Perturbations of the Voter Model in One-Dimension

We study the scaling limit of a large class of voter model perturbations in one dimension, including stochastic Potts models, to a universal limiting object, the continuum voter model perturbation. The perturbations can be described in terms of bulk and boundary nucleations of new colors (opinions). The discrete and continuum (space) models are obtained from their respective duals, the discrete net with killing and Brownian net with killing. These determine the color genealogy by means of reduced graphs. We focus our attention on models where the voter and boundary nucleation dynamics depend only on the colors of nearest neighbor sites, for which convergence of the discrete net with killing to its continuum analog was proved in an earlier paper by the authors. We use some detailed properties of the Brownian net with killing to prove voter model perturbations convergence to its continuum counterpart. A crucial property of reduced graphs is that even in the continuum, they are finite almost surely. An important issue is how vertices of the continuum reduced graphs are strongly approximated by their discrete analogues.


General voter model perturbations
Motivated by problems in Biology and Statistical Physics (see e.g., [MDDGL99] [NP00] [OHLN06]), Cox, Durrett, and Perkins [CDP13] considered a class of interacting particle systems on Z d , called voter model perturbations (or VMP), whose rates of transition are close to the ones of a classical voter model. Informally, they considered models whose transition rates are parametrized by with → 0, so that the transition rates c x,η (j) (the transition rate of site x to state j given a configuration η for the model with parameter ) converges to the rate of a (possibly non-nearest neighbor) zero-drift voter model. For d ≥ 3, and the number q of opinions or colors at each site equal to 2 (i.e. for spin systems) the authors show that, under mild technical assumptions, the properly rescaled local density of one of the colors converges to the solution of an explicit reaction diffusion equation. They are then able to combine this convergence result together with some percolation arguments to show some properties of the particle system when is small enough.
In this work, we consider a similar problem when d = 1. We will construct a natural continuous object -the continuum voter model perturbation (CVMP)-which will be seen to be the scaling limit of a certain type of voter model perturbation. As we shall see, such a limit can not be described in terms of a reaction-diffusion equation anymore (contrary to the case d ≥ 3), but directly in terms of a duality relation with the Brownian net and Brownian net with killing. The former is a family of one-dimensional branching-coalescing Brownian motions, as first introduced by Sun and Swart [SS08] and then studied further by Newman, Ravishankar and Schertzer [NRS08]; the latter is an extension introduced in [NRS15]. See also [SSS15] for a recent review of those objects.
We will use some of the properties of the Brownian net to define continuous versions of the voter model perturbations, and show some properties of these continuous objects. We expect that a large class of VMP's in one dimension in both discrete and continuous time settings will converge to (the universal limiting object) CVMP. Our goal in this paper is to prove the convergence of a particular class of (discrete time) VMP to CVMP. In the spirit of [CDP13], we hope that our results will provide some insights into a large class of interacting particle systems which are close enough to the voter model in dimension one.
with boundary noise {g i,j } i,j and bulk noise p .
We note that in this particular case, the transition rates only depend on the states of the two nearest neighbors, but not on the current state of the site under consideration. As we shall see, this hypothesis is made mostly for technical reasons and is done solely to take advantage of the known convergence result of coalescing-branching discrete (time and space) random walks to the Brownian net; such a result for the continuous time or non-nearest neighbor version has still to be established. The continuous time setting, together with the study of more general voter model perturbations in one dimension (including the non-nearest neighbor case and more general boundary nucleation mechanism) will be the subject of future work.
Before stating our main theorem, we recall that the dynamics we consider are all nearest-neighbor in that the update probabilities of η t (x) at a particular lattice site x depend only on (η t (x − 1), η t (x + 1)). This property (which is also the case for the standard nearest neighbor voter model) implies that the Markov chain is reducible with complete independence between {η t (x)} on the even (Z 2 even = {(x, t) : x + t is even}) and odd (Z 2 odd = {(x, t) : x + t is odd}) subsets of discrete space-time. Thus one may restrict attention to one of these irreducible components. Given a realization of the odd component of the voter model perturbation with parameter , we define θ (x, t) to be the color of site (x, t) at time t. In particular, θ (·) defines a random mapping from Z 2 odd ∩ {(x, t) ∈ Z 2 : t ≥ 0} to {1, · · · , q}. Analogously, we will define the CVMP as a mapping from R × R + to the set of subsets of {1, · · · , q}, allowing one space-time point to take several colors. The next theorem enumerates some of the properties of the CVMP. The precise construction and description will be carried out in Section 3. Theorem 1.1. Let λ, {g i,j }, p be a family of probability distributions on {1, · · · , q}, with g i,i = δ i . Let b, κ ≥ 0. There exists a random mapping (x, t) → θ(x, t) from R × R + to subsets of {1, · · · , q}, with the following properties: 1. For every deterministic (x, t), |θ(x, t)| = 1 a.s.. 2. (Coarsening) For every deterministic t, |θ(x, t)| ≤ 2 a.s.. Moreover, the set {x : |θ (x,t) | = 2} is locally finite and partitions the line into intervals of uniform color, i.e. the color of x → θ(x, t) between two consecutive points of this set remains constant.
3. (Scaling Limit) Let { n } be a sequence of positive real numbers converging to 0. Let θ n (·, ·) := θ n (·, ·) be the simple voter model perturbation characterized by the boundary noise and bulk noise ({g n i,j } 1≤i,j≤q , p n ), with branching and killing parameters (b n , κ n ) such that (i) there exists b, κ ≥ 0 such that b n / n → b, and κ n / 2 n → κ as n → ∞, (ii) there exists ({g i,j } 1≤i,j≤q , p) a family of probability distributions on {1, · · · , q} such that ∀i, j, g n i,j → g i,j , and p n → p as n → ∞, (iii) the initial distribution of the particle system is given by a product measure ν (not depending on n) with one-dimensional marginal given by λ.
(1.4) Remark 1.2. In the Appendix, we show that the stochastic Potts model in one-dimension at low temperature -i.e., high β -is a VMP. Further, we shall see that the scaling in Theorem 1.1(3)(i) emerges naturally in this setting. See Remark 5.2 in the Appendix.

Duality to branching-coalescing-killling random walks
In this section, we start by introducing a percolation model on Z 2 odd (already introduced in [MNR13]) and we show how a random coloring algorithm of this percolation configuration is dual to simple voter model perturbations.
An oriented percolation model. Let odd -where x is interpreted as a space coordinate and t as a time coordinate -has two nearest neighbors with higher time coordinates -v r = (x + 1, t + 1) and v l = (x − 1, t + 1) -and v is randomly (and independently for different v's) connected to a subset of its neighbors v r and v l according to the following distribution.
• With probability , draw the arrow (v → v l ) (i.e. starting at v and ending at v l ); • with probability draw the arrow (v → v r ); • with probability b, draw the two arrows (v → v r ) and (v → v l ); • finally do not draw any arrow with probability κ. See Fig. 1.
If we denote by E b,κ the resulting random arrow configuration, the random graph G b,κ = (Z 2 odd , E b,κ ) defines a certain type of 1 + 1 dimensional percolation model oriented forward in the t-direction. In this percolation model, the vertices with two outgoing arrows will be referred to as branching points whereas points with no arrow will be referred to as killing points.
By definition, a path π along E b,κ will denote a path starting from any site of Z 2 odd and following the random arrow configuration until getting killed or reaching ∞. More precisely, a path π is the graph of a function defined on an interval in R of the form [σ π , e π ], with σ π , e π ∈ Z ∪ {∞} such that e π = ∞ or (π(e π ), e π ) is a killing point, for every t ∈ [σ π , e π ), (π(t), t) connects to (π(t + 1), t + 1) and π is linear between t and t + 1.
Considering the set of all the paths along E b,κ , one generates an infinite family -denoted by U b,κ -that can loosely be described as a collection of graphs of one dimensional coalescing simple random walks, that branch with probability b and get killed with probability κ. Roughly speaking, a walk at space-time site v can create two new walks (starting respectively at v l and v r ) with probability b and can be killed with probability κ; two walks move independently when they are apart but become perfectly correlated (i.e., they coalesce) upon meeting at a space-time point. In the following, U b,κ will be referred to as a system of branching-coalescing-killing random walks (or in short, BCK) with parameters (b, κ), or equivalently, and in analogy with their continuum Backward discrete net. Our percolation model is oriented forward in time. We define the backward BCK -denoted byÛ b,κ -as the backward in time percolation model obtained from the BCK by a reflection about the x-axis. See Fig. 2. A coloring algorithm. We now describe how simple voter model perturbations (as described in Section 1.3) are exactly dual to the backward BCK (i.e.,Û b,κ ). To describe this duality relation, we consider the subgraph ofĜ b,κ (the backward percolation model) whose set of vertices is given by Z 2 odd ∩ {(x, t) : t ≥ 0}. Then, we equip each vertex z ∈ Z 2 odd ∩ {(x, t) : t ≥ 0} with an independent uniform (on [0, 1]) random variable U z . We start by coloring the leaves as follows (see Fig. 3). The color assigned to a point z with time coordinate equal to 0 -denoted by θ (z) -is the only integer in {1, · · · , q} such that (With the convention that λ(0) = 0.) The color assigned to a killing point is the integer in {1, · · · , q} such that where we recall that p (·) is the probability distribution determining the transition when a bulk nucleation occurs (with the convention that p (0) = 0). For every other vertex z ∈ Z 2 odd ∩ {(x, t) : t ≥ 0}, we consider the component of our (backward) oriented percolation model originated from z and restricted to the upper half plane. This defines a certain acyclic directed graphĜ z = (V z ,Ê z ) whose vertices have out-degree at most 2 -see Fig. 5. For every point inV z , we assign a color to each of the nodes sequentially from the leaves to the root by applying the following algorithm, whereV z n will denote the set of color-assigned vertices at step n of the algorithm.
• Step 1V z 1 is the set of leaves, whose colors have already been assigned. • Step n> 1 IfV z \V z n = ∅ stop. Otherwise, pick a vertex z inV z such that z is connected toV z n−1 , i.e., all its nearest (directed) neighbor(s) belong toV z n−1 . Next, if z connects to a single vertex or if the color of its two neighbors match, assign to it the color of its neighbor(s). If the colors of the two neighbors disagree, with two colors k = l, then the site is assigned a color i such that g k,l (i + 1)).
We note that this specific choice for the bulk and boundary noises correspond to the discrete time stochastic Potts model considered in the Appendix (see Section 5.2).
Remark 1.3. At step n of the previous algorithm, there can be multiple z ∈ V z \V z n , such that all of its (directed) neighbors are inV z n . (In Fig 6, there are two such vertices -one connecting to the top white circle and one connecting to the top black circle). However, we let the reader convince herself that the final coloring of the graph is independent of the choice of z at step n. Remark 1.4. Our coloring is consistent in the sense that if z ∈V z , the color assigned to z in the coloring algorithm applied toV z is identical to θ (z ) -i.e., the coloring algorithm applied toV z .
Proof. Since coalescing random walks are the dual of the standard voter model [HL75] [ L04], it is clear that in the absence of killing and branching the color genealogy of the VMP is given by coalescing random walks. At a bulk nucleation site (killing point) a color independent of the previous color evolution is assigned to the site. This leads to the conclusion that if a path in the color genealogy evolution hits a bulk nucleation point then it should be killed at that point. The color at a boundary nucleation point is determined by the colors of the two adjacent sites. The colors of these two sites are determined by    following the color genealogy evolution starting at these sites until reaching time zero or hitting a killing point as given by the dual percolation model starting at these two sites. The uniform random variables at the bulk and boundary nucleation sites along with the rules for determining the color at these sites given earlier ensure that the colors at these sites are chosen with the distribution p and g k,l respectively.

Scaling limit
In [NRS15], we introduced the Brownian net with killing (or N b,κ where b and κ are continuum branching and killing parameters) that was shown to emerge as the scaling limit of a BCK system with small branching and killing parameters (see Theorem 2.6 below for a precise statement).
In light of the previous section, it is natural to construct the CVMP as a process dual to the Brownian net with killing; the duality relation between the two models being described in terms of the random coloring of the "continuum graphs" induced by the Brownian net.
The main difficulty in defining the CVMP lies in the fact that the set of vertices of this "continuum graph" is dense in R × R + . In order to deal with this extra difficulty, we now discuss some remarkable features of the coloring algorithm described in the previous section.
Definition 1.1 (Relevant Separation Point and Reduced Graph - Fig. 9-10). Let G = (V, E) be a finite acyclic oriented graph. We will say that an intermediate node z (i.e. a node which is neither a root nor a leaf) of the graph is relevant iff there exists two paths of G passing through z and reaching the leaves such that they do not meet between z and the leaves. Any other intermediary point will be called irrelevant.
We define the reduced version of G, denoted byG = (Ṽ ,Ẽ), as the oriented graph of G obtained after "skipping" all irrelevant points in V . More precisely, the vertices ofṼ are obtained from V by removing the set of irrelevant separation points and placing a directed edge between two points z, z inṼ iff there exists a path π = (z 1 , · · · , z n ) in G with z 1 = z and z n = z and such that z i is irrelevant for i = 1, n.  Proposition 1.6. Let G be a finite rooted acyclic oriented graph whose vertices have at most out degree 2 and let us assume that the vertices z of the graph are equipped with i.i.d. random variables {U z }, uniformly distributed on [0, 1]. For a given coloring of the leaves, our coloring algorithm (as described in Section 1.4) applied separately to G and to the reduced graphG induces the same coloring of the verticesṼ (using the same set of i.i.d. uniform random variables for the reduced graph).
Proof. If a vertex z connects to two edges with the same color, then z must align to this color (by definition of the boundary nucleation). The same holds if z only connects to a single point. It is then easy to see that any path in the graph G that does not contain any relevant separation points is uniformly colored, and as consequence, in our coloring algorithm, irrelevant separation points can be "skipped" to deduce the color of the verticesṼ (see Figs 5-8 for a concrete example).
Significantly, even if the graph induced by the Brownian net paths starting from a point is infinite, its reduced version is finite (see Fig. 13 below). This will allow us to apply our coloring algorithm at the continuum level and to define the CVMP by imitating the duality relation described in Section 1.4.

Perspectives
The discrete VMP's considered in Section 1.3 and Theorem 1.1 are dual to a system of coalescing-branching nearest neighbor random walks, that are known to converge (under proper rescaling) to the Brownian net with killing [NRS15].
Let us now reconsider the discrete VMP alluded to in Section 1.2, and whose transition probabilities are defined in (1.3). As for the nearest neighbor case, at least when f x,η = f x,η , it is not hard to see that such systems are dual to branching-coalescing random walks. However, the transition probabilities of such walks are more complex: (1) walks move according to the transition kernel K as described in (1.1); (2) they are killed with probability κ ; and finally (3) they branch with probability b , and upon branching, the location of the N new particles are chosen according Q (shifted by the current location of the particle under consideration).
It is plausible that (under some restrictions on the moments of the kernels K and Q) such systems of particles also converge to the Brownian net with killing. If so, the limiting object alluded to in Theorem 1.1 -and whose construction will be carried in Section 3 -should also be the scaling limit of a large class of discrete VMP's. However, it remains unclear what should be the limiting branching and killing mechanism at the continuum. Both the limit of generalized branching-coalescing random walks, and their relation to the CVMP are challenging questions, and would certainly require techniques that go beyond the scope of this paper.
Finally, it would be natural to also consider the continuous time analog of the models alluded above (as in [CDP13] for higher dimensions). In particular, some recent work by Etheridge, Freeman and Straulino [EFS15] where they considered the paths generated by the genealogy of a spatial Fleming-Viot process in one dimension could be relevant to make progress in this direction.

Outline of the rest of the paper
The rest of the paper is organized as follows. In Section 2, we recall the definition of the scaling limit of the BCK: the Brownian net with killing introduced in [NRS15], which can be loosely described as an infinite family of one dimensional coalescing-branchingkilling Brownian motions. In Section 3, this is used to construct the continuum voter model perturbation (CVMP) by mimicking at the continuum level the discrete duality relation of Section 1.4. Properties (1)-(2) of Theorem 1.1 will be shown in Proposition 3.7. Finally, we show the convergence result of Theorem 1.1 (Property (3)) in Section 4.

The Brownian net with killing
In Proposition 1.5, the nearest-neighbors discrete voter model perturbations were seen to be dual to an infinite family of BCK random walks. Hence, before defining a continuous analog of the voter model perturbation, it is natural to start with the construction of the BCK scaling limit: the Brownian net with killing (as introduced in [NRS15]), denoted by N b,κ -where b and κ are two non-negative numbers playing a role analogous to the branching and killing parameters in the discrete setting.

The space (H, d H )
As in [FINR04], we will define the Brownian net with killing as a random compact set of paths. In this section, we briefly outline the construction of the space of compact sets. For more details, the interested reader may refer to [FINR04] and [NRS15].
Next, let C[t 0 , t 1 ] denote the set of continuous functions from [t 0 , t 1 ] to [−∞, +∞]. From there, we define the set of continuous paths in R 2 (with a prescribed starting and ending point) as Finally, we equip this set of paths with a metric d, defined as the maximum of the sup norm of the distance between two paths, the distance between their respective starting points and the distance between their ending points. (In particular, when no killing occurs, as in the forward Brownian web, the ending point of each path is {∞}). More precisely, if for any path π, we denote by σ π the starting time of π and by e π its ending time, we have whereπ is the extension of π into a path from −∞ to +∞ by settinḡ π(t) = π(σ π ) for t < σ π , π(e π ) for t > e π .
Finally, let H denote the set of compact subsets of (Π, d) and let d H denote the Hausdorff metric, In [FINR04], it is proved that (H, d H ) is Polish. In the following, we will construct the Brownian net with killing as a random element of this space.

The Brownian web
As mentioned in the introduction, the Brownian web ([TW98] [FINR04] [SSS15]) is the scaling limit of the discrete web under diffusive space-time scaling and is defined as an element of (H, d H ). The next theorem, taken from [FINR04], gives some of the key properties of the BW.
Theorem 2.1. There is an (H, F H )-valued random variable W 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 B (x,t) starting from (x, t). (i) for any deterministic, dense countable subset D of R 2 , almost surely, W is the closure in (Π, d) of {B (x,t) : (x, t) ∈ D}. (ii) for any deterministic n and (x 1 , t 1 ), . . . , (x n , t n ), the joint distribution of B (x1,t1) , . . . , B (xn,tn) is that of coalescing Brownian motions from those starting points (with unit diffusion constant).
Note that (i) provides a practical construction of the Brownian web. For D as defined above, construct coalescing Brownian motion paths starting from D. This defines a skeleton for the Brownian web that is denoted by W(D). W is simply defined as the closure of this precompact set of paths.

Idea of the construction
When κ = 0, Sun and Swart give a construction of the Brownian net which is based on the construction of two coupled drifted Brownian webs (W l , W r ) that interact in a sticky way. For instance, for every point (x, t), there is a unique pair (l, r) ∈ (W l , W r ) starting from (x, t), with l ≤ r, and where where l is distributed as the path of a Brownian motion with drift −b, and r is a Brownian motion with drift +b. The difference between the two paths is a Brownian motion with sticky reflection at 0 (see [SS08] and a recent review paper [SSS15] for more details).
The Brownian net N b is then constructed by concatenating paths of the right and left webs. More precisely, given two paths π 1 , π 2 ∈ Π, let σ π1 and σ π2 be the starting times of those paths. For t > σ π1 ∨ σ π2 (note the strict inequality), t is called an intersection time of π 1 and π 2 if π 1 (t) = π 2 (t). By hopping from π 1 to π 2 , we mean the construction of a new path by concatenating together the piece of π 1 before and the piece of π 2 after an intersection time. Given the left-right Brownian web (W l , W r ), let H(W l ∪ W r ) denote the set of paths constructed by hopping a finite number of times between paths in W l W r . N b is then defined as the closure of H(W l W r ). After taking this limit, W l and W r delimit the net paths from the left and from the right, in the sense that paths in N b can not cross any element of W l (resp., W r ) from the left (resp., from the right).
Finally, we cite from [SS08] a crucial property of the Brownian net that will be useful for the rest of this paper.
Proposition 2.2. Let S < T and define the branching-coalescing point set For almost every realization of the Brownian net, the set ξ S (T ) is locally finite.

Special points of the standard Brownian net
In this section, we recall the classification of the special points of the Brownian net, as described in [SSS09]. As in the Brownian web, the classification of special points will be based on the local geometry of the Brownian net. Of special interest to us will be the points with a deterministic time coordinate.
Definition 2.1 (Equivalent Ingoing and Outgoing Paths). Two paths π, π ∈ (Π, d) are said to be equivalent paths entering a point (x, t) = z ∈ R 2 , or in short π ∼ z in π , iff there exists a sequence {t n } converging to t such that t n < t and π(t n ) = π (t n ) for every n. Equivalent paths exiting a point z, denoted by π ∼ z out π , are defined analogously by finding a sequence t n > t with {t n } converging to t and π(t n ) = π (t n ).
Despite the notation, these are not in general equivalence relations on the spaces of all paths entering resp. leaving a point. However, in [SSS09], it is shown that that if (W l , W r ) is a left-right Brownian web, then a.s. for all z ∈ R 2 , the relations ∼ z in and ∼ z out actually define equivalence relations on the set of paths in W l ∪ W r entering (resp., leaving) z, and the equivalence classes of paths in W l ∪ W r entering (resp., leaving) z are naturally ordered from left to right. Moreover, the authors gave a complete classification of points z ∈ R 2 according to the structure of the equivalence classes in W l ∪ W r entering (resp., leaving) z.
In general, such an equivalence class may be of three types. If it contains only paths in W l then we say it is of type l, if it contains only paths in W r then we say it is of type r, and if it contains both paths in W l and W r then we say it is of type p, standing for pair. To denote the type of a point z ∈ R 2 in a Brownian net N b , we list the incoming equivalence , and all of these types occur.
3. Every path π ∈ N b starting from the line R×{t}, is squeezed between an equivalent pair of right-most and left-most paths, i.e. there exists l ∈ W l and r ∈ W r so that l ∼ z out r such that l ≤ π ≤ r on [t, ∞). 4. Any point (x, t) entered by a path π ∈ N b with σ π < t is of type (p, p). Moreover, π is squeezed between an equivalent pair of right-most and left-most paths, i.e. there exist l ∈ W l , r ∈ W r with r ∼ z in l and > 0 and such that l ≤ π ≤ r on [t − , t].
In the Brownian net, a separation point (or branching point) will refer to a point z = (x, t) such that there exist two paths π r and π l ∈ N b with σ π l , σ πr < t so that (1) π l (t) = π r (t) = x and (2) there exists > 0 so that π l (s) < π r (s) for every s ∈ (t, t + ]. Proposition 2.4 (Geometry of the Net at a Separation Point). If z = (x, t) is a separation point in the standard Brownian net N b then 1. z is of the type (p,pp). 2. Every path π starting from z is squeezed between some pair of equivalent rightmost and left-most paths -i.e. there exists l ∈ W l and r ∈ W r (which will depend on π) with l ∼ z out r such that l ≤ π ≤ r on [t, ∞). 3. Every path π entering the point z is squeezed between a pair of equivalent rightmost and left-most paths -i.e., there exist l ∈ W l , r ∈ W r , l ∼ z in r, entering the point z and > 0 such that l ≤ π ≤ r on [t − , t].

The Brownian net with killing
Construction. Having the standard Brownian net at hand, we now need to turn on the killing mechanism. To do that, we recall the definition of the time length measure of the Brownian net as introduced in [NRS15].

Definition 2.2 (Time Length Measure of the Brownian Net). For almost every realization
of the standard Brownian net, T is the unique σ-finite measure such that for every Borel set E ⊂ R 2 , where (p, p) refers to the set of (p, p) points for the net N b .
Given a realization of the standard Brownian net N b , we define the set of the killing marks M κ as a Poisson point process with intensity measure κT . Finally, we define the Brownian net with killing as the union of (1) all the paths π ∈ N b killed at and (2) for every z ∈ R 2 , the trivial path whose starting point and ending point coincide with z. In the following, this construction will be denoted as N b,κ .
Let E ⊆ R 2 . We will denote by N b,κ (E) the subset of paths in N b,κ (E) with starting points in E. Finally, M κ (S) will denote the set of killing points that are attained by π ∈ N b,κ with σ π = S. We cite the following result from [NRS15] (Proposition 2.14 therein).
Coupling between the net and the net with killing. Our construction of N b,κ induces a natural coupling between N b and N b,κ , and in the following, N b will refer to the net used to construct the net with killing N b,κ . Let U b ,κ denote the discrete BCK (or discrete net with killing) with branching and killing parameters (b , κ ), and U b := U b ,0 .
Like in the continuum, there is a natural and analogous coupling between the discrete objects U b and U b,κ , and again, U b will always refer to the discrete net coupled with U b,κ . (This coupling is obtained by starting from U b and by removing all the outgoing arrows independently at every vertex with probability κ).

Convergence.
In [NRS15], we show that the killed Brownian net is the scaling limit of the BCK system for small values of the branching and killing parameters. In the following, for every element U ∈ H, S (U ) is the set paths obtained after scaling the x-axis by and the t-axis by 2 .
Theorem 2.6. (Invariance Principle) Let b, κ ≥ 0 and let {b n } n>0 and {κ n } n>0 two sequences of non-negative numbers such that b n +κ n ≤ 1 and such that lim n→∞ b n −1 n = b and lim n→∞ κ n −2 n = κ. Then, as n → ∞, Proof. In [NRS15], we showed that S n (U bn,κn ) converges to N b,κ in law. However, a closer look at the proof shows that the result was shown by constructing a coupling between {(S n (U bn ), S n (U bn,κn ))} n>0 and (N b , N b,κ ) such that S n (U bn ) converges to N b a.s., and such that under this coupling, the convergence of the second discrete coordinate to its continuum counterpart holds in probability. This allows to extend the invariance principle derived in [NRS15] to Theorem 2.6.

The continuum voter model perturbation (CVMP)
As described in the introduction, we aim at constructing the CVMP as a map from R × R + onto finite subsets of {1, ..., q}. This "quasi-coloring" will be defined in terms of the duality relation given in Section 1.4, i.e, we see the CVMP as dual to the Brownian net with killing. The key observation is that even if the continuous graph starting from a "generic" point z is infinite (what we mean by generic will be discussed below in more details), the reduced version of this graph is locally finite. (Recall that a reduced graph is constructed after removal of all the non-relevant separation points in the original graph -see Definition 1.1).
We now give an outline of the present section. In Section 3.1, we introduce the notion of relevant separation points and of reduced graph for the Brownian net with killling and prove some of their properties. Those results are simple extensions of the results already known in the case κ = 0 [SSS09,SSS14]. In Section 3.2, we construct the CVMP.
In Section 3.3, we prove properties 1 and 2 listed in Theorem 1.1.

The reduced graph
We first define a notion of graph isomorphism adapted to our problem.
Definition 3.1. Let G = (V, E) and G = (V , E ) be two finite directed acyclic graphs. We will say that G and G are isomorphic if there exists a bijection ψ from V to V such that for every z 1 , In other words, two graphs will be isomorphic if they coincide modulo some relabelling of the nodes.
Let us consider the set of all finite directed acyclic graphs whose vertices are labelled by points in R 2 . The graph isomorphism property is an equivalence relation partitioning this set of graphs into equivalence classes. In the following, we will denote by (Γ, G) the set of all equivalence classes endowed with its natural σ field. In this section, we construct a natural reduced graph generated by the Brownian net with killing paths (starting from a point z until a certain time horizon T ) as an element of (Γ, G).
As in the standard Brownian net, in the Brownian net with killing, a separation point (or branching point) will refer to a point z = (x, t) such that there exist two paths π r and π l ∈ N b,κ with σ π l , σ πr < t and e π l , e πr > t so that (1) π l (t) = π r (t) = x and (2) there exists > 0 so that π l (s) < π r (s) for every s ∈ (t, min(t + , e r , e l )]. We note that in the coupled pair (N b , N b,κ ) the separation points of the killed Brownian net and the ones of the standard Brownian net coincide a.s..

Definition 3.2 (Relevant Separation Point).
A point z = (x, t) ∈ R 2 with t ∈ (S, T ) is said to be an (S, T )-relevant separation point of the Brownian net N b,κ , if and only if there exist two paths π 1 , π 2 ∈ N b,κ with starting time σ π1 = σ π2 = S and ending times e π1 , e π2 > t, and such that π 1 (t) = π 2 (t) but π 1 (u) = π 2 (u) for u ∈ (t, inf(T, e 1 , e 2 )) . (3.2) In the following, we will denote by R κ (S, T ) the set of (S, T )-relevant separation points in the Brownian net with killing N b,κ (with no b superscript in R κ (S, T ) to ease the notation). Finally, R(S, T ) will denote the same set when κ = 0 (Proposition 6.1. in [SSS14]).
Proposition 3.1. For every S < T , the set R κ (S, T ) is locally finite a.s.
Proof. As already mentioned, this property for κ = 0 has been established in [SSS14]. It remains to extend the property for κ = 0. Let E ⊂ R 2 be a bounded set and let us show that R κ (S, T ) ∩ E is finite a.s..
Let L > 0 and define B L to be the set of realizations such that In particular, the equicontinuity of the paths in N b implies that P(B L ) goes to 1 as L → ∞. Next, we inductively define θ 0 ≡ θ 0,L = S and a sequence of killing times {θ i ≡ θ i,L } i≥1 as follows: where M κ (t) is the set of killing points attained by the set of paths π ∈ N b,κ with σ π = t. From the very definitions of the θ i 's, it is not hard to see that We now claim that In order to prove (3.5), it is enough to prove that (a) when θ i < T , R(θ i , T ) is locally finite, and (b) R(θ i , θ i+1 ) is also locally finite. For property (a), we first note that R(U, T ) is a.s. locally finite for every deterministic U with U < T , implying that the set of times U for which R(U, T ) is locally finite has full Lebesgue measure. Combining this with the definition of the random intensity measure used in our Poisson marking (see Definition 2.2) and using Fubini's theorem show that property (a) must hold.
For Property (b), the strong Markov Property and stationarity in time show that it is enough to prove the result for the special value i = 0. For any deterministic times U , we know that R(S, U ) is locally finite a.s.. Reasoning as in the previous paragraph completes the proof of (b), and thus (3.5).
We are now ready to complete the proof of Proposition 3.1. First of all, M κ (S) is locally finite by Proposition 2.5. Thus we get that θ 1 − θ 0 > 0 a.s.. Furthermore, using the strong Markov property of the standard Brownian net and its stationarity in time, the sequence (θ i+1 − θ i ) is a sequence of i.i.d. strictly positive random variables. As a consequence, for every realization of N b,κ , the sequence {R(θ i ∧ T, θ i+1 ∧ T )} i≥0 is empty after some finite value of i. Our proposition then immediately follows from (3.4) and (3.5) and the fact that P(B L ) goes to 1 as L goes to ∞.
(3.6) Definition 3.3 (Continuous Reduced Graph). Let S < T be deterministic or random. Let z = (x, S) and let V z (T ) be the union of the singleton z with the subset of vertices For every T -finite point z = (x, S), the reduced graph starting from z on the time interval [S, T ] -denoted by G z (T ) -is the random directed graph whose vertices are given by V z (T ) and whose edges are generated by the Brownian net with killing Proposition 3.2. Let S < T be two deterministic times. Almost surely for every realization of the Brownian net with killing, every point (deterministic or random) in R × {S} is T -finite.
Proof. By equicontinuity of the paths in N b,κ , we only need to show that is locally finite a.s.. First, R κ (S, T ) is locally finite using Proposition 3.1, and M κ (S) is also locally finite by Proposition 2.5. Finally, when κ = 0, L 0 (S, T ) is locally finite by Proposition 2.2. Since L κ (S, T ) is stochastically dominated by L 0 (S, T ), this ends the proof of Proposition 3.2.

Construction of the continuum voter model perturbation
In this section, we construct the CVMP alluded to in Theorem 1.1. It will depend on following quantities (1) the branching and killing parameters b, κ, (2) the boundary nucleation mechanism {g i,j } i,j≤q , (3) two probability distributions p and λ on {1, · · · , q}, where p is the bulk nucleation mechanism and λ is the analog of the one-dimensional marginal of the initial distribution for the discrete voter model perturbation.
Recall that in order to determine the color of a vertex (x, t) in our discrete voter model perturbation {θ (x, t)} (x,t)∈Z 2 odd , we start with a random coloring of the leaves of the graph induced by the backward BCK, starting from (x, t) as a root. The color of (x, t) is then determined by applying the algorithm described in Section 1.4 to the reduced graph rooted at (x, t) (see Proposition 1.6).
The backward Brownian net with killing. As in the discrete case, the backward Brownian net with killingN b,κ is obtained from the Brownian net with killing by a reflection about the x-axis. In the following, we will use identical notation for the backward Brownian net with killing, adding only a hat sign to indicate that we are dealing with the backward object. For instanceR κ (S, T ) will refer to the (S,T)-relevant separation points for the backward net between times S and T .
Step 0. Assignment of i.i.d. random variables. In the continuum, we consider a realization of the backward Brownian net with killlingN b,k and consider the random set of points consisting of the union of 1. all the separation points inN b,κ , 2. all the killing points inN b,κ , Those three sets are countable for almost every realization of the Brownian net with killing for the following reasons: 1. First, for any separation point z, there exists p < q ∈ Q such that z ∈R κ p,q . From Proposition 3.1,R κ p,q is locally finite a.s., which implies that the set of separation points is countable a.s.
2. Since, conditional on N b , the set of killing points is defined as a Poisson Point Process, this set is also countable a.s..
3. Finally, for the third set, we note that whereξ s (0) is the backward version of the branching-coalescing point set as defined in Proposition 2.2. Sinceξ s (0) is locally finite for every deterministic s, the third item is also countable.
Finally, since the random set consisting of the union of the sets 1, 2 and 3 above is a.s.
countable, given a realization ofN b,κ , i.i.d. uniformly distributed random variables U z in [0, 1] can be assigned to each of the points z in the three sets described above.
Step 1. Pre-coloring of the leaves. We consider the set of leaves, consisting of the killing points and the intersection points ofN b,κ with {t = 0}. For each such point z, we deduce the pre-coloring of z, denoted byθ(z), from the variable U z according to the following rule.
• If the leaf z is an intersection point between a path fromN b,κ and {t = 0} (i.e. a point from set 3),θ(z) is the unique integer in {1, · · · , q} such that with the convention that λ(0) = 0. • If the leaf z is a killing point inN b,κ (i.e. from set 2),θ(z) is the unique integer in {1, · · · , q} such that with the convention that p(0) = 0.
Step 2. Pre-coloring of the "nice" points. Given a point (x, S) and a time T > S, recall the definition of the reduced graph G (x,S) (T ) in the forward Brownian net with killing N b,κ (see Definition 3.3). Similarly, for any S > 0, one can define a reduced grapĥ G (x,S) (0) whenever the point (x, S) is 0-finite in the backward Brownian net with killing.
In particular, if we consider D, a countable deterministic dense set of R × R + , Proposition 3.2 (in its backward formulation) ensures that all the points in D are nice for almost every realization of the Brownian net with killing.
Definition 3.5. A graph is said to be simply rooted if the out-degree of the root is 1.
For nice points with a simply rooted graph, we assign the color determined by the coloring algorithm described in Section 1.4 using the family of distributions {g i,j } and the U z 's at separation points (as in (1.5)). For every such point, this defines a color that we denote byθ(z).
For graphs with more complicated root structure (i.e. with the root connected to more than one point), we decompose the finite graph into simply rooted graphs. (For instance, if the root connects to two distinct points z and z , the first component is the subgraph whose set of vertices is the union of the root with the vertices that can be accessed from z ; the second component is defined analogously.)θ(z) is then defined as the union of the coloring induced by each of the simply rooted subgraphs in the decomposition.
Remark 3.3. By construction, |θ(z)| is less or equal to the out-degree of the root.
Step 3. Horizontal Limits. The coloring of the half-plane R × R + is generated by taking horizontal limits. Our procedure is based on the following lemma.
Proof. Let us consider a dense countable deterministic set D ⊂ R × R + . By Proposition 3.2 in its backward formulation, every point in D is nice a.s.. Furthermore, any point z belonging to the trace ofN b,k (D) (the backward Brownian net with killing paths starting from D), must also have a finite graph representation, since every relevant separation point for z is also relevant for some z ∈ D, and every leaf for z is also a leaf for z. Hence, for every every realization of the Brownian net, for every t (random or deterministic) it is sufficient to prove thatN b,κ (D) intersects the set R × {t} at a dense set of points.
Remark 3.4. It is natural to ask whether one could define the CVMP by taking limits in any directions. It is not hard to show that for every deterministic (x, t), and every color c, there exists a sequence (x, t n ) with t n → t such thatθ(x, t n ) = c. As a consequence, if we allowed for vertical limits, every deterministic point will be trivially colored with {1, · · · , q}. This property was established in the case where the boundary noise is absent

Properties of the CVMP
We will now prove some basic properties of our mapping (Theorem 1.1(1)-(2)). In order to do so, we first elaborate on the properties of the reduced graph at deterministic times.

Reduced graphs at deterministic times
In this subsection, we derive some properties of the reduced graph at deterministic times. We formulate these results "forward in time" -i.e., for the original Brownian net, and not its backward version. When needed (in the forthcoming sections), these results will be used in their backward formulation.
Before stating the main proposition of this section (see Proposition 3.5 below), we start with some definitions. Let z 0 and z 1 be two T -finite points. We will say that G z0 (T ) and G z1 (T ) only differ by their root iff they are identical up to relabeling of their root, i.e., when the mapping ψ, defined by  is a locally finite set and is made of (o, pp) points. Further, between two consecutive points of S 2 (S, T ), the graphs only differ by their root. I.e., if x, y are such that x 1 < x ≤ y < y 1 , where x 1 and y 1 are two consecutive points in S 2 (S, T ), then G (x,S) (T ) and G (y,S) (T ) only differ by their root.
The proof of this result relies on some properties of the special point of the Brownian net. Let (r, l) ∈ (W r , W l ) and let z, z be two vertices in the graph G z0 (T ). We will write z := (x, t) → l,r z := (x , t ) iff t < t , l ∼ z out r and l ∼ z in r, and z and z are connected by the paths l and r, in the sense that l and r do not encounter any element of V z0 (T ) in the time interval (t, t ).
Among other things, the following technical lemma will allow to relate the out-degree of a root to its type (see Corollary 3.1).
Lemma 3.2. Let S < T be two deterministic times. For almost every realization of the Brownian net with killing, for every z 0 = (x 0 , S) (with x 0 random or deterministic), the following properties hold.
1. Suppose z ∈ V z0 (T ) is not a leaf. For every equivalent outgoing pair (l, r) from z, there exists a unique z ∈ V z0 (T ) such that z → l,r z .
2. For every z, z ∈ V z0 (T ) such that z → z , there exists a unique pair (l, r) ∈ (W r , W l ) such that z = (x, t) → l,r z = (x , t ). 3. Let z − = (x − , S) and z + = (x + , S) with x − ≤ x + such that there exist two distinct equivalent outgoing paths (l, r ) and (l , r) starting respectively from z − and z + , with the convention that when x − = x + , (l, r ) and (l , r) are ordered from left to right. Finally, let z l and z r such that z − → l,r z l and z + → l ,r z r .
Then z l = z r if and only if l and r do not meet on (S, U l,r ∧ U l ,r ], where U l,r := T ∧ inf{u ≥ S : ∃π ∈ N b,κ s.t. σ π = S, e π = u, and l ≤ π ≤ r } (3.9) and U l ,r is defined analogously using l and r.
Proof. We start with the proof of item 1. Let z := (x, t) ∈ V z0 (T ), for every equivalent outgoing pair (l, r) at z, define U l,r analogously to (3.9), i.e., U l,r = T ∧ inf{u ≥ t : ∃π ∈ N b,k s.t. σ π = t, e π = u, and l ≤ π ≤ r}. to be the last time l and r separate before time U l,r . We first claim that the point z := (l(τ l,r ), τ l,r ) = (r(τ l,r ), τ l,r ) (3.11) belongs to V z0 (T ) and that r ∼ z in l. In order to prove this claim, we distinguish two cases.
Let us first assume that τ l,r = U l,r . There are only two possibilities: If U l,r = T , then z is a leaf and must be of type (p, p) by Proposition 2.3(4); if U l,r = T , the point z must be a killing point (ands thus a leaf) which is also of type (p, p) (this follows directly from the structure of the intensity measure (2.1) used in the Poissonian construction of the killing points). Hence, when τ l,r = U l,r , the point z (as defined in (3.11)) is always a leaf of V z0 (T ) and is always of type (p, p), which implies that l ∼ z in r. Let us now assume that τ l,r < U l,r . It is not hard to show that z is an (S,T)-relevant separation point in G z0 (T ). Since separation points are of type (p, pp) (by Proposition 2.4(1)), z is a point of V z0 (T ) (a separation point) with the property l ∼ z in r. In the two previous paragraphs, we showed that the point z = (l(τ l,r ), τ l,r ) belongs to V z0 (T ) and that l ∼ z in r. Furthermore, the definition of τ l,r directly implies that r and l do not encounter any relevant separation point or any killing point before entering the point z . Combining those two facts implies that z → l,r (l(τ l,r ), τ l,r ). This ends the proof of item 1.
We now proceed with the proof of the second item of Lemma 3.2. Let z, z ∈ V z0 (T ) be such that z → z . Let π ∈ N b be a path connecting z and z . z is not a leaf and thus it can either be (1) a separation point, or (2) a root. In case (1), Proposition 2.4 implies that π is squeezed between a unique equivalent pair (l, r) such that l ∼ z out r. In case (2), since we are considering the case of a reduced graph rooted at a deterministic time, the same property must hold by Proposition 2.3. Let τ l,r be defined as in (3.10). By definition of τ l,r , π can not access any point of V z0 (T ) on the interval (t, τ l,r ). Since we proved that (l(τ l,r ), τ l,r ) ∈ V z0 (T ), z must coincide with (l(τ l,r ), τ l,r ). Finally, since we showed that z → l,r (l(τ l,r ), τ l,r ), this ends the proof of item 2. We now proceed with the proof of the third item of Lemma 3.2. The existence of z l and z r is guaranteed by item 1 of Lemma 3.2. Further, from what we showed earlier, z l = (l(τ l,r ), τ l,r ) and z r = (r(τ l ,r ), τ l ,r ) and thus z l = z r ⇐⇒ τ l,r = τ l ,r , and l(τ l,r ) = r(τ l ,r ).  T ). The leftmost and rightmost graphs are generated from two consecutive points in S 2 (S, T ). Grey (resp., black) paths represent paths from W l (resp., W r ).
Using the non-crossing property of the Brownian web paths (in W l or W r ), if l and r meet before time U l,r ∧ U l ,r , then the condition on the RHS of the former equivalence is satisfied. Conversely, if the condition on the RHS of the former equivalence is satisfied, it is clear that l and r meet before time U l,r ∧ U l ,r . This ends the proof of Lemma 3.2.
As a direct corollary of Lemma 3.2(1) and (2), we deduce the following result.
Corollary 3.1. Let S < T be two deterministic times. For almost every realization of the Brownian net with killing, for every z 0 = (x 0 , S) (with x 0 random or deterministic), the out-degree of a point z ∈ V z0 (T ) is at most equal to the number of pairs (l, r) ∈ (W l , W r ) such that l ∼ z out r.
Proof of Proposition 3.5. The fact that z 0 is T-finite is the content of Proposition 3.2. We now turn to the proof of item 1. By Proposition 2.3, for any point of type (o, p) or (p, p), there exists a unique pair l, r such that l ∼ z0 out r. By Corollary 3.1, this proves that the degree of z 0 is at most 1. Since z 0 is a.s. not a killing point, the out-degree is at least 1, thus showing that the finite graph representation at any point of type (o, p) or (p, p) is simply rooted. Finally, since every deterministic point is a.s. of type (o, p) (by Proposition 2.3), the same property must hold at any deterministic point a.s.. This ends the proof of item 1.
We now turn to the proof of items 2 and 3. We first claim that any node of type ( * , pp) -where * can be either be o (for roots of type (o,pp)) or 1 (corresponding to separation point by Proposition 2.3 and Proposition 2.4) -connects to one or two points.
By definition of a point of type ( * , pp), there exists exactly two distinct pairs (l, r ) and (l , r) in (W l , W r ) starting from z such that l ∼ z out r and r ∼ z out l. By Corollary 3.1, the out-degree is at most two, thus proving item 2. Further, by definition of a relevant separation point, the out-degree for such a point is at least two, and thus the out-degree of a relevant separation point is exactly equal to 2. This ends the proof of item 3. It remains to show item 4. The fact that S 2 (S, T ) is a set of (o, pp) points follows from Corollary 3.1 and the fact that at deterministic times, points are either of type (o, p), (p, p) or (o, pp). Next, we show that S 2 (S, T ) is locally finite. Lemma 3.2(3) (in the special case x − = x + ) easily implies that if x is in S 2 (S, T ) then there exists a time t x -either a killing time for a path starting from time S, or t x = T -such that the rightmost and leftmost paths l, r starting from (x, S) do not meet before time t x .
Let L > 0. Since M κ (S) is locally finite (by Proposition 2.5), the previous argument and the equicontinuity of the net paths imply that there exists a rational number q > 0 such that [−L, L] ∩ S 2 (S, T ) is a subset of {x : the leftmost and rightmost path in N b starting from (x, S) do not meet on (S, S +q)}; (3.12) e.g., take q < inf{s > S : (y, s) ∈ M κ and ∃π ∈ N b , σ π = S, π(S) ∈ [−L, L], π(s) = y}.
Since a.s. for every realization of the net, the set (3.12) is locally finite for every deterministic q, this proves that [−L, L] ∩ S 2 (S, T ) is finite a.s. and thus, S 2 (S, T ) is locally finite. (This can easily be seen as a direct consequence from the fact that the dual branching-coalescing point set (the dual branching-coalescing point set being defined from the dual Brownian net as defined in [SS08]) starting from t + q is finite at time t by Proposition 2.2.) To conclude the proof of item 4, it remains to show that G z− (T ) and G z+ (T ) are identical (up to some relabeling of the roots) if z − = (x − , S) and z + = (x + , S) are located (strictly) between two consecutive elements of S 2 (S, T ). We give an argument by contradiction and assume that the roots of G z− (T ) and G z+ (T ) do not connect to the same point (i.e., z l = z r ). We now show that this implies the existence of a point z 0 = (x 0 , S) between z − and z + (i.e. x − ≤ x 0 ≤ x + ) that connects to two distinct points.
For i = 1, 2, going to a subsequence if necessary, π n i converges to a net path π ∞ i starting from (x 0 , S) and passing trough the point z i . We will now show that π ∞ i connects z 0 to z i (in order to get the desired contradiction). In order to do so, it is sufficient to prove that π ∞ i does not encounter any element in L κ (S, T ) ∪ R κ (S, T ) ∪ K κ (S, T ) before entering z i . Again, we argue by contradiction. Let us assume that π ∞ i connects (x 0 , S) to a point z = (x,t) before entering z i . Since z 0 →z, by the second item of Lemma 3.2, there must exist a left-right pair (l, r) such that l ≤ π ∞ i ≤ r on [0,t]. Let q be any rational number in (S,t). By Proposition 2.2, the set ξ S (q) is locally finite a.s., and since π n 1 converges to π ∞ 1 , π n 1 (q) and π ∞ 1 (q) coincide for n large enough. This implies that π n 1 is "trapped" between l and r after a finite rank, i.e., that l ≤ π n i ≤ r on [q,t]. (Here we used the property that a net path can not cross a left-most path from the right, and a rightmost path from the left -see [SS08]). This implies that π n i enters the pointz for n large enough, which is the desired contradiction. This completes the proof of item 4 of Proposition 3.5.
Let us take a point x in S 2 (S, T ). By Proposition 3.5, (x, S) must be of type (o, pp), and connects to two distinct points. By Lemma 3.2(3), there exist two points z , z with z = z , such that if (l, r ) and (l , r) are the two distinct outgoing equivalent paths in (W l , W r ) exiting the point (x, S), ordered from left to right, then (x, S) → l,r z = (x , t ) (x, S) → l ,r z = (x , t ).
One can always decompose G (x,S) into two simply rooted graphs, a left and a right component G (T ) as the subgraph whose set of vertices is the union of (x, S) with the vertices of the (directed) subgraph originated from z . G (x,S) r (T ) is defined analogously using z . (Note that the two sub-graphs can share a lot of common vertices, and that this decomposition is not a partition of the vertices). For a pictorial representation of the next proposition, we refer the reader to Fig. 13. Proof. We only prove the first relation. The second relation can be proved along the same lines. In the following, as before, (l, r ) and (l , r) will denote the two outgoing equivalent paths in (W l , W r ) exiting the point (x, S), ordered from left to right. In particular, it is not hard to see from Proposition 2.3, that l (resp., r) is the leftmost (resp., rightmost) path starting from (x, S). Take x n ↑ x, and for every n, let (l n , r n ) be an equivalent pair of outgoing paths starting from (x n , S). By Proposition 3.5(4), all the points in (x l , x) have the same finite graph representation, up to relabeling of their root. As a consequence, we need to prove that if x n ↑ x then G (xn,S) (T ) and G (x,S) l (T ) coincide up to their root, after some rank, i.e., we need to show that (x n , S) → (ln,rn) (x , t ) (after some rank), where z = (x , t ) is the only point in G (x,S) l (T ) connected to (x, S). According to Lemma 3.2(3), and since U ln,r ≤ U ln,rn ∧ U l,r (as defined in (3.9)), we need to prove that l n and r meet before time U ln,r = T ∧ inf{u ≥ t : ∃π ∈ N b,k s.t. σ π = t, e π = u, and l n ≤ π ≤ r }.
Let us first show that l n coalesces with l at some time τ n → S. By compactness of the Brownian web, l n converges to some pathl ∈ W l starting from (x, S). Since l n does not cross the path l, we have l n ≤ l, implying thatl ≤ l. Since l is the leftmost path starting from (x, S), this yieldsl = l, and thus l n converges to l. Using an argument analogous to the one given in the last paragraph of the proof of Proposition 3.5, it is not hard to show that l n coalesces with l at some τ n → S, as claimed earlier. Next, since r n can not cross r , we have r n ≤ r . This implies that for any s ≥ τ n , l(s) = l n (s) ≤ r n (s) ≤ r (s).
On the other hand, since l ∼ z out r , we can choose a timet arbitrarily close to S, such that l(t) = r (t). In particular, for n large enough such that τ n ≤t, the previous inequality implies that l n and r must have met before timet. Finally, since the set of killing points M κ (S) is locally finite, and sincet can be chosen arbitrarily close to S, we can choose n large enough such that l n and r meet before any path squeezed between those two paths encounters a killing point, i.e., before time U ln,r . As explained earlier, by Lemma 3.2(3), this implies that for n large enough, the graphs G (xn,S) (T ) and G (x,S) l (T ) coincide modulo their root. This ends the proof of Proposition 3.6.

Proof of Theorem 1.1 (1)-(2)
Let t be a deterministic time. Proposition 3.2 implies that for any point (x, t) (with x deterministic or random) is "nice" and has a well defined pre-coloringθ(x, t). Lemma 3.3. At any deterministic time t, for almost every realization of the CVMP, for every x (deterministic or random), the pre-coloringθ(x, t) coincides with the coloring θ(x, t).
Proof. The key point of our proof is that if two graphs only differ by their root, they must have the same pre-coloring (by the very definition of the coloring algorithm as defined in Section 1.4). We denote byŜ 2 (0, t) the backward analog of S 2 (0, t) as defined in Proposition 3.5(4), i.e. S 2 (0, t) = {x :Ĝ (x,t) (0) is not simply rooted.} We first show that Lemma 3.3 holds for z = (x, t) / ∈Ŝ 2 (0, T ). SinceŜ 2 (0, T ) is a.s. locally finite, Proposition 3.5(4) -in its backward formulation -implies that there exists As a consequence, for any x n → x,θ(x n , t) is stationary after a certain rank, and thus Let us now consider z ∈Ŝ 2 (0, T ). First, by definition of the precoloringθ(·), we must haveθ(z) = {c l } ∪ {c r }, with c l (resp., c r ) being the color given by our coloring algorithm applied on the graphĜ x,t l (0) (resp.,Ĝ x,t r (0)) -those two graphs being defined as in Proposition 3.6 in the backward context. (Note that those two colors are potentially equal.) Let us now consider a subsequence x n → x, with x n = x. We can decompose this sequence into two subsequences x l n < x and x r n > x. For the sequence x l n , Proposition 3.6, implies that for n large enough,θ(x l n , t) = c l , while for the sequence x r n , we must haveθ(x r n , t) = c r . As a consequence, we get θ(z) =θ(z).  2. For every x, |θ(x, t)| ≤ 2.
3. The set of points such that |θ(x, t)| = 2 is locally finite. Moreover, the color between two consecutive points of this set remains constant.
Proof. The first item of Proposition 3.7 is a direct consequence of Proposition 3.5(1) and Lemma 3.3. For the second item, we first note that Proposition 3.5 implies that for any point (x, t) (with x deterministic or random), the out-degree of the root ofĜ (x,t) (0) is either 1 or 2. Furthermore, the definition of our pre-coloring implies |θ(x, t)| is less or equal to the out-degree of the reducedĜ (x,t) (0). The second item then follows by a direct application of Lemma 3.3.

Convergence to the continuum model. Proof of Theorem 1.1 (3)
Let { n } be a sequence of positive numbers such that n → 0. As in Theorem 1.1, we assume that there exists b, κ ≥ 0, and two positive sequences {b n } and {κ n } such that b n / n → b and κ n / 2 n → κ. We also assume that the boundary and bulk mechanisms {g n i,j } i,j≤q and p n converge to a sequence of limiting probability distributions {g i,j } i,j≤q and p when n → ∞.
Let z 1 n , · · · , z k n ∈ S n (Z 2 odd ) be such that for every i = 1, · · · , k, lim n→∞ z i n = z i for some z i ∈ R 2 . Our goal is to show that the distribution of the coloring of the points (z 1 n , · · · , z k n ) for the discrete model labelled by n converges to the distribution of the coloring of (z 1 , · · · , z k ) at the continuum.
By Proposition 3.7(1), the coloring of a deterministic point z in the CVMP is obtained by applying the coloring algorithm to the reduced graphĜ z (0) (the backward of the object introduced in Definition 3.3). By Proposition 1.6, the color of a point in the discrete model can be recovered by applying the same procedure to the discrete backward net with killing.
From the previous observation, and since the same property holds at the discrete level (see Proposition 1.6), our convergence result easily follows from the convergence (in is the backward reduced graph starting from z i n with time horizon 0 and constructed from the (backward) discrete net with killing S n (Û bn,κn ) restricted to the upper half-plane. In this section, and for the sake of clarity, we will only prove the result for k = 1 (i.e., the convergence of one dimensional marginals of the coloring). The general case involves more cumbersome notation, but can be treated along exactly the same lines.
Let z n ∈ S n (Z 2 odd ) such that z n converges to z = (x, T ). We need to show that G zn n (0) converges toĜ z (0). By reversing the direction of time and using translation invariance (in the x and t directions), it is equivalent to show the following result in the forward-in-time setting: Proposition 4.1. Let z n ∈ S n (Z 2 odd ) such that z n → (0, 0). For every T > 0, where G zn n (T ) is the reduced graph at z n with time horizon T for the (rescaled) forward discrete net with killing S n (U bn,κn ).

Proof of Proposition 4.1
We start by introducing some notation. Define and for any path π ∈ N b,κ (Σ 0 ), let r(π) be the (a.s. finite) sequence of points in (as defined in (3.6)) visited by the path π, the points in the sequence being listed in their order of visit. (In particular, the final point of r(π) must be a leaf, and every other point is a relevant separation point). In the following, r(π) will be referred to as the finite graph representation of π.
Analogously to the continuum level, we assume that the discrete net with killing U bn,κn is coupled with a discrete net U bn (where κ n = 0). (See Section 2.4 for more details). S n (U bn,κn (Σ 0 )) will denote the discrete analog of N b,κ (Σ 0 ), and for any path π n ∈ S n (U bn,κn (Σ 0 )), r n (π n ) will denote the finite graph representation at the discrete level.
For any point (x, t) and s ≤ t, define d s ((x, t)) := inf{|y − x| : y = x and y ∈ ξ s (t)}.  In words,R(s, t) records the location of the (s, t)-relevant separation points (in the standard Brownian net (with no killing)), together with a measure of "isolation" around each of those points. Analogously, we defineR n (s, t) for the rescaled discrete net S n (U bn ). Let L > 0. Recall the definition of θ i given in (3.3) for S = 0, i.e. θ 0 = 0 and for every where we recall that M κ (S) is the set of killing points attained by some path in N b,κ starting at time S. Define Note that when θ i+1 < T , L i+1 is a single killing point a.s.. Analogously to (4.2), for every i ≥ 1, we also definē (To ease the presentation, L and T are not explicit in the notation). Finally, θ i n , N n , R i n ,R i n and L i n ,L i n will denote the analogous quantities for the rescaled process S n (U bn , U bn,κn ).
then any intermediary point (resp., leaf) in G must be a point in ∪ i R i (resp., ∪ i L i ). See Fig. 12. The analogous property holds at the discrete level. The next two results will be instrumental in the proof of Proposition 4.1. Informally, Proposition 4.2 states that relevant separation points are isolated from the left and from the right. Proposition 4.3 implies that the same holds at the discrete level; furthermore it shows that each continuum separation point is the limit of a single discrete separation point, which is essential for the proof of Proposition 4.1, the convergence of reduced graphs.

Proposition 4.2. For almost every realization of
Proof. By Proposition 2.3 and 2.4 and the construction of the killing points, z is of type (p, * ) -with * being equal either to p and pp. Proposition 4.2 is then the content of Proposition 3.11(c) in [SSS09]. S n (U bn,κn ), S n (U bn,κn (Σ 0 )), where the convergence is meant in the product topology H × H ×P ×P. By the Skorohod Representation Theorem and Proposition 4.3, for every L > 0, there exists a coupling between the discrete and continuum levels such that S n (U bn,κn (Σ 0 )), Until the end of the proof, we work under this coupling. For any n ∈ N and any realization in A L , for any discrete point z n ∈ ∪ i≥1 (R i n ∪ L i n ), define ψ n (z n ) to be the closest point in ∪ i≥1 (R i ∪ L i ) to z n (it is easy to see that ψ n (z n ) is well defined a.s.). As argued above, we need to show that for almost every realization in A L , for n large enough, the graphs G n and G are isomorphic under this mapping.
We claim that conditional on A L , for any subsequence of discrete paths {π n } n≥1 with π n ∈ S n (U bn,κn ((0, 0))) converging to π ∈ N b,κ ((0, 0)), the discrete and continuum paths have the same finite graph representation (up to ψ n ) after a finite rank. Once this claim is obtained, the proof of Proposition 4.1 goes by contradiction: it is rather easy to see that if Proposition 4.1 was not satisfied, this would contradict the latter claim. Thus, the rest of the proof is dedicated to the proof of this statement.
Proof. We first show that A(z 1 , · · · , z κ ) is a compact set of paths. Since N b is compact, it is is sufficient to prove that for any {p l } l≥1 in A(z 1 , · · · , z κ ) converging to π in N b (Σ 0 ), the finite graph representation r(π) is given by z 1 , · · · , z κ . On the one hand, it is clear that π must hit the points z 1 , · · · , z κ . On the other hand, if π touches somez = (x,t) ∈ V distinct from z 1 , · · · , z κ , we would have p l (t) =x, but p l (t) →x as l → ∞, which would imply that d 0 (z) = 0. However, by Lemma 4.1, d 0 (z) > 0 a.s.. This shows that A(z 1 , · · · , z κ ) is a compact set of paths a.s.
Next, since sets of the form A(z 1 , · · · ,z κ ) with (z 1 , · · · ,z κ ) ∈ ∪ l∈N * (R 2 ) l are pairwise disjoint, the distance between A(z 1 , · · · , z κ ) and any finite union of sets of the form A(z 1 , · · · ,z m ) = ∅ with (z 1 , · · · ,z m ) = (z 1 , · · · , z κ ) is strictly positive. Furthermore, by equicontinuity of the paths in N b and the local finiteness of V , the same property holds for any infinite union of those sets since one can restrict attention to a finite space-time box. Combining this with the fact that B(z 1 , · · · , z k ) can be written as the union of all the non-empty A(z 1 , · · · ,z m )'s with (z 1 , · · · ,z m ) = (z 1 , · · · , z κ ) completes the proof of Corollary 4.1.
Let P be the space of Radon measures on R × R + equipped with the vague topology.
From Proposition 6.14 in [SSS14], it is already known that (S n (U bn ), R n (0, T )) → (N b , R(0, T )) in law, where the convergence is meant in the product topology H × P. (4.5) Proof. From the Skorohod representation theorem, there exists a coupling between the discrete and continuum levels such that S n (U bn ) → N b a.s..
Let a < b ∈ R. The previous convergence statement entails that under our coupling: On the other hand, using the same technic as in Proposition 4.3. in [NRS15], one can show that Combing the two previous results, for every interval [a, b], we get which ends the proof of the lemma.
We will need one more ingredient that is a simple extension of Lemma 6.7 in [SSS14], whose proof is left as an exercise to the reader. Lemma 4.3. Let E be a Polish space. Let {F i } i∈I and {G i } i∈I be two finite or countable collections of Polish spaces and for each i ∈ I, let f i : E → F i and g i : E → G i be measurable functions. Let X, X k , Y, Y k and U k,i , V k,i be random variables {k ≥ 1, i ∈ I} such that X, X k , Y, Y k take values in E and U k,i , V k,i takes values in F i and G i respectively; for every k, (X k , Y k , U k,i , V k,i , i ∈ I) are defined on the same probability space; (X, Y ) are defined on the same probability space. Then where the convergence is meant in the product topology.
By Lemma 6.12 in [SSS14], (S n (U bn ), S n (U bn (Σ 0 ))) converges to (N b , N b (Σ 0 )) in (H, d H ). Using Lemma 4.3 (in the special case X = Y = N b , and U k,i = V k,i ) and the Skorohod Representation Theorem, (4.4) and (4.5) imply that there exists a coupling between the discrete and the continuum levels such that From now until the end of this subsection, we will work under this coupling. In particular, if V n denotes the analog of V (as defined in Lemma 4.1), then V n =⇒ V a.s. under our coupling.
In the next lemma, A n (z 1 , · · · , z k ) will refer to the discrete analog of A(z 1 , · · · , z k ) (i.e., the set of paths in the rescaled net S n (U bn (Σ 0 )) with discrete finite graph representation (z 1 , · · · , z k )).
Proof. The existence of the z i n 's is a direct consequence of the fact that V n =⇒ V a.s.. The rest of the proof (i.e., the convergence A n (z 1 n , · · · , z k n ) to A(z 1 , · · · , z k )) is decomposed into two steps.
In this step, we will show that after a finite rank, for any π n ∈Ā n (z 1 , · · · , z k ), the finite graph representation of π n is a subsequence of (z 1 n , · · · , z k n ), and that A n (z 1 n , · · · , z k n ) ⊆ A n (z 1 , · · · , z k ).
As a consequence, after a finite rank, for any π n ∈Ā n (z 1 , · · · , z k ), the (discrete) finite graph representation of π n is a subsequence of (z 1 n , · · · , z k n ). (In words, some z i n 's can be "missed", but any point in the finite graph representation of π n must be in (z 1 n , · · · , z k n )).
Secondly, by reasoning along the same lines, one can deduce that after a finite rank, the finite graph representation of any path π n ∈B n (z 1 , · · · , z k ) must be distinct from (z 1 n , · · · , z k n ). SinceĀ n (z 1 , · · · , z k ) andB n (z 1 , · · · , z k ) form a partition of the set of discrete net paths starting at time 0 (for large enough n), it follows that A n (z 1 n , · · · , z k n ) ⊆ A n (z 1 , · · · , z k ) after a finite rank.
Step 2. We now show thatĀ n (z 1 , · · · , z k ) = A n (z 1 n , · · · , z k n ) for n large enough. Since we showed in Step 1 that after a finite rank, for any π n ∈Ā n (z 1 , · · · , z k ), the finite graph representation of π n is a subsequence of (z 1 n , · · · , z k n ), and since A n (z 1 n , · · · , z k n ) ⊆ A n (z 1 , · · · , z k ), it is sufficient to show that for n large enough, any π n ∈Ā n (z 1 , · · · , z k ) must pass through z 1 n , · · · , z k n .
First, for n large enough, is is easy to see that any π n ∈Ā n (z 1 , · · · , z k ) must go through z k n . This is a direct consequence of the fact that ξ 0 (T ) is locally finite, ξ 0 n (T ) =⇒ ξ 0 (T ) a.s. under our coupling, and the factĀ n (z 1 , · · · , z k ) converges to A(z 1 , · · · , z k ) a.s. (see Step 1).
Let us now argue that after some rank, any π n ∈Ā n (z 1 , · · · , z k ) must also go through the point z i n for i < k. In fact, we will show a little more: We will show by induction on i ∈ N, that for any m > i and anyz 1 , · · ·z m such that A(z 1 , · · ·z m ) = ∅, for n large enough, any π n ∈Ā n (z 1 , · · · ,z m ) must go throughz 1 n , · · · ,z i n , where {z i n } is defined analogously to {z i n } (i.e., as a sequence of points in V n approximatingz i ). The case i=1: Let us first show the property for i = 1. We argue by contradiction.
Going to a subsequence if necessary, let us assume that there exists an infinite sequence {π n } n with π n ∈Ā n (z 1 , · · · ,z m ) (with m > 1) and such that π n does not go through the pointz 1 n (i.e., ifz 1 n = (x 1 n ,t 1 n ) then π 1 n (t 1 n ) =x 1 n ). Going to a further subsequence, we can assume w.l.o.g. that {π n } converges to π ∈ A(z 1 , · · · ,z m ). (The fact that π ∈ A(z 1 , · · · ,z m ) easily follows from Corollary 4.1.) Since m > 1, the pointz 1 = (x 1 ,t 1 ) is a relevant separation point. From this observation, one can construct a path π ∈ N b (Σ 0 ) such that π = π up tot 1 and the paths π and π separate aftert 1 . (This is done by concatenating π with one of the paths in (W r , W l ) starting fromz 1 : for instance, following the notation of Proposition 2.4 and Fig. 11, if π is squeezed between l and r , we concatenate π with r. The constructed path belongs to the net using the stability of the net under hopping -See [SS08].) Since S n (U bn (Σ 0 )) converges to N b (Σ 0 ), there is a sequence {π n } n in S n (U bn (Σ 0 )) converging to π . Let q ∈ Q, with q <t 1 . ξ 0 (q) is a.s. locally finite and under our coupling ξ 0 n (q) =⇒ ξ 0 (q) a.s.. Since lim n→∞ |π n (q) − π n (q)| = |π(q) − π (q)| = 0, it follows that π n (q) = π n (q) for n large enough. Finally, since π and π separate atz 1 , π n and π n must separate at some point (x n , t n ) with lim sup t n ≤t 1 . However, in Step 1, we showed that after a finite rank, the finite graph representation of π n is a subset of {z 1 n , · · · ,z k n }. Since lim sup t n ≤t 1 , it follows that for large n, (x n , t n ) =z 1 n and so π n must go throughz 1 n (for n large enough), which yields a contradiction. This shows the induction hypothesis for i = 1.
The induction step from i-1 to i: Let us now assume that our property holds at rank i − 1 with i ≥ 2. Again we argue by contradiction and we assume that there exists a subsequence π n ∈Ā n (z 1 , · · · ,z m ) (with m > i) such that π n (t i n ) =x i n (wherē z i n = (x i n ,t i n )). By arguing as in the case i = 1, we can assume w.l.o.g. that π n converges to π ∈ A(z 1 , · · · ,z m ), and that there exists π ∈ N b such that π and π coincide up tot i and separate afterwards. Let (z 1 , · · · ,z l ) be the finite graph representation of π . By construction, l > i andz j =z j for j ≤ i. SinceĀ n (z 1 , · · · ,z l ) → A(z 1 , · · · ,z l ) (by Step 1), there exists π n ∈Ā n (z 1 , · · · ,z l ) such that {π n } converges to π . By using our induction hypothesis, π n and π n must go through the pointz i−1 n =z i−1 n after a finite rank. Further, since π and π separate atz i , the two paths π n and π n must separate at some time t n with lim sup t n ≤t i . However, for n large enough, the finite graph representation of π n is a subset of (z 1 n , · · · ,z k n ), and since π n and π n coincide at t i−1 n , π n and π n must separate atz i n . As a consequence, π n goes through the pointz i n , which yields the desired contradiction. This ends the proof of Lemma 4.4.
Let m n denote the discrete analog of m, and let z n , z 1 n , · · · , z k n ∈ V n converging to z, z 1 , · · · , z k ∈ V such that A(z 1 , · · · , z k ) = ∅. Lemma 4.4 implies that m n (z 1 n , · · · , z k n ), z n converges to m (z 1 , · · · , z k ), z . The latter convergence statement extends by taking the infimum over finitely many m (z 1 , · · · , z k ), z 's and m n (z 1 n , · · · , z k n ), z n . Proposition 4.5 is then a direct consequence of the equicontinuity of the paths in N b and ∪ n S n (U bn ).

Proof of Proposition 4.3
In the course of the proof of Proposition 3.1, we already proved that N < ∞ (where N is defined as in (4.3)) and for every 1 ≤ i ≤ N , the random point measureR i is finite a.s.. When θ i < T ,L i is a singleton a.s., and when θ i > T , the number of atoms inL i coincides with the cardinality of {x : x ∈ ξ 0 (T ) ∩ (−L, L)}. By Proposition 2.2, this shows that thatL i is also a finite measure a.s.. In this section, to prove Proposition 4.3, our main task will be to prove the following two lemmas: Lemma 4.5. (S n (U bn,κn ),L 1 n ) → (N b,κ ,L 1 ) in law.
We note that in both lemmas, the convergence is meant in the product topology H × P.
By using the strong Markov property and the stationarity in time of the killed net (both at the discrete and continuum level), Lemmas 4.5 and 4.6 will imply that where the convergence is again meant in the product topology. Since N (resp., N n ) coincides with the first i such thatR i (resp.,R i n ) is empty, this implies S n (U bn,κn ), Using Theorem 2.6, it is not hard to prove that S n (U bn,κn , U bn,κn (Σ 0 )) converges (in law) to (N b,κ , N b,κ (Σ 0 )) (this is done in Lemma 6.19 in [SSS14] for the case κ n , κ = 0. Proof of Lemma 4.5. In Lemma 4.2, we showed that (1) (ξ 0 n (T ), S n (U bn )) converges to (ξ 0 (T ), N b ) in law.
Furthermore, by Theorem 2.6, S n (U bn , U bn,κn ) converges to (N b , N b,κ ). By the Skorohod representation theorem and Lemma 4.3 with (X, Y ) = (N b , N b,κ ), this implies that there exists a coupling between the discrete and continuum levels such that (1) S n (U bn ) converges to N b a.s., (2)L 1 n converges toL 1 a.s., (3) the convergence statement {(S n (U bn,κn ), θ 1 n , ξ 0 n (θ 1 n ))} hold a.s., and (4) such thatR n (0, t) =⇒R(0, t) a.s. for any positive rational value of t and for t = T . We will work under this coupling for the rest of the proof.
Next, we first note that (4.8) (This is a direct consequence of the fact that R(0, t) is locally finite a.s. and translation invariance of the Brownian net along the x-axis). Since ∀t ∈ Q + , t = T,R n (0, t) =⇒ R(0, t) a.s., this implies that ∀t ∈ Q + , t = T, (4.9) (4.9) when t = T entails that Lemma 4.6 holds for almost every realization on {θ 1 > T }.
In order to complete the proof of Lemma 4.6, it remains to show that (4.10) We will proceed in three steps.
Step 1. Since N b is a set of equicontinuous paths a.s., for almost every realization of our coupling, there exists L ∈ Q + with L > L (and L random) such that any path in N b (Σ 0 ) hitting a point in R(0, θ 1 ) ∩ L 0,θ 1 remains in the box L 0,θ 1 between time 0 and θ 1 , and further, under our coupling, the same holds at the discrete level for n large enough.
In particular, the sets ξ 0 n (θ 1 n ) are sparse, in the sense that distinct points must remain at a macroscopic distance from each other. Let us assume w.l.o.g. that the realization under consideration belongs to the set under which the previous limiting statement holds.
Since the two distinct points x 1 n , x 2 n alluded to earlier belong to ξ 0 n (θ 1 ), this entails that the family of paths {π 1 n , π 2 n } n is not equi-continuous (the two paths π 1 n and π 2 n separate at time arbitrarily close to θ 1 n and take two macroscopically distinct values at time θ 1 n ). We already argued that ∪ n S n (U bn ) is a set of equicontinuous paths a.s. As a consequence, the event {lim sup n→∞ inf{x < θ 1 n : R n (0, θ 1 n ) ∩ L x,θ 1 n = ∅} = θ 1 } has measure 0. It follows that (4.12) (and thus (4.11)) holds a.s., as claimed earlier.
Step 2. For every q, L ∈ Q + , define E q,L to be the set of realizations such that (1) for n large enough, any path in S n (U bn )(Σ 0 ) hitting a point in R(0, θ 1 n ) ∩ L 0,θ 1 remains in the box L 0,θ 1 and (2) θ 1 > q, and (3) (4.11) holds. Note that for any realization in E q,L , any point in R(0, θ 1 ) ∩ L 0,θ 1 must belong to the rectangle L 0,q . In Step 2, our goal is to show that for almost every realization in E q,L , (4.13) First, under our coupling, (4.14) Next, on E q,L , since q < θ 1 , R(0, θ 1 ) ∩ L 0,q is a subset of R(0, q) ∩ L 0,q : if t < q, any two paths separating at t up to time θ 1 must also separate up to time q. Let us now show that for almost every realization in E q,L : ∀z n = (x n , t n ) ∈ R n (0, q) s.t. z n → z = (x, t) ∈ R(0, q), then z ∈ R(0, θ 1 ) iff z n ∈ R(0, θ 1 n ) for infinitely many n. Combining (4.14) with (4.15) yields (4.13). The rest of Step 2 is dedicated to the proof of (4.15) (thus showing (4.13)).
First, let us assume that z ∈ R(0, q) and that z also belongs to R(0, θ 1 ). We now show that z n (as defined in (4.15)) belongs to R(0, θ 1 n ) for n large enough. Let π 1 , π 2 ∈ N b (Σ 0 ) passing through z and separating up to θ 1 . Since S n (U bn (Σ 0 )) converges to N b (Σ 0 ) a.s., there are two sequences π 1 n and π 2 n in S n (U bn (Σ 0 )) such that π i n converges to π i , for i = 1, 2. By Lemma 4.1, d 0 (z) > 0 a.s. and further, under our coupling d 0 n (z n ) converges to d 0 (z) a.s.: in other words, z n must remain macroscopically isolated from the left and from the right. Since π 1 n (t n ) − π 2 n (t n ) → 0, this implies that π 1 n (t n ) = π 2 n (t n ) for n large enough.
Next, recall that π 1 and π 2 separate at z up to time θ 1 . By the latter property, π 1 n and π 2 n separate up to θ 1 n for large enough n at a point z n with ρ(z n , z n ) → 0 as n → ∞. However, because R(0, q) is locally finite and R n (0, q) =⇒ R(0, q) a.s., z n and z n must coincide after a finite rank. This proves that if z ∈ R(0, q), but z also belongs to R(0, θ 1 ), then z n also belongs to R(0, θ 1 n ) for n large enough. Next, let us assume that z n ∈ R n (0, q) and also belongs to R n (0, θ 1 n ) for infinitely many n. Going to a subsequence if necessary, we can assume that z n belongs to R n (0, θ 1 n ) for every n. Let π 1 n and π 2 n in S n (U bn (Σ 0 )) separating at z n up to time θ 1 n . In order to prove (4.15), we need to show that z belongs to R(0, θ 1 ). Going to a subsequence if necessary, π i n converges to π i ∈ N b (Σ 0 ) for i = 1, 2. Furthermore, since d 0 (z) > 0 a.s., arguing as in the previous paragraph, the π i 's must go through the point z. It remains to prove that π 1 and π 2 separate up to time θ 1 after passing through z. First, since ξ 0 (θ 1 ) is locally finite a.s., and since ξ 0 n (θ 1 n ) converges to ξ 0 (θ 1 ) a.s., the set of ξ 0 n (θ 1 n ) remains "sparse" as n goes to ∞. Since π 1 n (θ 1 n ) = π 2 n (θ 1 n ), this entails lim n π 1 n (θ 1 n ) − π 2 n (θ 1 n ) > 0, which is equivalent to π 1 (θ 1 ) = π 2 (θ 1 ).
We now need to prove that π 1 and π 2 never coincide on the time interval (t, θ 1 ) (where t is the time coordinate of z). Let us assume by contradiction that π 1 and π 2 meet again on this time interval. Since π 1 (θ 1 ) = π 2 (θ 1 ), the two paths must separate at another point z = (x , t ), with t < t < θ 1 . Since we are only considering realizations in E q,L , we must have t < q (as already mentioned at the beginning of Step 2). Let z n ∈ R n (0, q) converging to z . Since d 0 (z ) > 0 and d 0 n (z n ) converges to d 0 (z ), this implies that π n 1 and π n 2 must also go through z n after a finite rank. But this would contradict the fact that π 1 n and π 1 n separate at z n . This ends the proof of (4.13).
Step 3. We are now ready to prove (4.10). For every q, L ∈ Q + , and for almost every realization in E q,L : Finally, since P(∪ q,L ∈Q + E q,L ) = 1 by Step 1, (4.10) follows. This completes the proof of Lemma 4.6.

Proof of Proposition 4.4
Going to a subsequence, we can assume w.l.o.g. that the sign of t n remains constant as n varies. In the following, we will treat the case ∀n ∈ N, t n ≤ 0. The other case can be handled along the same lines. We decompose the proof of Proposition 4.4 into three lemmas. Then lim inf γ→0 lim inf n→∞ P (D γ,n ) = 1.
Proof. This is a direct consequence of the equicontinuity of the paths in ∪ n S n (U bn ).
Finally, the conclusion follows from the equicontinuity of ∪ n S n (U bn,κn ) and the fact that θ n 1 goes to θ 1 , with θ 1 > 0 a.s..
We are now ready to complete the proof of Proposition 4.4. It is not hard to check that for every γ > 0, Letting successively n and γ go to ∞ and 0 respectively completes the proof of Proposition 4.4.

Appendix
In this section we present three models that can be written as a voter model perturbation (VMP). In the following, we use the same notation as in Section 1.2.

Spatial stochastic Lotka-Volterra model
As in [CDP13], we consider a discrete version of the stochastic Lotka-Volterra model as introduced by Pacala and Neuhauser [NP00]. It is a spin system with each site taking value in {0, 1}. At each time step, the dynamics can easily be described by a two steps procedure (between t and t + 1). First, the particle at x dies with probability α(1 − m (1 − η(x)) f x,η (1 − η(x))), where m is a function from {0, 1} to (0, 1) and α ∈ [0, 1].
Secondly, if it dies, it is replaced by the type of one its neighbor chosen according to the transition kernel K. One can get a symmetric expression for η(x) = 1. From there, one can directly check that (5.1) holds.

The q colors Potts model
In this example, we show how our particular scaling for the voter model perturbation emerges naturally when considering a stochastic Potts model at low temperature. where η(x) is the x coordinate of the configuration η. Interpreting {1, ..., q} as a set of colors, H simply counts the number of boundaries between the color-clusters of the system. In dimension 1, it is well known that for β < ∞ the Gibbs measure is unique, and that there is no phase transition for the (static) Potts model. We will primarily be concerned with the following discrete time model in which at each integer time, the values of η t (x) for x ∈ Z all update simultaneously with the following probabilities w β = 2(e β − 1) q + 2(e β − 1) , b β = (e β − 1) 2 q ((e 2β − 1) + q)(q + 2(e β − 1)) , κ β = q e 2β + (q − 1) This particular dynamics qualifies as a perturbation of the voter model, with the parameter corresponding to e −β and further k β e 2β → q and b β e β → q/2. Later in this section, we will discuss the relation of the Potts model Gibbs measure to this discrete time process. But first, we motivate the choice of transition probabilities by considering a continuous time voter model perturbation with transition rates given by w β , b β and k β Proposition 5.3. Let us consider the continuous time Markov process with transition rates (w β , b β , κ β ) -the transitions corresponding to a walk, branch and kill move respectively (see end of Section 1.2) -and let {B x,η } for η(x − 1) = η(x + 1) and p be the uniform distribution on {1, · · · , q} and B x,η = δ η(x−1) when η(x − 1) = η(x + 1). This model defines a contiuous time q-states Potts model at inverse temperature which is reversible with respect to Gibbs measure.
Proof. When a Poisson clock rings at site x at time t that site is updated based on the values of spins at the two nearest neighbor sites. Therefore in the detailed balance equations needed for reversibility η(x − 1) and η(x + 1) are fixed while η t− (x) makes a transition to η t (x). For c i = (c , c i , c r ) the color configuration in {1, . . . , q} 3 of (η t− (x − 1), η t− (x), η t− (x + 1)), c f the color of η t (x), with c f = (c , c f , c r ), the detailed balance equations for the rates q of the transitions c i → c f and c f → c i are q(c i → c f )/q(c f → c i ) = exp(−β∆H(c i → c f )) ,  Before returning to our discrete time model where all vertices update simultaneously, we consider a model in which, say, even (resp., odd) vertices update at even (resp., odd) integer times. We leave it as an exercise for the reader to convince herself that the detailed balance calculation in the proposition above for continuous time model implies the same for each time step in this (alternating) discrete time model.
For the (simultaneously updating) discrete model, we note two elementary properties: (A) if one restricts attention to the even (resp., odd) space-time sublattice of Z × N, these two restricted stochastic processes are independent of each other. (B) The odd sublattice restriction of this dynamics is identical to the odd sublattice restriction of the alternating discrete model. These two properties imply that an invariant measure for simultaneously updating process is P × P where P (resp., P ) is the Gibbs measure on Z restricted to Z even (Z odd ).

Noisy biased voter model
We start with a general description of this model. Let (κ , b , α) be three non-negative numbers with α ∈ [0, 1]. In a continuous time setting, the noisy biased voter model is a spin system with transition rates at x: transition to 2 f 2,η (x) + κ α.