Two-dimensional Poisson trees converge to the brownian web

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Two-dimensional Poisson Trees converge to the Brownian web

P.A. Ferrari

, L.R.G. Fontes

, Xian-Yuan Wu

aInstituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo – SP, Brazil bDepartment of Mathematics, Capital Normal University, Beijing 100037, China

Received 20 October 2003; received in revised form 6 April 2004; accepted 8 June 2004 Available online 5 February 2005

Abstract

The Brownian web can be roughly described as a family of coalescing one-dimensional Brownian motions starting at all times inRand at all points ofR. The two-dimensional Poisson tree is a family of continuous time one-dimensional random walks with uniform jumps in a bounded interval. The walks start at the space–time points of a homogeneous Poisson process inR²and are in fact constructed as a function of the point process. This tree was introduced by Ferrari, Landim and Thorisson.

By verifying criteria derived by Fontes, Isopi, Newman and Ravishankar, we show that, when properly rescaled, and under the topology introduced by those authors, Poisson trees converge weakly to the Brownian web.

2005 Elsevier SAS. All rights reserved.

Résumé

La « toile brownienne » peut approximativement être décrite comme une famille coalescente de mouvements browniens unidimensionnels commençant, en tout temps de la droite réelle, à partir de tout point de la droite réelle. On montre qu’elle peut être approchée en un sens faible par une famille d’arbres poissonniens bidimensionnels.

2005 Elsevier SAS. All rights reserved.

MSC: 60K35; 60F17

Keywords: Brownian web; Poisson tree; Coalescing one-dimensional Brownian motions

* Corresponding author.

E-mail addresses: [email protected] (P.A. Ferrari), [email protected] (L.R.G. Fontes), [email protected], [email protected] (X.-Y. Wu).

1 Partially supported by CNPq grants 520811/96-8, 300576/92-7 and 662177/96-7 (PRONEX) and FAPESP grant 99/11962-9.

2 Partially supported by FAPESP grant 01/02577-6, NSFC grant 10301023 and Foundation of Beijing Education Bureau.

0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2004.06.003

1. Introduction and results

LetS be a two-dimensional homogeneous Poisson process of parameterλ.Sis a random subset ofR²,s∈S has coordinatess1,s2.

Forx=(x₁, x₂)∈R²,tx₂andr >0, letM(x, t, r)be the following rectangle M(x, t, r):=

(x₁, x₂): |x₁−x₁|r, x₂x₂ t

. (1.1)

Astgrows, the rectangle gets longer. The first timetthatM(x, t, r)hits some (or another, whenx∈S) point ofS is calledτ (x, S, r); this is defined by

τ (x, S, r):=inf

t > x₂: M(x, t, r)∩ S\ {x}

= ∅

. (1.2)

The hitting point is the pointα(x)∈Sdefined by α(x):=M

x, τ (x, S, r), r

∩ S\ {x}

, (1.3)

which consists of a unique point almost surely. Ifx=somes∈S, we say thatα(x)=α(s)is the mother ofsand thats is a daughter ofα(s). Letα⁰(x)=x and iteratively, forn1,αⁿ(x)=α(αⁿ⁻¹(x)). For the case ofx= somes∈S,αⁿ(x)=αⁿ(s)is thenth grand mother ofs.

Now let G=(V , E)be the random directed graph with vertices V =S and edges E= {(s, α(s)): s∈S}. Ferrari, Landim and Thorisson [4] proved thatGis a tree with a unique connected component and called it the two-dimensional Poisson tree. The drainage networks of Gangopadhyay, Roy and Sarkar [9] can be viewed as a discrete space, long range version of the Poisson tree.

The Poisson tree induces sets of continuous paths. For anys=(s₁, s₂)∈S, define the pathX^s inR²as the linearly interpolated line composed by all edges{(αⁿ⁻¹(s), αⁿ(s)): n∈N}ofG. Let

X:= {X^s: s∈S}, (1.4)

which we also call the Poisson web.

ClearlyXdepends onλ >0 andr >0; if necessary we denote it byX(λ, r). Takeλ=λ₀=√

3/6,r=r₀=√ 3, and let

X₁:=X(λ₀, r₀); X_δ:=

(δx₁, δ²x₂)∈R²: (x₁, x₂)∈X₁

, δ∈(0,1]. (1.5)

Namely,X_δis the diffusive rescaling ofX₁.

Our main result is a proof thatX_δconverges in distribution to the Brownian web characterized in [7]. [7] in- troduces a metric space(Π, d)of continuous paths firstly, and then defines the Hausdorff metric space(H, d_H) of compact subsets of(Π, d), whered_Handd are the corresponding metric functions. Denote byF_Hthe corre- sponding Borelσ-algebra generated byd_H. The Brownian web is characterized there as a(H,F_H)-valued random variableW (or its distributionµ_W) whose “finite-dimensional distributions” (in a sense made precise in [7]) are coalescing one-dimensional Brownian motions.

Theorem 1.1. The rescaled Poisson treesXδconverge in distribution to the standard Brownian web asδ→0.

Givent₀∈R,t >0,a < b, and a(H,F_H)-valued random variableV, letη_V(t₀, t;a, b)be the{0,1,2, . . . ,∞}- valued random variable giving the number of distinct points inR× {t₀+t}that are touched by paths inV which also touch some point in[a, b] × {t₀}. By the weak convergence criteria given in [7], for any(H,F_H)-valued random variables{X_n}^∞_n=1with noncrossing paths, to prove thatX_nconverges to the standard Brownian web, one may verify the following: For some countable dense setDinR²,

(I₁) There existθ_n^y∈Xnsuch that for any deterministicy₁, . . . , ym∈D,θ_n^y1, . . . , θ_n^ym converge in distribution as n→ ∞to coalescing Brownian motions (with unit diffusion constant) starting aty₁, . . . , ym;

(B₁) lim sup_n→∞sup_(a,t

0)∈R²P(η_Xn(t₀, t;a, a+ )2)→0 as →0+; (B₂) ⁻¹lim sup_n→∞sup_(a,t

0)∈R²P(η_Xn(t₀, t;a, a+ )3)→0 as →0+.

To prove the main result, we show in Section 2 that the Poisson websX_δ satisfy the three hypotheses above.

The verification of I₁, on Subsection 2.1, relies on a comparison with independent paths and on the almost sure coalescence of the Poisson web paths with each other. See Lemma 2.3.

In Subsection 2.2, an FKG inequality enjoyed by the distribution of a single Poisson web path (Lemma 2.6) and the O(t^−1/2)decay of the coalescence time of two such paths (Lemma 2.7), combined with I₁, yield both B₁and B₂. The argument is similar in spirit to the one for establishing weak convergence of coalescing random walks to the Brownian web in [7]. The details are nonetheless substantially different, more involved here, due to the dependence between the paths of the Poisson tree before coalescence. See the remark at the end of the paper.

See also [3] for more details.

Working out a second example of a process in the basin of attraction of the Brownian web (the first example is ordinary one-dimensional coalescing random walks) that is natural on one side, and that requires substantial technical attention on another side, is the primary point of this paper. Its main result may have an applied interest, e.g. in the context of drainage networks. The convergence results here may lead to rigorous/alternative verification of some of the scaling theory for those networks. See [13]. Ordinary one-dimensional coalescing random walks starting from all space–time points have also been proposed as model of a drainage network [14], so the latter remark applies to them as well. Another application would be in obtaining aging results from the scaling limit results for systems that could be modeled by Poisson webs, like drainage networks. For the relation between aging and scaling limits, see e.g. [7,5,6,8], and references therein.

2. Proofs

Coalescing random walks. LetS be the Poisson process with parameterλ >0, fix some r >0. For anyx = (x₁, x₂)∈R², letτⁿ(x)= [αⁿ(x)]2,n0, be the second coordinate ofαⁿ(x)and consider{ξ^x(t ): tx₂}as the continuous time Markov process defined by

ξ^x(t )= αⁿ(x)

1,the first coordinate ofαⁿ(x); t∈

τⁿ(x), τⁿ⁺¹(x)

, n0. (2.1)

We remark that for any fixed{xⁱ}^m_i=1, withxⁱ =(x₁ⁱ, x₂ⁱ)∈R² fori=1, . . . , m,{(ξ^xi(t ): tx₂ⁱ),i=1, . . . , m} defines a finite system of coalescing random walks starting at the space–time pointsx¹, . . . , x^m.

Forx=(x₁, x₂)∈R², letx_δ=(δ⁻¹x₁, δ⁻²x₂),δ∈(0,1]. For the single random walk starting atx=(x₁, x₂), ξ^x(·), defined in the last paragraph, the diffusive rescaling is

ξ_δ^x(t ):=δξ^xδ(δ⁻²t ), fortx₂; δ∈(0,1]. (2.2)

Since the characterizing theorem and the weak convergence criteria given in [7] apply to continuous paths only, we need to replace the original processes by their linearly interpolated versions:

ξ¯_δ^x(t )=δ

ξ^xδ τⁿ(x_δ)

+ δ⁻²t−τⁿ(x_δ) τⁿ⁺¹(x_δ)−τⁿ(x_δ)

ξ^xδ

τⁿ⁺¹(x_δ)

−ξ^xδ

τⁿ(x_δ)

, (2.3)

fortx₂such thatδ⁻²t∈ [τⁿ(x_δ), τⁿ⁺¹(x_δ)),n0;δ∈(0,1],x∈R². Denote byξ¯_δ^xthe corresponding continu- ous path inR²and note thatξ¯₁^s is justX^s in (1.4) withs∈S. It is straightforward to see thatξ¯_δ^x∈X_δ, the Poisson web defined by (1.5), if and only ifxδ∈S.

Let

θ_δ^x:= ξ¯_δ^x ifx_δ∈S,

ξ¯_δ^{(δ[α(xδ)]1,δ2[α(xδ)]2)} otherwise.

(2.4)

In this way, for allx∈R² andδ∈(0,1],θ_δ^x∈X_δ. Note that the paths defined by (2.3) and (2.4) depend on the choice ofλ >0 andr >0. In case of necessity, we denote them byξ¯_δ^x(λ, r)andθ_δ^x(λ, r).

The following is an application of the classical Donsker’s theorem [2] to our case.

Lemma 2.1. Ifλ=λ₀=√

3/6,r=r₀=√

3, thenξ¯_δ^xconverges in distribution asδ→0 toB^x, the Brownian path with unit diffusion coefficient starting from space–time pointx=(x₁, x₂)∈R².

For anyx¹, . . . , x^m∈R²,m∈N, regard(¯ξ_δ^x1, . . . ,ξ¯_δ^xm)and(θ_δ^x1, . . . , θ_δ^xm)as random variables in the product metric space(Π^m, d^∗m), whered^∗mis a metric onΠ^msuch that the topology generated by it coincides with the corresponding product topology. Here we choose and define

d^∗m

(ξ¹, . . . , ξ^m), (ζ¹, . . . , ζ^m)

= max

1imd(ξⁱ, ζⁱ), (2.5)

for all(ξ¹, . . . , ξ^m), (ζ¹, . . . , ζ^m)∈Π^m, whered was defined in [7]. The next result follows immediately from the definition.

Lemma 2.2.

Pλ

d^∗m(¯ξ_δ^x1, . . . ,ξ¯_δ^xm), (θ_δ^x1, . . . , θ_δ^xm)

→0, asδ→0 (2.6)

for all >0,λ >0,r >0, andx¹, . . . , x^m∈R², m∈N, wherePλis the probability distribution ofS, the Poisson process with parameterλ.

2.1. Convergence in finite-dimensional cases: verification of condition I₁

LetDbe a countable dense set of points inR², to verify condition I₁, by Lemma 2.2, we only need to prove the following.

Lemma 2.3.(¯ξ_δ^y1, . . . ,ξ¯_δ^ym)converges in distribution asδ→0 to coalescing Brownian motions (with unit diffusion constant) starting aty¹, . . . , y^m(∈D).

For the finite system of coalescing random walks defined in the last subsection, Ferrari, Landim and Thorisson [4] proved that, for anyx, y∈R², the random walksξ^x(t )andξ^y(t ),tx₂∨y₂ will meet and then coalesce almost surely. This also follows from Lemma 2.7 below. The following is a corollary of this result.

Lemma 2.4. For anyλ >0,r >0,Pλ{sup_tx

2|¯ξ₁^x(t )− ¯ξ₁^y(t )|σ} →0, asσ→ ∞for allx, y∈R²withx₂=y₂. Now, for anymdistinct pointsy¹, . . . , y^m∈Dandδ∈(0,1]. Letξ¯_δ^y1, . . . ,ξ¯_δ^ym be themrescaled continuous random paths defined in (2.3) from the same Poisson process with λ=λ₀=√

3/6 and r=r₀=√

3. Having (¯ξ_δ^y1, . . . ,ξ¯_δ^ym)as a random element inΠ^m, we want to define a functionfδ from it toΠ^m. This is our main idea for the verification of condition I₁: we define what we call “δ-coalescence” of the random pathsξ¯_δ^y1, . . . ,ξ¯_δ^ym in such a way that, in the systemf_δ(ξ¯_δ^y1, . . . ,ξ¯_δ^ym), before anyδ-coalescence, the paths involved are independent.

We definef_δby renewing the whole system step by step as follows. Consider(ξ_δ^y1, . . . , ξ_δ^ym), the rescaled finite system of coalescing random walks starting at the space–time pointsy¹, . . . , y^m. Letγδ,0=min(y₂¹, . . . , y₂^m), and we assume thatδ >0 is close enough to 0, so that, in particular, the following stopping times we need are well defined.

For(ξ_δ^y1, . . . , ξ_δ^ym), ast (> γ_δ,0)grows, letγ_δ,k, 1km−1, be the time when thekthδ-coalescence occurs.

We say aδ-coalescence occurs at timet₀, ift₀ is the time when two particles get within a distance smaller than 2√

3δ. Once aδ-coalescence occurs, we renew the system by coalescing the two particles to be the left one and then wait for the nextδ-coalescence.

Denote the linearly interpolated versions of the resulting object after renewingm−1 times byf_δ(¯ξ_δ^y1, . . . ,ξ¯_δ^ym) and, with that, finish the definition of the functionf_δ. Clearly, forδ∈(0,1]small enough, we have

−∞< γ_δ,0< γ_δ,1<· · ·< γ_δ,m−1<∞ (2.7)

and the functionf_δis well defined almost surely.

Now, suppose that ξ˜_δ^yi has the same distribution as ξ¯^yi, 1 i m, and, as a random element in Π^m, (˜ξ_δ^y1, . . . ,ξ˜_δ^ym) has independent components. It is easy to see that the function f_δ is also well defined for the random paths(˜ξ_δ^y1, . . . ,ξ˜_δ^ym). LetC_δ⊂Π^mbe such that

P(˜ξ_δ^y1, . . . ,ξ˜_δ^ym)∈C_δ

=1 (2.8)

and, onC_δ,f_δis well defined.

For anyy¹, . . . , y^m∈D, letB^y1, . . . , B^ym be mindependent Brownian paths starting at space–time points y¹, . . . , y^m, respectively. As Arratia did in [1], we construct a set{ ˜B^y1, . . . ,B˜^ym}ofmone-dimensional coalescing Brownian motions starting aty¹, . . . , y^mby defining an almost surely continuous functionf fromΠ^mtoΠ^mas follows. The first Brownian path of the set,B˜^y1, isB^y1 itself. Once we have{ ˜B^y1, . . . ,B˜^yk−1}, we defineB˜^yk to be equal toB^yktill that path first hits any ofB˜^y1, . . . ,B˜^yk−1, sayB˜^yˆ; thence on it coincides withB˜^yˆ. This procedure is a.s. well defined, and the system resulting afterm−1 steps is the so-called one-dimensional coalescing Brownian motions starting at space–time pointsy¹, . . . , y^m, which we denote byf (B^y1, . . . , B^ym), as a function of the m independent Brownian motionsB^y1, . . . , B^ym.

Lemma 2.5. Let(¯ξ_δ^y1, . . . ,ξ¯_δ^ym)be themrescaled continuous random paths defined in (2.3) from the same Poisson process withλ=λ₀=√

3/6 andr=r₀=√

3, and(˜ξ_δ^y1, . . . ,ξ˜_δ^ym)have independent components andξ˜_δ^yihave the same distribution asξ¯^yi for all 1im. Then,

(a) f_δ(¯ξ_δ^y1, . . . ,ξ¯_δ^ym)has the same distribution asf_δ(˜ξ_δ^y1, . . . ,ξ˜_δ^ym);

(b) f_δ(˜ξ_δ^y1, . . . ,ξ˜_δ^ym)converges in distribution tof (B^y1, . . . , B^ym)asδ→0;

(c) for any >0,P{d^∗m[fδ(¯ξ_δ^y1, . . . ,ξ¯_δ^ym), (¯ξ_δ^y1, . . . ,ξ¯_δ^ym)] } →0,asδ→0.d^∗mwas defined in (2.5).

Proof. (a), (c) Immediate from the definition offδand Lemma 2.4. For (b), it is straightforward to check that, for anyc=(c¹, . . . , c^m)∈C,c_δ=(c_δ¹, . . . , c_δ^m)∈C_δ,d^∗m(c_δ, c)→0 impliesd^∗m(f_δ(c_δ), f (c))→0 asδ→0. Thus, an extended continuous mapping theorem of Mann and Wald [11], Prohorov [12] (see also Theorem 3.27 of [10]) gives (b). 2

Lemma 2.3 is an immediate consequence of Lemma 2.5. Thus, condition I₁for the Poisson webX_δ, δ∈(0,1] follows from Lemma 2.2.

2.2. Verification of conditions B₁and B₂

Consider the Poisson processSwith parameterλ >0 and the corresponding Poisson treeX:=X(λ, r)defined in (1.4) with respect to some fixed r >0. Given t₀∈R, t >0, a, b∈Rwith a < b, let η_X(t₀, t;a, b)be the

{0,1,2, . . . ,∞}-valued random variable defined right after the statement of Theorem 1.1. Let η¯_X(t₀, t;a, b)be another{0,1,2, . . . ,∞}-valued random variable defined as the number of distinct pointsy=(y₁, y₂)∈R×{t₀+t} such that there existss∈S withs₂t₀,ξ^s(t₀)∈ [a, b] andξ^s(t₀+t )=ξ^s(y₂)=y₁, whereξ^s is the Markov process defined in (2.1). It is straightforward to see that, for any fixedn∈N,

¯

η_X(t₀, t;a, b)n⇒η_X(t₀, t;a−2r, b+2r)n⇒ ¯η_X(t₀, t;a−4r, b+4r)n. (2.9) This implies that, to verify conditions B1, B2for the Poisson treesXδ, we only need to verify the following B₁ and B₂.

(B₁) lim sup_n→∞P(η¯δ_n(0, t;0, )2)→0 as →0+; (B₂) ⁻¹lim sup_n→∞P(η¯_δn(0, t;0, )3)→0 as →0+

for any sequence of positive numbers(δ_n)such that lim_n→∞δ_n=0, whereη¯_δ= ¯η_Xδ, and we have used the space homogeneity of the Poisson point process to eliminate the sup_(a,t

0)∈R² and puta=t₀=0.

Here, we firstly introduce an FKG inequality for probability measures on the path space, which will play an important role in our proofs. Letξ=ξ^(0,0)be the random path starting at the origin defined in (2.1); denote byΠ the space of paths whereξtakes value. We define a partial order “” onΠas follows. Givenπ₁, π₂∈ Π,

π₁π₂ if and only if π₁(t )−π₁(s)π₂(t )−π₂(s) for allts0. (2.10) Define increasing events inΠas usual. Denote byµ_ξ the distribution ofξonΠ.

Lemma 2.6 (FKG Inequality). µ_ξ satisfies the FKG inequality, namely, for any increasing events A, B ⊆ Π, µ_ξ(AB)µ_ξ(A)µ_ξ(B).

Proof. Follows by discretizingξ as a discrete time random walk, and then using FKG for its i.i.d. increments.

Details can be found in [3]. 2

Letξ^(0,0),ξ^{(γ ,0)},γ r, be two of the random walks defined in (2.1). Denote by_γ the difference between them. Then,_γ is a (space inhomogeneous) jump process in[0,∞)with absorbing state 0: Inx∈ [r,∞), it has rates(2r+x∧(2r))λand jump laws

νx:=2r−x∧(2r)

2r+x∧(2r)δ_{−x}+ 2(x∧(2r)) 2r+x∧(2r)U

r−x∧(2r), r

, (2.11)

whereδ_{−x}is the usual Dirac measure andU[r−x, r]is the uniform distribution on[r−x, r]. LetT =inf{t >0:

_2r(t )=ξ^(0,0)(t )−ξ^(2r,0)(t )=0}.

Lemma 2.7. There exists a constantc >0 such thatP(T > t )c/√

t for anyt >0, wherecdepends onrandλ only.

Proof. Follows by a coupling of2r and a space homogeneous process which has the same transition distribution as2r outside a neighborhood of the origin. Inside that neighborhood they are not too different, so that the result for2r follows from that the analogous one for the homogeneous process, which is a random walk, and for which that result follows from standard arguments. Details can be found in [3]. 2

Now, we begin to verify conditions B₁and B₂. By Lemma 2.3, it is straightforward to get that, lim sup

n→∞ P

ηδ_n(0, t;0, )2

=2φ /√

2t

−1, (2.12)

whereδ_n is any sequence of positive numbers converging to 0 as n→ ∞,η_δ=η_Xδ, andφ(x) is the standard normal distribution function. By (2.9), (2.12) withη¯X_δ replacingηX_δ also holds. This gives B₁.

Verifying B₂for the Poisson webXδis equivalent to checking that for anyt >0

−1lim sup

N→∞ P

¯ ηX₁

0, t N;0, √

N

3

→0 as →0+, (2.13)

whereX₁is defined in (1.5).

For that, fixt >0. On the Poisson fieldSwith parameterλ=λ₀=√

3/6, chooser=r₀=√

3, and then define X1as in (1.5). We first condition the probability in (2.13) on the set of points of intersection, in increasing order, of the pathsξ^s,s∈S, with[0, √

N], denoted{K₁, . . . , K_J}, whereJ, K₁, . . . , K_J are random variables, withJ an integer which can equal 0 (in which case set of intersection points is empty by convention). We note that by the definition ofξ^s,s∈S, no two distinctK_i’s can be at distance smaller thanr₀. For{x₁, . . . , x_n} ⊂ [0, √

N], letξ_j:=ξ^(xj,0), 1jn, as in (2.1). Letη=η(x₁, . . . , x_n)= |{ξ_j(t N ): 1j n}|(conventioned to be 0 if {x₁, . . . , xn} = ∅). Clearly, J, K₁, . . . , KJ depend only on the points ofS below time 0. Thus, sinceη depends only on the points ofSabove and at time 0 for all{x1, . . . , xn} ⊂ [0, √

N], givenJ=n,K1=x1, . . . , KJ=xn, the probability in (2.13) equalsP(η3).

We derive next an upper bound for the latter probability which is independent of{x₁, . . . , x_n}. First, we enlarge, if necessary, the set{x₁, . . . , x_n}to make sure thatx₁=0,x_n= √

N, andr₀x_j−x_j−12r₀. This also ensures thatn √

N /r₀+1, and the enlargement can only increase the probability to be estimated. Ifη3, then there should be some 1jn−1 such thatξ_j−1(t N ) < ξ_j(t N ) < ξ_n(t N ). Hence,

P(η3)

n−1

j=2

P

ξj−1(t N ) < ξj(t N ) < ξn(t N )

=

n−1

j=2

Π_j

P

ξ_j−1(t N ) < ξ_j(t N ) < ξ_n(t N )|ξ_j=π

µ_ξj(dπ )

=

n−1

j=2

Π_j

P

ξ_j−1(t N ) < ξ_j(t N )|ξ_j=π P

ξ_j(t N ) < ξ_n(t N )|ξ_j=π

µ_ξj(dπ ), (2.14)

whereΠj is the state space of ξj, andµξ_j its distribution. In the latter equality, we used the independence of ξ_j−1(t N ) < ξ_j(t N )andξ_j(t N ) < ξ_n(t N )conditioned onξ_j=π.

We claim now thatP(ξ_j−1(t N ) < ξ_j(t N )|ξ_j =π )decreases inπ andP(ξ_j(t N ) < ξ_n(t N )|ξ_j =π )increases inπ. (The reader should check it.) This and the FKG Inequality forµ_ξj (Lemma 2.6) imply that the right-hand side of (2.14) is bounded above by

n−1

j=2

P

ξj−1(t N ) < ξj(t N ) P

ξj(t N ) < ξn(t N )

P

ξ0(t N ) < ξn(t N )ⁿ⁻¹

j=2

P

ξj−1(t N ) < ξj(t N )

sinceP(ξ_j(t N ) < ξ_n(t N )) is clearly nonincreasing in j. Now the probabilities inside the sum are all bounded above byP(ξ^(0,0)(t N ) < ξ^(2r0,0)(t N )), and we get

P(η3)nP

ξ^(0,0)(t N ) < ξ^(2r0,0)(t N ) P

ξ^(0,0)(t N ) < ξ⁽

√N ,0)(t N )

√N

r₀ P(T > t N )P(T ,N> t N ),

whereT is the time when ξ^(0,0) and ξ^(2r0,0) meet and coalesce, andT ,N is the analogue time forξ^(0,0) and ξ⁽

√N ,0).

Now, let us consider the items at the right-hand side of the latter equation. By Lemma 2.3, lim sup

N→∞ P(T ,N> t N )=P(T ,B> t ),

whereT ,B is the time when two i.i.d. Brownian motions starting at the same time at distance apart meet and coalesce. Thus the latter probability is an O( )for everyt >0 fixed. By Lemma 2.7,P(T > t N )c/√

t N. These estimates imply that

lim sup

N→∞ P

¯ η_X1

0, t N;0, √ N

3

lim sup

N→∞

√N

r₀ P(T > t N )P(T ,N> t N )=O ₂ ,

and we get B₂forXδ.

Remark. The argument for B2in [7] for establishing weak convergence of coalescing random walks to the Brown- ian web also relies on an FKG property of the path distributions. But in that case, it is a stronger FKG property than in the present case. It allows boundingP(η3)above by[P(η2)]², and then the use of B₁. In particular, it is not necessary in that case to have an estimate of a microscopic quantity likeP(T > t N ), on which we had to rely in here.

Acknowledgements

This work was begun when one of us (X.-Y. Wu) was visiting the Statistics Department of the Institute of Mathematics and Statistics of the University of São Paulo. He is thankful to the probability group of IME-USP for hospitality. We much thank Rongfeng Sun for pointing out important incorrections on Subsection 2.2 of an earlier version, as well as for subsequent discussions on our fixing of them.

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