Convergence of Coalescing Nonsimple Random Walks to the Brownian Web

The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $\R\times\R$. It was first introduced by Arratia, and later analyzed in detail by T\'{o}th and Werner. More recently, Fontes, Isopi, Newman and Ravishankar gave a characterization of the BW, and general convergence criteria allowing either crossing or noncrossing paths, which they verified for coalescing simple random walks. Later Ferrari, Fontes, and Wu verified these criteria for a two dimensional Poisson Tree. In both cases, the paths are noncrossing. In this thesis, we formulate new convergence criteria for crossing paths, and verify them for non-simple coalescing random walks (both discrete and continuous time) satisfying a finite fifth moment condition. This is the first time convergence to the BW has been proved for models with crossing paths. Several corollaries are presented, including an analysis of the scaling limit of voter model interfaces that extends a result of Cox and Durrett.


Chapter 1 Introduction and Results
The idea of the Brownian Web (BW) dates back to Arratia's thesis [1] in 1979, in which he constructed a process of coalescing one-dimensional Brownian motions starting from every point in space R at time zero. In a later unpublished manuscript [2], Arratia generalized this construction to a process of coalescing Brownian motions starting from every point in space and time R × R, which is essentially what is now often called the Brownian Web. He also defined a dual family of backward coalescing Brownian motions equally distributed (after a time reversal) with the BW which is now called the Dual Brownian Web. Unfortunately, Arratia's manuscript was incomplete and never published, and the BW was not studied again until a paper by Tóth and Werner [23], in which they constructed and analyzed versions of the Brownian Web and Dual Brownian Web in great detail and used them to construct a process they call the True Self Repelling Motion.
In both Arratia's and Tóth and Werner's constructions of the BW, some semicontinuity condition is imposed to guarantee a unique path starting from every space-time point. (For example, in Arratia's [1] construction of coalescing Brownian motions starting from every point on R at time 0, when multiple paths start from the same point, a unique path is chosen so that if we regard the collection of paths as a mapping from R to the space of continuous paths, then it is right continuous with left limits.) More recently, Fontes, Isopi, Newman and Ravishankar [12,13] gave a different formulation of the BW which provides a more natural setting for weak convergence, and they coined the term Brownian Web. Instead of imposing semicontinuity conditions, multiple paths are allowed to start from the same space-time point. Further, by choosing a suitable topology, the BW can be characterized as a random variable taking values in a complete separable metric space whose elements are compact sets of paths. In [13], they gave general convergence criteria allowing either crossing or noncrossing paths (i.e., almost surely, paths in the random set may cross each other in the crossing paths case, and can never cross each other in the noncrossing paths case), and they verified the criteria for the noncrossing paths case for coalescing simple random walks. Recently, Ferrari, Fontes, and Wu [11] verified the same criteria for a two dimensional Poisson tree also with noncrossing paths.
The main result of this thesis is the formulation of new convergence criteria for the crossing paths case, and we verify these criteria for both discrete time and continuous time one-dimensional coalescing nonsimple random walks satisfying a finite fifth moment condition, which are models with crossing paths. The main technical distinction between the noncrossing paths case and the crossing paths case is that if paths cannot cross, then they form a totally ordered set, and one expects certain correlation inequalities to hold, which is not the case if paths can cross. We will also present some corollaries for one-dimensional coalescing nonsimple random walks and their dual non-nearest-neighbor voter models.
In the remaining sections of this chapter, we will present some background on coalescing random walks, their dual voter models, and the Brownian web. We will also state the main results of this thesis.

Coalescing random walks and voter models
Let Y , a random variable with distribution µ Y , denote the increment of an irreducible aperiodic random walk on Z. Throughout this thesis, all random walk increments will be distributed as Y . We will always assume E[Y ] = 0 and E[Y 2 ] = σ 2 < +∞ unless a weaker hypotheses is explicitly stated. For our main result, we will also need to assume E[|Y | 5 ] < +∞. For continuous time random walks, we further assume that P(Y = 0) = 0, and the random walk increments occur at rate 1.

Coalescing random walks
The process of discrete time coalescing random walks with one walker starting from every space-time lattice site is defined as follows. One walker starts at every site on the space-time lattice Z × Z (first coordinate space, second coordinate time), and makes jumps at integer times (including the time when it is born). All walkers have i.i.d. increments distributed as Y , and two walkers move independently until they jump to the same site at the same time, then they coalesce. The random walk paths are piecewise constant, right continuous with left limits, and have discontinuities at integer times. A continuous version of the random walk path is obtained by linearly interpolating between the consecutive space-time lattice sites from where the random walk makes its jumps. Note that for non-simple random walks, two interpolated random walk paths can cross each other many times before they actually coalesce. If Y was such that the random walks had period d = 1, as in the case of simple random walks where d = 2, then we would just have d different copies of coalescing random walks on different space-time sublattices, none of which interacts with the other copies. We will let X 1 (with distribution µ 1 ) denote the random realization of such a collection of interpolated coalescing random walk paths on Z × Z, and let X δ (with distribution µ δ , 0 < δ < 1) be X 1 rescaled with the usual diffusive scaling of δ/σ in space, δ 2 in time. Sometimes we will also need the piecewise constant version of X δ , which we denote by Γ δ , i.e., replace each path in X δ by its piecewise constant counterpart.
The continuous time analog of X δ ,X δ is defined as follows. One walker starts from every point on Z × R and undergoes rate 1 jumps with increments distributed as Y . The jump times are given by independent rate 1 Poisson clocks at each integer site i ∈ Z. Two walkers coalesce when they first occupy the same site at the same time. Clearly all walkers starting at the same site between two consecutive Poisson clock rings will have coalesced by the time of the second clock ring. If we call the time and location at which a Poisson clock rings a jump point, then the path of a random walk is piecewise constant with discontinuites at the jump points. We can also define an interpolated version of the random walk path, which consists of first a constant position line segment connecting the point of the walker's birth to its first jump point, and then linearly interpolating between consecutive jump points along its path. For random walks born at a jump point, we will take two paths, one starting with a constant position line segment, and one without.X 1 (with distribution µ 1 ) is then defined to be the random variable consisting of all the interpolated coalescing random walk paths, andX δ (with distributionμ δ ) isX 1 diffusively rescaled. We will denote the piecewise constant version ofX δ byΓ δ .
For a system of d-dimensional (d ≥ 1) discrete time coalescing random walks starting from a space-time subset A ⊂ Z d ×Z, we will denote the set of occupied sites in Z d at time n by ξ A n . (To be consistent with our definition of X 1 , we also define the random walks so that they jump at the time of birth. But the random walks' positions at time n is now taken to be the positions of the interpolated random walk paths at time n.) In the special case when A = B × {n 0 } for some B ⊂ Z d , we will denote it by ξ B,n 0 n ; and when n 0 = 0, we simply denote it by ξ B n . For most of this thesis, we will only deal with one-dimensional coalescing random walks. To simplify the notation, we will use ξ A s also to denote the continuous time analogue of ξ A n , where A ⊂ Z d × R and the random walks jump with rate 1 and increments distributed as Y .
The main result of this thesis (see Theorem 1.3.3 below) is that if E[|Y | 5 ] < +∞, then X δ andX δ converge in distribution to the BW as δ → 0.

Voter models
The voter model was first introduced in the probability literature by Clifford and Sudbury [8], and Holley and Liggett [16]. In population genetics, a variant of the voter model, the stepping stone model, was first introduced by Kimura [19]. A two type d-dimensional discrete time voter model is defined as follows. The state space is {0, 1} Z d with product topology. A metric that generates the product topology is η−ζ = x∈Z d 2 −|x|∞ |η(x)−ζ(x)|. We will denote the state at time n of the voter model by φ n (with distribution ν n ). φ n (x), the value of φ n at site x, can be regarded as the opinion, or political affiliation, of the voter at site x at time n, hence the name voter model. The initial configuration is specified by φ 0 . If we also use Y to denote the increment of a d-dimensional random walk, then at each time n ≥ 1, we update φ n (x) by setting φ n ( Given any realization of {Y x,n } x∈Z d ,n∈N , if we trace backward in time the geneaologies of the opinions of all sites at all times, then the collection of all such geneaology trees is exactly a realization of coalescing random walk paths running backward in time, with one walker starting at every site in Z d at every positive integer time. This provides a natural coupling between coalescing random walks and voter models, and leads to the duality relation where A ⊂ Z d × Z + is a set of space-time lattice sites with positive time, and ξ A 0 is the set of occupied sites at time 0 of a system of coalescing random walks running backward in time, with one walker starting at every site in A. For more details on the voter model, see, e.g., [16,20]. The definition above easily generalizes to a multitype voter model with state space S Z d , where S can be any finite or countable set. For continuous time, the opinion at each site is updated independently according to a rate 1 Poisson clock. Whenever a clock rings at a site x, another site y ∈ Z d is picked with (y − x) distributed as the random walk increment Y , and the opinion at site x changes to that of site y. The dual of this continuous time voter model is the process of rate 1 continuous time coalescing random walks running backward in time, with one walker starting at every site in Z d × R + . The two models are coupled through the realization of the jump points and the random walk increment associated with each jump point. We will denote the continuous time voter model also by φ t (with distribution ν t ). The duality relation (1.1.1) also holds for continuous time.

Brownian web
One way of thinking about the Brownian web is to view it as the diffusive scaling limit of the system of coalescing random walks X 1 . Heuristically, we expect to obtain as the limit a set of coalescing Brownian motions with one Brownian motion starting from every space-time point. The main conceptual difficulty is that there are uncountable number of space-time points, and in general we would like to deal only with a countable number of Brownian motions because of the countable additivity of measures. Fortunately, it turns out that the limiting object (the Brownian web) is fully determined by a countable number of coalescing Brownian motions starting from a countable dense subset D of R 2 (see Theorem 1.2.1 below). Coalescing Brownian motion paths starting from space-time points off the dense set D can be constructed by taking limits of paths starting from D using the noncrossing property of coalescing Brownian motions. See Arratia [2] and Tóth and Werner [23] for two such constructions. The Brownian web intrinsically contains space-time points from which multiple paths start out. In both Arratia's and Tóth and Werner's construction, only one path is retained starting from every space-time point satisfying some semicontinuity conditions. Here we follow a different approach by Fontes, Isopi, Newman and Ravishankar [12,13], who coined the term Brownian Web. Their approach is to regard the Brownian web as a random variable taking values in the space of sets of paths. Thus there is no need to throw away paths when multiplicity arises. Another advantage is that, by choosing the topology approriately for the space of sets of paths, the Brownian web takes values in a complete separable metric space, thus providing a natural setting for establishing weak convergence results, which is the central theme of this thesis.
We now recall Fontes, Isopi, Newman and Ravishankar's [12,13] choice of the metric space in which the Brownian Web takes its values.
Let (R 2 , ρ) be the completion (or compactification) of R 2 under the metric ρ, where We can think of the mapping as first squeezeR 2 to the square [−1, 1] × [−1, 1] by the mapping (tanh x, tanh t), and then the x coordinate is squeezed even further depending on its time coordinate such that the top and bottom edge of the square is squeezed to two points (0, ±1).
where (f, t 0 ) ∈ Π represents a path inR 2 starting at (f (t 0 ), t 0 ). For(f, t 0 ) in Π, we denote byf the function that extends f to all [−∞, ∞] by setting it equal to f (t 0 ) for t < t 0 . Then we take i.e., after applying the mapping (Φ, Ψ) to (f 1 (t), t) and (f 2 (t), t), the distance between the two original paths is then taken to be the maximum of (i) the supremum norm distance between the two image paths, and (2) the absolute difference in the starting times of the two image paths. Note that (Π, d) is a complete separable metric space.
Let now H denote the set of compact subsets of (Π, d), with d H the induced Hausdorff metric, i.e., (1.2.5) is also a complete separable metric space. Let F H denote the Borel σ-algebra generated by d H .
In [12,13], the Brownian Web (W with measure µW ) is constructed as a (H, F H ) valued random variable, with the following characterization theorem. Theorem 1.2.1 There is an (H, F H )-valued random variableW whose distribution is uniquely determined by the following three properties.
(o) from any deterministic point (x, t) in R 2 , there is almost surely a unique path W x,t starting from (x, t).
(i) for any deterministic n, (x 1 , t 1 ), . . . , (x n , t n ), the joint distribution of W The Brownian webW uniquely determines a dual (backward) Brownian web W, which is equally distributed with the standard Brownian webW except for a time reversal. The pair (W,Ŵ) forms what is called the double Brownian webW D with the property that, almost surely, paths inW andŴ reflect off each other and never cross. The double Brownian web is most easily recognized as the limit of coalescing simple random walks. For coalescing simple random walks, only walks starting from lattice sites (x, m) ∈ Z 2 with x + m having the same parity (even or odd) interact with each other. Take the set of walks that start from (x, m) with x + m even. Then any realization of such forward coalescing random walk paths uniquely determines a set of backward (in time) coalescing simple random walk paths with one path starting from every site (y, n) ∈ Z 2 with y + n odd, provided that we require the backward paths never cross the forward paths. The resulting system of backward coalescing simple random walks is equally distributed with the forward coalescing system except for a time reversal. Under the diffusive scaling limit, it is then seen that the joint distribution of the forward and backward systems of coalescing simple random walks converge in distribution to the double Brownian web. For more on the Brownian web and the double Brownian web, see [1,2,23,22,12,13].

Convergence criteria and main result
In [13], a set of general convergence criteria were formulated for measures supported on compact sets of paths which can cross each other. However, one of the conditions (B ′ 2 ) turns out to be difficult to verify for the coalescing nonsimple random walks {X δ } and {X δ }. In our modified general convergence theorem 1.3.2, we will replace condition (B ′ 2 ) by an alternative condition (E 1 ), and we will verify (E 1 ) and the other convergence criteria for {X δ } and {X δ } under the assumption that E[|Y | 5 ] < +∞.
We first recall the convergence criteria formulated in [13] (for the crossing paths case) for a family of (H, F H )-valued random variables {X n } with distributions {µ n }.
(I 1 ) There exist single path valued random variables θ y n ∈ X n , for y ∈ R 2 , satisfying: for D a deterministic countable dense subset of R 2 , for any deterministic y 1 , . . . , y m ∈ D, θ y 1 n , . . . , θ y m n converge jointly in distribution as n → +∞ to coalescing Brownian motions (with unit diffusion constant) starting at y 1 , . . . , y m . Let to be the event (in F H ) that K (in H) contains a path touching both R(x 0 , t 0 ; u, t) and (at a later time) the left or right boundary of the bigger rect-angle R(x 0 , t 0 ; 17u, 2t) (the number 17 is chosen to avoid fractions later). Then the following is a tightness condition for {X n }: for every u, L, T ∈ (0, +∞), As shown in [13], if (T 1 ) is satisfied, one can construct compact sets G ǫ ⊂ H for each ǫ > 0, such that µ n (G c ǫ ) < ǫ uniformly in n. G ǫ consists of compact subsets of Π whose image under the map (Φ, Ψ) are equicontinuous with a modulus of continuity dependent on ǫ.
For K ∈ H a compact set of paths in Π, define the counting variable Let l t 0 (resp., r t 0 ) denote the leftmost (resp., rightmost) value in [a, b] with some path in K touching (l t 0 , t 0 ) (resp., (r t 0 , t 0 )). Also define to be the subset of N t 0 ,t ([a, b]) due to paths in K that touch (l t 0 , t 0 ) (resp., (r t 0 , t 0 )). The last two conditions for the convergence of {X n } to the Brownian Web are The general convergence theorem of [13] is the following, Theorem 1.3.1 Let {X n } be a family of (H, F H ) valued random variables satisfying conditions (I 1 ), (T 1 ), (B ′ 1 ) and (B ′ 2 ), then X n converges in distribution to the standard Brownian WebW.
Condition (B ′ 1 ) guarantees that for any subsequential limit X of {X n } (with distribution µ X ), and for any deterministic point y ∈ R 2 , there is µ X almost surely at most one path starting from y. Together with condition (I 1 ), this implies that for a deterministic countable dense set D ⊂ R 2 , the distribution of paths in X starting from finite subsets of D is that of coalescing Brownian motions. This shows X contains at least as many paths as the Brownian webW. Conditions (B ′ 1 ) and (B ′ 2 ) together imply that for the family of counting random variables η(t 0 , t; a, b) πt for all t 0 , t, a, b ∈ R with t > 0, a < b. By Theorem 4.6 in [13] and the remark following it, this fact implies that X contains no extra paths besides the Brownian webW, thus X is equidistributed withW. For the systems of coalescing random walks {X δ } and {X δ }, we have not yet been able to verify condition (B ′ 2 ), but an examination of the proof of Theorem 4.6 in [13] shows that we can also use the dual family of counting random variableŝ By a duality argument [23] (see also [1,2,13]),η and η−1 are equally distributed for the Brownian WebW. We can then replace ( With this change, we obtain our modified general convergence theorem, and (E 1 ), then X n converges in distribution to the standard Brownian WebW.
The main result of this paper is Condition (E 1 ) in our general convergence theorem 1.3.2 may seem strong, but as we will show in our proof of (E 1 ) for {X δ } and {X δ } in Section 3.5, the key ingredients are the Markov property of the random walks and an upper bound of the type lim sup δ↓0 E[η X δ (t 0 , t; a, b)] ≤ C for some finite constant C depending on t, a, b.
In this chapter, we first introduce some notation, and then list some basic facts about random walks that will be used throughout the rest of the thesis. Once acquainted with the basic notation, the reader may skip the rest of the chapter until the results in this chapter are referred to.
For both discrete time and continuous time, we will denote the piecewise constant version of the path of a random walk (which by definition is right continuous with left limits in time) starting at position x at time t 0 by π x,t 0 (s). We will denote the linearly interpolated version of the random walk path by κ x,t 0 (s). Denote the event that the path of a random walk π x,t 0 (s) (either discrete or continuous time) starting at (x, t 0 ) stays inside the interval [a, b] containing x up to time t by B x,t 0 [a,b],t . Given any r ∈ Z, we also define the following stopping times associated with either a discrete or a continuous time random walk π x,t 0 When the time coordinate in the superscripts of π, κ, B, τ, τ+ is 0, we will suppress it. We will use P x and E x to denote probability and expectation for a random walk process starting from x at time 0. P x,y and E x,y will correspond to two independent random walks starting at x and y at time 0.
The following lemmas are stated for random walk paths π x and π y , which can be interpreted for both discrete and continuous time. The random walk increment Y is always assumed to be that of an irreducible and aperiodic random walk with E(Y ) = 0 and E[Y 2 ] < +∞, unless a different moment condition is explicitly stated. For continuous time random walks, we always assume it jumps with rate 1 unless otherwise explicitly stated.
Proof. Letπ y−x (t) = π y (t) − π x (t). Thenπ y−x is an irreducible aperiodic symmetric random walk starting at y −x. For discrete time,π y−x has increment distributed as µ Y * µ −Y ; for continuous time,π y−x is a rate 2 random walk with increment distributed as 1 2 The lemma is simply a consequence of the recurrence ofπ y−x , which requires less than finite second moment of Y .
Proof. Letπ y−x (t) = π y (t) − π x (t) as in the proof of Lemma 2.0.1. LetP y−x denote probability for this random walk, and letτ y−x 0 denote the stopping time when the random walkπ y−x first lands at the site 0. Then P x,y (τ x,y > t) = P y−x (τ y−x 0 > t). When |x − y| = 1, it is a standard fact (see, e.g., Proposition 32.4 in [21]) that this probability is bounded by C √ t . When |x − y| > 1, we can without loss of generality assume x < y and regard {π x , π y } as a subset of the system of coalescing random walks ξ {x,x+1,...,y} up to time τ x,y . Then which establishes the lemma.
Lemma 2.0.3 Let u > 0 and t > 0 be fixed, and let π(s) = π 0,0 (s) be a random walk starting from the origin at time 0. Letũ,t and the event B 0,0 be defined as at the beginning of this chapter (note that they depend on δ), and let If B s is a standard Brownian motion starting from 0, then Proof. The limit follows from Donsker's invariance principle for random walks. The first inequality is trivial, and the second inequality follows from a wellknown computation for Brownian motion using the reflection principle.
Lemma 2.0.4 Let u, t,ũ,t be as before. Let π x , π y and τ x,y be as in Lemmas < +∞, and δ is sufficiently small, then we have for some constant C(t, u) depending only on t and u.
Proof. Let z = x − y < 0. Note that x, y, z are fixed whileũ,t → +∞ as δ → 0. For the difference of the two walksπ z (s) = π x (s) − π y (s), we denote the first time whenπ z (s) = 0 byτ z 0 , and the first time whenπ z (s) ≥ũ byτ z u + . We are using the bar· to emphasize the fact that we are studying the symmetrized random walks. The inequality then becomes For simplicity, we will only prove (2.0.3) for the discrete time case. For the continuous time case, only the notations will be different. We will first prove that, for δ sufficiently small, By the strong Markov property, . If δ is sufficiently small, the last inequality is valid by Lemma 2.0.3. Also by Lemma 2.0.2,P together they give (2.0.4).
To show (2.0.3), we condition at the first time whenπ z (s) ≥ 0, which we denote byτ z 0 + . Then by the strong Markov property and (2.0.4), The last inequality follows from our assumption E[|Y | 3 ] < +∞ and the following two lemmas.
Proof.This is a standard fact from renewal theory, see e.g. Proposition 24.7 in [21].
Lemma 2.0.6 Let π x , Y and Z be as in the previous lemma.
Proof. We may assume π x is a discrete time random walk, since the continuous time random walk is just a random time change of the discrete time walk, which does not change the overshoot distribution. Note that if we let γ x denote the discrete time random walk starting from x < 0 at time 0 with increment distributed as Z, then γ x simply records the successive maxima of the random walk π x , so the overshoots π x (τ x 0 + ) and γ x (τ x 0 + ) are equally distributed. By a last passage decomposition for γ x , giving uniform integrability. The rest then follows from Lemma 2.0.5 and dominated convergence.
Lemma 2.0.7 Let ξ Z t be a system of coalescing random walks (either discrete or continuous time) starting from every site of Z at time 0, whose random walk for some constant C independent of the time t.
Remark 2.0.5 We present two proofs, the first of which works for both discrete and continuous time, and is an adaptation of the argument used by Bramson and Griffeath [5] to establish similar upper bounds for continuous time coalescing simple random walks in Z d , d ≥ 2. The second proof is special to continuous time walks, and can also be found in the paper of Bramson and Griffeath [5]. In Corollary 4.1.1 below, we will prove that in Since M − |ξ B M t | is at least as large as the number of nearest neighbor pairs in B M that have coalesced by time t, we may take expectation and apply Lemma 2.0.2 to obtain Therefore p t < 1/M + C/ √ t. Since M can be arbitrarily large for any fixed t, we The process A t is then a continuous time Markov chain on the space of finite subsets of Z. A t undergoes jumps The rate at which |A t | increases by 1 is x∈A c y∈A P(Y = y − x); the rate at which |A t | decreases by 1 is x∈A y∈A c P(Y = y − x). By translation invariance, it is not difficult to see that these two rates are the same, and by our assumption P(Y = 0) = 0 for continuous time random walks, the sum of the two rates is at least 2. Therefore, |A t | is a continuous time simple symmetric random walk with a random rate bounded below by 2, and |A t | is absorbed at 0. P(|A t | ≥ 1) is then bounded from above by the probability that a rate 2 simple symmetric random walk starting from 1 does not hit 0 by time t, which by Lemma 2.0.2 is bounded above by C √ t . By the duality relation (1. , the lemma then follows. Note that this proof only works for continuous time. Proof. The continuous time version of this lemma is due to Arratia (see Lemma 1 in [3]). Arratia's proof uses a theorem of Harris [15], which breaks down for discrete time because there are transitions between states that not comparable to each other with respect to some partial order. However, this can be easily remedied by using an induction argument, which we present below.
We can assume A, B, C are all finite sets, since otherwise we can approximate by finite sets, and the relevant probabilities will all converge. The main tool in the proof is again the duality between coalescing random walks and voter models. For any pair of finite disjoint sets B, be the event that some site in A is assigned the value +1 (resp., −1). Then by the duality relation (1.
We can define a partial order on the state space X by setting η ≤ ζ ∈ X whenever η( ). An event E is called increasing (resp., decreasing) if 1 E is an increasing (resp., decreasing) function. Clearly, for finite A, 1 E + A is a continuous increasing function and 1 E − A is a continuous decreasing function. Inequality (2.0.7) will follow if we show that ν B,C n has the FKG property (see, e.g, [14,20]), i.e., for any two continuous increasing functions f and g, f g dν B,C n ≥ f dν B,C n g dν B,C n . We prove this by induction. For any pair of finite disjoint sets B, C ⊂ Z d , ν B,C 0 has the FKG property because the measure is concentrated at a single configuration. Observe that ν B,C 1 is a product measure and therefore also has the FKG property (this is a simple special case of the main result of [14]). We proceed to the induction step, which is a fairly standard argument [17]. Assume that for all disjoint finite sets B and C, and for all 0 ≤ k ≤ n − 1, Recall that Γ δ andΓ δ denote the piecewise constant version of X δ andX δ . We can extend the definition of d(·, ·) (resp., d H (·, ·)) to path (resp., sets of paths) that are right continuous with left limits. Then we have

Proof of the Main Result
In this chapter, we first establish the almost sure pre-compactness of X δ and X δ , so that the almost sure closures of X δ andX δ are (H, F H )-valued random variables. We will then proceed to verify conditions (B ′ 1 ), (T 1 ), (I 1 ) and (E 1 ) for {X δ } and {X δ }, thus establishing the main result of this thesis, Theorem 1.3.3.
Proof. We prove the lemma only for X 1 andX 1 , since the proof for X δ and X δ is identical. We will show that under the mapping (Φ, Ψ), X 1 andX 1 are almost surely equicontinuous. Note that by the properties of (Φ, Ψ), this reduces to showing the almost sure equicontinuity of points of a random walk, and we are only interested in line segments that intersect Λ L . We will show that almost surely, we can choose L ′ sufficiently large such that line segments inX 1 starting from points outside Λ L ′ do not intersect Λ L . Since almost surely there are only a finite number of jump points inside Λ L ′ , and hence only a finite number of non-constant-position line segments intersecting Λ L (note that almost surely none of the line segments is constant in time),X 1 restricted to Λ L must be equicontinuous.
Let L be fixed. Let I L ′ (L ′ > L) denote the event that some line segment in X 1 starting from some point outside Λ L ′ intersects Λ L . Since I L ′ is a descreasing family of events as L ′ increases, it suffices to show thatμ 1 (I L ′ ) → 0 as L ′ → +∞.
We divide the complement of Λ L ′ into six regions as illustrated in The event I 2 L ′ only occurs when the random walk increment associated with some jump point (x, t) in Region 2 exceeds −x−L, and at the landing site there is no Poisson clock ring during the time interval (t, −L). Since the intensity of the Poisson process at each site is 1, we can estimateμ 1 (I 2 L ′ ) by By the assumption E[|Y |] < +∞,μ 1 (I 2 L ′ ) → 0 as L ′ → +∞. Similarlyμ 1 (I 4 L ′ ) → 0 as L ′ → +∞. An analogous calculation shows thatμ 1 (I 3 L ′ ) ≤ 2L ′ e −L ′ +L , which also tends to 0 as L ′ → +∞. Since I L ′ ⊂ ∪ 5 i=0 I i L ′ , the lemma then follows.
We now complete the proof of Lemma 2.0.9. Proof of Lemma 2.0.9 for continuous time. In Figure 3

Verification of (B ′ 1 )
Verification of (B ′ 1 ). We first treat the discrete time case. The continuous time case will be similar. Fix t 0 , a ∈ R, β > 0, t > β, ǫ > 0. Also fix a δ and lett 0 ,t,ã andǫ be defined from t 0 , t, a and ǫ by diffusive scaling. Then we have Ift 0 = n 0 ∈ Z, then the contribution to N is all due to walkers starting from [ã −ǫ,ã +ǫ] ∩ Z at time n 0 . Thus we have The first inequality follows from the observation that if the collection of walkers starting from [ã −ǫ,ã +ǫ] ∩ Z at n 0 has not coalesced into a single walker by n 0 +t, then there is at least one adjacent pair of such walkers which has not coalesced by n 0 +t. The next inequality follows from Lemma 2.0.2. Now supposet 0 ∈ (n 0 , n 0 + 1) for some n 0 ∈ Z. Note that a walker's path can only cross [ã −ǫ,ã +ǫ] × {t 0 } due to the increment at time n 0 . After the increment, at time n 0 + 1, it will either land in [ã − 2ǫ,ã + 2ǫ], or else outside that interval. In the first case, the contribution of the walker's path to N is included in ξ [ã−2ǫ,ã+2ǫ]∩Z,n 0 +1 t 0 +t , the probability of which by our previous argument is bounded by 4Cσǫ √ t times a prefactor which approaches 1 as δ → 0. In the second case, either a walker in (−∞,ã +ǫ] jumps to the right ofã + 2ǫ, or a walker in [ã −ǫ, +∞) jumps to the left ofã − 2ǫ, the probability of which is bounded by The next to last inequality in (3.2.2) is valid if we take δ to be sufficiently small. The bounds in (3.2.1) and (3.2.2) are independent of t 0 , t > β and a. Taking the supremum over t > β, t 0 and a, and letting δ → 0 + , we establish (B ′ 1 ) for {µ δ }.
The continuous time case is similar to the discrete time case. We first scalȇ X δ back to the Z×R lattice, and note that the contribution to Nt 0 ,t ([ã−ǫ,ã+ǫ]) is only due to interpolated random walk paths intersecting [ã −ǫ,ã +ǫ] at timẽ t 0 . Thereforeμ 1 (|Nt 0,t ([ã−ǫ,ã+ǫ])| > 1) can be estimated by the union of three events: (i) event A, for some site x ∈ [ã − 2ǫ,ã + 2ǫ] ∩ Z, there is no Poisson clock ring during the time interval [t 0 ,t 0 +t]; (ii) event B, the set of coalescing random walks starting from [ã−2ǫ,ã+2ǫ]∩Z at timet 0 has not coalesced into a single walker by timet 0 +t; (iii) event C, some interpolated random walk paths intersects [ã −ǫ,ã +ǫ] at time t 0 , and after the intersection does not land at a site in [ã − 2ǫ,ã + 2ǫ] ∩ Z. The event {|Nt 0 ,t ([ã −ǫ,ã +ǫ])| > 1} is contained in the union of the events A, B and C.μ 1 (A) is bounded by 4ǫe −t and tends to 0 as δ → 0.μ 1 (B) can be estimated exactly as the computation in (3.2.1), and we obtain the desired factor of ǫ in the limit as δ → 0. The event C plays the same role as the event whose probability was estimated in (3.2.2), andμ 1 (C) → 0 as δ → 0, but we will defer its proof to the next section on tightness, where we need to estimate a similar, but more general quantity (see the paragraph above Remark 3.3.1).
Corollary 3.2.1 Assume X (with distribution µ) is a subsequential limit of X δ (orX δ ), then for any deterministic point y ∈ R 2 , X has almost surely at most one path starting from y.

Proof. It was shown in the proof of Theorem 5.3 in [13] that (B
the corollary then follows. is the process of coalescing random walks (either discrete or continuous time) starting from a subset A δ of the rescaled lattice, and Z A δ δ converges in distribution to a limit Z, then by the same argument as above, for any deterministic point y ∈ R 2 , Z has almost surely at most one path starting from y.

Verification of (T 1 )
In this section, we verify condition (T 1 ) for the measures {µ δ } and {μ δ } under the assumption E[|Y | 5 ] < +∞. At the end of this section, we will also show that for {X 0 T δ } and {X 0 T δ }, the set of interpolated coalescing random walk paths starting with one walker at every site in the rescaled lattice at time 0, E[|Y | 3 ] < +∞ will be sufficient to guarantee tightness.
Define A + t,u (x 0 , t 0 ) to be the event that K contains a path touching both R(x 0 , t 0 ; u, t) and (at a later time) the right boundary of the bigger rectangle R(x 0 , t 0 ; 17u, 2t), and similarly define the event A − t,u (x 0 , t 0 ) corresponding to the left boundary of the bigger rectangle. Then A = (A + ∪ A − ), and writing (T 1 ) in terms of µ 1 , we argue that it is sufficient to prove 2ũ 6ũ 8ũ 16ũ Figure 3.2: The random walks π 1 , π 2 , π 3 and π 4 start from 3ũ, 7ũ, 11ũ and 15ũ at time 0 and each stays within a distance ofũ from its initial position. The random walk π x,m starts from (x, m) inside the rectangle R(ũ,t) and exits the right boundary of the rectangle R(17ũ, 2t) at time τ 4 without coalescing with π 1 , π 2 , π 3 and π 4 on the way.
The sup over x 0 , t 0 has been safely omitted because µ 1 is invariant under translation by integer units of space and time. Whenx 0 ,t 0 / ∈ Z, we can bound the probability from above by using larger rectangles with vertices in Z × Z and base centered at (0, 0). Since the argument establishing the analogous tightness condition (T − 1 ) for the event A − is identical to that for (T + 1 ), (T 1 ) for the measures {µ δ } follows from (T + 1 ). Simlarly, (T 1 ) for the measures {μ δ } follows from (T + 1 ) with µ 1 replaced byμ 1 . Before we prove (T + 1 ), and hence (T 1 ) for µ 1 andμ 1 , we introduce some simplifying notation. We will abbreviate A + t,u (0, 0) by A + t,u , or just A + , and abbreviate R(0, 0; u, t) by R(u, t). Denote the random walks (either discrete or continuous time) starting at time 0 from x 1 = ⌈3ũ⌉, x 2 = ⌈7ũ⌉, x 3 = ⌈11ũ⌉, x 4 = ⌈15ũ⌉ by π 1 , π 2 , π 3 , π 4 (with their paths taken to be the piecewise constant version). Denote the event that π i , (i = 1, 2, 3, 4) stays within a distanceũ of x i up to time 2t by B i (see Figure 3.2). For a random walk starting from (x, m) ∈ R(ũ,t), denote the stopping times when the walker's path π x,m (s) first exceeds 5ũ, 9ũ, 13ũ and 17ũ by τ x,m . We also define τ x,m 0 = m, and τ x,m 5 = 2t. Denote the event that π x,m does not coalesce with π i before time 2t by C i (x, m). As we shall see, the reason for choosing four paths π i is because each path contributes a factor of δ to our estimate of the µ 1 (resp., µ 1 ) probability in (T + 1 ), and an overall factor of δ 4 is needed to outweigh the O(δ −3 ) number of lattice points (resp., jump points) in the rectangle R(ũ,t) from where a random walk can start. We are now ready to prove (T + 1 ) for the discrete time case µ 1 . The proof of (T + 1 ) for the continuous time caseμ 1 is similar and will be discussed afterwards.
Verification of (T + 1 ) for µ 1 . First we can assumet ∈ Z, since we can always replacet by ⌈t⌉ which only enlarges the event A + . The contribution to the event A + is either due to random walk paths that originate from within R(ũ,t), or paths that cross R(ũ,t) without landing inside it after the crossing. Denote the latter event by D(ũ,t). Then To estimate the third term in (3.3.1) (see Figure 3.2 for an illustration of the event), we first treat the case of a fixed (x, m) ∈ R(ũ,t). Suppressing (x, m) from π x,m , C i (x, m) and τ x,m i , we have The first part is bounded by where ω(δ) → 0 as δ → 0. The last inequality is due to the uniform integrability of the third moment of the overshoot distribution, which follows from our assumption E[Y 5 ] < +∞ and Lemma 2.0.6. For the second µ 1 probability in (3.3.2), denote the event that none of the conditions listed are violated by time t by G t . If τ 1 > t, we interpret an inequality like π(τ 1 ) ≤ 5 1 2ũ as not having been violated by time t. G t is then a nested family of events, and the second probability in (3.3.2) becomes Denote the history of the random walks π x,m , π 1 , π 2 , π 3 and π 4 up to time t by Π t , and denote expectation with respect to the conditional distribution of Π t conditioned on the event G t by E t . Then for k = 1, 2, 3, 4, where the µ 1 probability on the right hand side is conditioned on a given realization of Π τ k−1 ∈ G τ k−1 , which is a positive probability event. For any Π τ k−1 ∈ G τ k−1 , we have by the strong Markov property that where the inequality follows from Lemma 2.0.4 for δ sufficiently small. Thus where 2ũt = O(δ −3 ) and hence ω ′ (δ) → 0 as δ → 0. Thus the last two terms in (3.3.1) go to 0 as δ → 0, and the first term is of order o(t) after taking the limit δ → 0. Together they give (T + 1 ) for the measure µ 1 . Verification of (T + 1 ) forμ 1 . Analogous to (3.3.1), the event A + is contained in the union of three events: , one of the four walks π 1 , · · · , π 4 fails to stay within a distanceũ of its starting point x i before time 2t; (ii) the eventD(ũ,t; 3 2ũ , 2t), some interpolated random walk path first intersects R(ũ,t), and then lands at a jump point outside R( 3 2ũ , 2t); (iii) or the event that π 1 , · · · , π 4 all stay within a distanceũ of its starting point x i before time 2t, and some random walk path π x,m starting from one of the jump points (x, m) ∈ R( 3 2ũ , 2t) (by definition, there are two random walk paths starting from any jump point, here we take the random walk path that jumps immediately) exits from the right boundary of R(17ũ, 2t) without first coalescing with any of the π ′ i s. Using the continuous time version of Lemma 2.0.3, the probability of the first event ∪ 4 i=1 B c i is of order o(t) after taking the limit δ → 0. For the third event, note that conditioned on the existence of a jump point at (x, t) ∈ R( 3 2ũ , 2t), by the Markov property of Poisson process, we can apply the computations in (3.3.2)-(3.3.4) to find that the probability of a random walk starting from (x, t), jumpping immediately, and exiting the right boundary of R(17ũ, 2t) while the events B i all hold is of order o(δ 3 ). Since the expected number of jump points in R( 3 2ũ , 2t) is of order δ −3 , the probability of the third event is of the order δ −3 o(δ 3 ), which tends to 0 as δ → 0. To conclude the proof of (T + 1 ) forμ 1 , it then only remains to show that the probability of the second event, , 2t)] is essentially the same as that forμ 1 (I L ′ ) in our earlier proof of the almost sure precompactness ofX 1 in Lemma 3.1.1. In Figure 3.1, we replace the inner square Λ L by R(ũ,t), and the outer square Λ L ′ by R(0, −t; 3 2ũ , 3t). We can assume that for all there is at least one poisson clock ring during the time interval [t, 2t], since the probability of the complentary event tends to 0 as δ → 0. Then no constantposition line segment inX 1 can intersect R(ũ,t) without landing at a jump point in R( 3 2ũ , 2t). For non-constant-position line segments inX 1 that originate from jump points inside R(0, −t; 3 2ũ , 3t) and intersect R(ũ,t) without landing at jump points in R( 3 2ũ , 2t), the probability is bounded by the expected number of jump points in R(0, −t; 3 2ũ , 3t), which is of order δ −3 , times the probability that the random walk increment Y has |Y | >ũ/2. Since E[|Y | 3 ] < +∞, this product tends to 0 as δ → 0. To estimate the probability of having line segments in X 1 that originate outside R(0, −t; 3 2ũ , 3t) and intersect R(ũ,t), the computation is exactly the same as that forμ 1 (I L ′ ) in our earlier proof of the almost sure precompactness ofX 1 . Assuming E[|Y | 3 ] < +∞, we find that the probability of such events also tend to 0 as δ → 0.
After translation in space and time, the event C defined in Section 3.2 at the end of the verification of (B ′ 1 ) for the continuous time case is then easily seen to be a subset of the eventD(ǫ,t/2; 2ǫ,t). Thereforeμ 1 (C) → 0 as δ → 0.
Let X 0 T δ with distribution µ 0 T δ (resp.,X 0 T δ andμ 0 T δ ) denote the (H, F H )valued random variable consisting of interpolated discrete time (resp., continuous time) coalescing random walk paths on the rescaled lattice starting with one walker at every site in (δ/σ)Z at time 0. We expect tightness for {X 0 T δ } and {X 0 T δ } to hold under much weaker moment assumptions on the random walk increment Y . Indeed, form a tight family of (H, F H ) valued random variables.
Proof. We only prove the lemma for Recalling the arguments leading to the formulation of the tightness condition (T 1 ) in [13], a sufficient condition for the family of measures {µ 0 T δ } on (H, F H ) to be tight is that, for some nonnegative integer m. By examining the locations of the path (f, 0) at time mt and (m + 1)t, we see that there exists a nonnegative integer m 0 (either m or m + 1) and a time m 0 t < t ′ ≤ (m 0 +1)t (either t 1 , t 2 or (m+1)t), such that either (1) |f (m 0 t)| ≤ 2L and |f (t ′ ) − f (m 0 t)| ≥ u/4; or (2) |f (m 0 t)| > 2L and |f (t ′ )| ≤ L. We will call the events that X 0 T δ ∈ F u,t;L,T and X 0 T δ contains a path (f, 0) satisfying either condition (1) or condition (2) respectively event (1) and event (2). Then the event {X 0 T δ ∈ F u,t;L,T } is a subset of the union of events (1) and (2).
LetĀ t,u (x 0 , t 0 ) denote the event (in F H ) that K (in H) contains a path touching the bottom of R(x 0 , t 0 ; u/16, t) and the left or right boundary of R(x 0 , t 0 ; u/8, 2t). Then event (1) is a subset of ∪ (x 0 ,t 0 )∈L D ×T DĀ t,u (x 0 , t 0 ). By the same argument as in the verification of (T 1 ) for µ δ , we have which also holds under the assumption E[|Y | 3 ] < +∞. This is because in the verification of (T 1 ), E[|Y | 5 ] < +∞ is used in (3.3.3) to guarantee the random walk overshoot has finite third moment, which is then used in a Markov inequality to outweigh the O(δ −3 ) number of rescaled lattice points in R(x 0 , t 0 ; u, t).
For the eventĀ t,u (x 0 , t 0 ), we are only concerned with random walks starting at the bottom of R(x 0 , t 0 ; u/16, t), which contains O(δ −1 ) number of rescaled lattice points. Therefore we only need finite first moment for the random walk overshoot, which translates into finite third moment for Y .
which tends to 0 as t ↓ 0. On the other hand, recall the notation ξ B s for a system of coalescing random walks on Z × Z starting with one walker at every site in B ⊂ Z at time 0, we have lim sup where α is some positive constant depending only on the random walk increment Y . Observe that a nondegenerate random walk with mean zero and finite variance starting at 0 will at any later time have a minimal probability α > 0 (independent of time) of being on the negative axis. If we condition on the time and location when some walker in ξ which is also equivalent to the event that the right boundary of the corresponding voter model interface r s satisfies rt ≥L at timet (see Section 4.2 for more details on the voter model interface). A result of Cox and Durrett [7] states that if the random walk increment Y has finite third moment, then r s /(σ √ s) converges in distribution to a standard Gaussian variable as s → +∞. Therefore Therefore lim t↓0 lim sup δ↓0 µ 0 T δ [ event(2) ] = 0. Together with our previous estimate for the event (1), this establishes (T 0 ), and hence the lemma.

Verification of (I 1 )
Our verification of (I 1 ) follows a similar line of argument as in the paper of Ferrari, Fontes and Wu [11]. We define three sets of random walks: {π i δ } 1≤i≤m , a family of m independent random walks on the rescaled lattice (δ/σ)Z × δ 2 Z ((δ/σ)Z × δ 2 R for continuous time); {π i δ,f } 1≤i≤m , the family of m coalescing random walks constructed from {π i δ } by applying a mapping f to {π i δ } such that two walks coalesce as soon as their paths coincide (recall that π i δ denote the piecewise constant version of the random walk path); and {π i δ,g } 1≤i≤m , an auxiliary family of m coalescing walks constructed by applying a mapping g to {π i δ } such that two walks coalesce as soon as their paths cross (i.e., coincide or interchange relative order; note that random walks in {π i δ,g } coalesce earlier than they do in {π i δ,f }). Here {π i δ }, {π i δ,f } and {π i δ,g } all denote the piecewise constant version of the random walk paths. We will denote their linearly interpolated counterpart by {κ i δ }, {κ i δ,f } and {κ i δ,g }. If we pretend for the moment that weak convergence makes sense for piecewise constant paths without resorting to Skorohod topology, then by Donsker's invariance principle, {π i δ } "converge weakly" to a family of independent Brownian motions {B i } 1≤i≤m . As we will see, the mapping g is almost surely continuous with respect to {B i }, and {B i g } 1≤i≤m is distributed as coalescing Brownian motions. Therefore by the Continuous Mapping Theorem for weak convergence, {π i δ,g } "converge weakly" to the coalescing Brownian motions {B i g }. Finally to show that {κ i δ,f } also converges weakly to {B i g }, we will prove that the distance between the two versions of coalescing walks {π i δ,f } and {π i δ,g } converges to 0 in probability, and the distance between the linearly interpolated version {κ i δ,f } and the piecewise constant version {π i δ,f } also converges to 0 in probability.
We introduce more notation. Let D be any deterministic countable dense subset of R 2 . Let y 1 = (x 1 , t 1 ), . . . , y m = (x m , t m ) ∈ D be fixed, and let B 1 , ..., B m be independent Brownian motions starting from y 1 , ..., y m . For a fixed δ, denote ⌈ỹ i ⌉ = (⌈x i ⌉, ⌈t i ⌉) (resp., ⌈ỹ i ⌉ = (⌈x i ⌉,t i ) for the continuous time case), wherex i = σδ −1 x i andt = δ −2 t i as defined in Chapter 2, and let y i δ denote ⌈ỹ i ⌉'s space-time position after diffusive scaling on the rescaled lattice (δ/σ)Z × δ 2 Z (resp., (δ/σ)Z × δ 2 R). Letπ i (i = 1, · · · , m) be independent random walks in the Z × Z (resp., Z × R) lattice starting from ⌈ỹ i ⌉. We regard (B 1 , ..., B m ), and (π 1 , ...,π m ) as random variables in the product metric space and d is defined in (1.2.4); thus d * m gives the product topology on Π m . We will also need the metric andd * m is defined in a similar way as d * m . If we denote the space of paths that are right continuous with left limits byΠ, and letΠ m be the product space, then d, d * m ,d andd * m are still well defined metric onΠ andΠ m . We now define a mapping g from (Π m , d * m ) to (Π m , d * m ) that constructs coalescing paths from independent paths. The construction is such that when two paths first cross (i.e., coincide or interchange relative order), the path with the higher index will be replaced by the path with the lower index after the time of intersection or order exchange. This procedure is then iterated until no more intersections take place. To be explicit, we give the following algorithmic construction.
Let (ξ 1 , . . . , ξ m ) ∈Π m , and let T i,j g denote the time when the two paths ξ i and ξ j first intersect or interchange relative order. We start with equivalence relations on the set {1, . . . , m} by setting i ≁ j ∀ i = j. We then define the one step iteration Γ on (ξ 1 , . . . , ξ m ) and the equivalence relations by 4.5) and update equivalence relations by assigning i ∼ i * . Iterate the mapping Γ, and label the successive intersection times τ g by τ k g . Then the iteration stops when τ k g = +∞ for some k ∈ {1, 2, . . . , m}, i.e., either there is no more crossing among the different equivalence classes of paths, or all the paths have coalesced and formed a single equivalence class. Denote the final collection of paths by g(ξ 1 , . . . , ξ m ) = (ξ 1 g , . . . , ξ m g ). Then it's clear by the strong Markov property, that (B 1 g , . . . , B m g ) has the distribution of coalescing Brownian motions. However (π 1 g , . . . ,π m g ) is not distributed as coalescing random walks, because for nonsimple random walks, paths can cross before the random walks actually coalesce (by being at the same space-time lattice site).
Proof. From the definition of d andd in (1.2.4) and (3.4.2), it is clear that d((f 1 , t 1 ), (f 2 , t 2 )) ≤d((f 1 , t 1 ), (f 2 , t 2 )) for any (f 1 , t 1 ), (f 2 , t 2 ) ∈Π. Therefore it is sufficient to prove the lemma with d * m replaced byd * m . In terms of random walks in the unscaled lattice, the lemma can be stated as ∀ ǫ > 0, P{d * m [(π 1 f , . . . ,π m f ), (π 1 g , . . . ,π m g )] ≥ǫ} → 0 as δ → 0 + . (3.4.7) We first prove (3.4.7) for m = 2. Note that for m = 2,π 1 f =π 1 g =π 1 , hencē d * 2 [(π 1 f ,π 2 f ), (π 1 g ,π 2 g )] =d (π 2 f ,π 2 g ). LetT 1,2 g denote the first time whenπ 1 and π 2 cross, and letT 1,2 f denote the first time when the two walks coincide. Also let l(0, n) denote the maximum distance over all time between two coalescing random walk paths π 0,0 and π n,0 starting at 0 and n at time 0. Then by the strong Markov property, and conditioning at timeT 1,2 g , The first probability in the summand converges to a limiting probability distribution as δ → 0 by applying Lemma 2.0.5 to (π 1 −π 2 ). The second probability converges to 0 for every fixed n by Lemma 2.0.1. This proves (3.4.7) for m = 2. For m > 2, letT i,j f andT i,j g denote respectively the first time when the two independent walksπ i andπ j coincide or interchange relative order. As usual, let T i,j δ,f = δ 2T i,j f and T i,j δ,g = δ 2T i,j g . By Donsker's invariance principle, the interpolated paths (κ 1 δ , · · · , κ m δ ) converge in distribution to (B 1 , · · · , B m ) as (Π m , d * m ) valued random variables. By Skorohod's representation theorem [4,10], we may assume this convergence is almost sure, i.e., d * m [(κ 1 δ , · · · , κ m δ ), (B 1 , · · · , B m )] → 0 almost surely. Then by the properties of standard Brownian motions, we also have d * m [(π 1 δ , · · · , π m δ ), (B 1 , · · · , B m )] → 0 almost surely. Note that the crossing times T i,j δ,g as functions from (Π m , d * m ) to R are almost surely continuous with respect to (B 1 , · · · , B m ). Therefore almost surely, T i,j δ,g → τ i,j as δ → 0, where τ i,j is the time of first crossing between B i and B j ; and {T i,j g } 1≤i<j≤m converge jointly in distribution to {τ i,j } 1≤i<j≤m . By the standard properties of Brownian motion, {τ i,j } 1≤i<j≤m are almost surely all distinct. By an argument similar to (3.4.8), we also have sup 1≤i<j≤m |T i,j δ,f −T i,j δ,g | → 0 in probability. Note that in our definition of the mapping g that constructs (π 1 g , · · · ,π m g ) from (π 1 , · · · ,π m ), the successive times of crossing {τ k g } 1≤k≤m−1 , are all times of first crossing between independent paths, i.e., {τ k g } 1≤k≤m−1 ⊂ {T i,j g } 1≤i<j≤m . The event in (3.4.7) can only occur due to: either (1) for some τ k g in the definition of g, with τ k g =T i,j g for some i and j, τ k+1 g ≤T i,j f ; or else, (2) whenever a coalescing takes place between two pathsπ i ,π j in the mapping g, the same two paths will coalesce in the mapping f before another coalescing takes place in the mapping g, and the event in (3.4.7) occurs because for some τ k g with τ k g =T i,j g , the distance between the two pathsπ i andπ j during the time interval [T i,j g ,T i,j f ] exceeds ǫ. The probability of the event (1) tends to 0 as δ → 0 by our observations that {T i,j δ,g } 1≤i<j≤m converges jointly in distribution to {τ i,j } 1≤i<j≤m , which are almost surely all distinct, and the fact that sup 1≤i<j≤m |T i,j δ,f − T i,j δ,g | → 0 in probability. The probability of the event (2) tends to 0 by our proof of (3.4.7) for m = 2. This proves (3.4.7) and Lemma 3.4.1.
Verification of (I 1 ). It is sufficient to show that for any sequence of δ n ↓ 0, we can find a subsequence δ ′ n ↓ 0, such that (κ 1

Verification of (E 1 )
As usual, we start with some notation. For an (H, F H )-valued random variable X, define X s − to be the subset of paths in X which start before or at time s, and for s ≤ t define X s − ,t T to be the set of paths in X s − truncated before time t, i.e., replacing each path in X s − by its restriction to time greater than or equal to t. When s = t, we denote X s − ,s T simply by X s T . Also let X(t) ⊂ R denote the set of values at time t of all paths in X. Note thatη X (t 0 , t; a, b) = |X t − 0 (t 0 + t) ∩ (a, b)|. We may sometimes abuse the notation and use X(t) also to denote the set of points We recall here the definition of stochastic domination as given in [13]. For two measures µ 1 and µ 2 on (H, F H ), µ 2 is stochastically dominated by µ 1 (µ 2 << µ 1 ) if for any bounded increasing function f on (H, When µ 1 , µ 2 are the distributions of two (H, F H )-valued random variables X 1 and X 2 , we will also denote the stochastic domination by X 2 << X 1 . The first step of our proof is to reduce (E 1 ) to the following condition: Lemma 3.5.1 Assuming {X n } is a tight family of (H, F H )-valued random variables, then (E ′ 1 ) implies (E 1 ).
where X ′ is equally distributed with X and Z t 0 +ǫ is the weak limit of X (t 0 +ǫ) − n ′ i . By Skorohod's representation theorem (see, e.g., [4,10]), we may assume the convergence is almost sure. Then almost surely, any path (f, t) ∈ X ′ with t ≤ t 0 is the limit of a sequence of paths (f n ′ i , t n ′ i ) ∈ X n ′ i with t n ′ i → t. Since (f n ′ i , t n ′ i ) is eventually in X (t 0 +ǫ) − n ′ i , we also have (f, t) ∈ Z t 0 +ǫ . Therefore X ′ t − 0 ⊂ Z t 0 +ǫ almost surely, and X ′ t − 0 << Z t 0 +ǫ . Since X ′ t − 0 is equally distributed with X t − 0 , the Lemma then follows.
We now cast the condition (E ′ 1 ) in terms of our random variables {X δ } and {X δ }. Let Z t 0 be any subsequential limit of X t − 0 δ (orX t − 0 δ ). Then the validity of (E ′ 1 ) for {X δ } and {X δ } is a consequence of the following two lemmas, which are also what one needs to establish to verify (E 1 ) for general models other than coalescing random walks {X δ } and {X δ }.
Lemma 3.5.2 Let Z t 0 (t 0 + ǫ) ⊂ R × {t 0 + ǫ} be the intersections of paths in Z t 0 with the line t = t 0 + ǫ. Then for any ǫ > 0, Z t 0 (t 0 + ǫ) is almost surely locally finite. , the set of paths in Z t 0 (which all start at time t ≤ t 0 ) truncated before time t 0 + ǫ, is distributed as B Zt 0 (t 0 +ǫ) , i.e., coalescing Brownian motions starting from the random set Z t 0 (t 0 + ǫ) ⊂ R 2 .
Since 0 < ǫ < t is arbitrary, letting ǫ → 0 establishes (E ′ 1 ) for {X δ } and {X δ }. Recall that Γ δ andΓ δ denote the piecewise constant version of X δ andX δ . Lemma 3.5.2 is a consequence of the following: Lemma 3.5.4 ∀ t 0 , t, a, b ∈ R with t > 0 and a < b, we have for some 0 < C < +∞ independent of t 0 , t, a and b. The same is true forΓ δ .
deterministic and ρ P (A δ , A) → 0 as δ → 0. Note that {X A δ δ } is tight since X A δ δ is almost surely a subset of X δ and {X δ } is tight. If Z is a subsequential limit of X A δ δ , then by (I 1 ) and the remark following Corollary 3.2.1, there is µ Z almost surely exactly one path starting from every y ∈ A, and the finite dimensional distributions of Z are those of coalescing Brownian motions. Therefore Z is equidistributed with B A , which proves the deterministic case. . Since A δ converges in distribution to A, by Skorohod's representation theorem [4,10], we can construct random variables A ′ δ and A ′ which are equidistributed with A δ and A, such that A ′ δ (ω) → A ′ (ω) in ρ P almost surely. Then for almost every ω in the probability space where A ′ δ and A ′ are defined, by the part of the proof already done (for deterministic A δ and A), X A ′ δ (ω) δ converges in distribution to B A ′ (ω) . Thus f δ (A ′ δ (ω)) = E[f (X initial configuration at time t 0 + ǫ, and U we have where in (4.1.3), we applied Lemma 2.0.8, and in (4.1.4), we applied Lemma 2.0.7. To apply Lemma 2.0.8, we have implicitly assumed δ −2 n ∈ N. If δ −2 n / ∈ N, then we need to approximate and use an argument similar to the one leading to the computation in (3.2.2). This establishes (4.1.2), thus proving any subsequential limit of X 0 T δn (1) must be a simple point process. We now show the convergence of the avoidance functions. Let A = n i=1 [a i , b i ] be the disjoint union of a finite number of finite intervals. By Lemma 4.1.1, X 0 T δ converges weakly toW 0 as (H, F H )-valued random variables, so by Skorohod's representation theorem, we may assume this convergence is almost sure. In particular, X 0 T δ (1) converges almost surely toW 0 (1) in ρ P as defined in (3.5.1). SinceW 0 (1) is a stationary simple point process with intensity 1/ √ π, P[∂A ∩W 0 (1) = ∅] = 0. It is then easy to see that 1 X 0 T δ (1)∩A=∅ → 1W0 (1)∩A=∅ almost surely. By the bounded convergence theorem, lim δ↓0 P(X 0 T δ (1) ∩ A = ∅) = P(W 0 (1) ∩ A = ∅), thus proving the the convergence of avoidance functions and the theorem.