemma is proved on multiplying through byPp(B|A^x,m(n)), and recalling (30).

7. Decomposition of hill-ranges

This section is devoted to the domination argument used in the proof of Theorem4. Let ω∈Ω be a configuration of the site percolation model onZ^d. Letv₁, v₂, . . .be an arbitrary but fixed ordering of the vertices inL, and write H_i =H_v_i for the hill atv_i, as defined in (11). We shall construct theH_i in an iterative manner, and observe the relationship between a hill thus constructed and the previous hills. To this end, we introduce some further notation.

Fori≥1, let H_i:=

j≤i

H_j.

We call aΛ-pathnon-negativeif it visits no sitewwithh(w) <0. Foru, v∈Z^d\H_i, writeu⁺ivif there exists an admissible non-negativeΛ-path fromutovusing no site ofHi. The ‘restricted’ hill atvi+1is given by

Hi+1:=

z∈Z^d: vi+1

+iz .

That is,H_i₊₁is given as before but in terms of non-negativeΛ-paths in the region obtained fromZ^d by removal of all hills already constructed. Ifv_i₊₁∈H_i, thenH_i₊₁:=∅. Finally, letH₁:=H₁.

Lemma 14. Forω∈Ω, H_i=

j≤i

H_j, i≥1. (33)

Note that the above is apointwisestatement in that it holds for all configurationsω. Its main application is as follows. In writing that two random variablesA,Bmay be coupled with a certain propertyΠ, we mean that there exists a probability space that supports two random variablesA,Bhaving the same laws asA,Band with propertyΠ. Let (Ji: i=1,2, . . .)be independent, identically distributed subsets ofZ^dsuch thatJi has the same law asHi.

Theorem 15. The families(H_i: i≥1)and(J_i: i≥1)may be coupled in such a way that the following property holds:

H_i⊆

j≤i

J_j, i≥1. (34)

(13)

Proof of Lemma14. We prove Eq. (33) by induction oni. It is trivial fori=1.

Suppose (33) holds fori=I≥1, and consider the casei=I +1. SupposeH_I has been found, and consider the vertexv_I₊₁. Ifv_I₊₁∈H_I, thenH_I₊₁⊆H_IandH_I₊₁=∅. In this case, (33) holds withi=I+1.

SupposevI+1∈/HI. SinceHI+1is the hill ofvI+1within a restricted domain, we haveHI+1⊆HI+1andHI+1⊇ HI∪HI+1. It remains to prove thatHI+1⊆HI∪HI+1, which holds ifHI+1\HI⊆HI+1. Lety∈HI+1\HI= H_I₊₁\H_I. Sincey∈H_I₊₁, there exists an admissible non-negativeΛ-pathπ fromv_I₊₁toy. Ifπ∩H_I =∅,πhas a first vertexzlying inHI. Since no admissible non-negativeΛ-path can exitHI, all points onπbeyondzbelong to H_I, and in particulary∈H_I, a contradiction. Therefore, there exists an admissible non-negativeΛ-path fromv_I₊₁ toynot intersectingHI, which is to say thaty∈HI+1. In this case, (33) holds withi=I+1.

In all cases, (33) holds withi=I+1, and the induction step is complete.

Proof of Theorem15. ForS⊆Z^d, write ΔuS= {x+ed: x∈S} \S.

By the definition of the hill H_y of a vertex y∈L, every vertex inΔ_uH_y is open. SinceH_y may be constructed by a path-exploration process, an event of the form{Hy=S}is an element of theσ-fieldFS generated by the random variablesω(s),s∈S∪Δ_uS. Furthermore, fori≥1, the event{H_i=S}lies inFS.

Suppose we are given thatHi=Sfor someiand someS. Any event defined in terms of admissible non-negative Λ-paths ofZ^d\Sis independent of the states ofΔ_uS, since no such admissibleΛ-path contains an upward step with second endvertex inΔuS.

Consider a sequence of independent site percolations on Z^d with parameterp. LetJ_i be the hill atv_i in theith such percolation model. In particular, for everyi,Ji has the same law asHi. We construct as follows a sequence(H_i) with the same joint law as(H_i)and satisfying

j≤i

H_j⊆

j≤i

J_j, i≥1,

and the claim of the theorem will follow by Lemma14. First, we takeH₁=J₁. Then we letH₂ be the subset ofJ₂ containing all endpoints of all admissible non-negativeΛ-paths fromv2in the second percolation model that do not intersectH₁. More generally, having foundH_j forj < i, we letH_ibe the subset ofJi containing all endpoints of all admissible non-negativeΛ-paths fromviin theith percolation model that do not intersectH₁∪ · · · ∪H_i₋₁. Proof of Theorem 4. By Theorem 15, there exists a probability space on which are defined random variables (H_y, J_y:y∈L)such that

(a) the family(H_y: y∈L)has the same joint law as(H_y: y∈L), (b) theJ_y are independent, and eachJ_y has the same law asH_y, (c) for allx∈L,

H_y: y∈Lwithx∈H_y

⊆

J_y: y∈Lwithx∈J_y . Let

F(x)=1+sup l_x

H_y

: y∈L , G(x)=1+sup

l_x J_y

: y∈L .

By (c) above,F(x)≤G(x)for allx∈L, and the claim follows sinceF(respectively,G) has the same law asF

(respectively,G).