Branching random walk with non‐local competition

We study the Bolker–Pacala–Dieckmann–Law (BPDL) model of population dynamics in the regime of large population density. The BPDL model is a particle system in which particles reproduce, move randomly in space and compete with each other locally. We rigorously prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, even infinite range, whence the term non‐local competition. This makes the particle system non‐monotone and of infinite‐range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands‐on approach. Some ideas in the proof are inspired by works on the non‐local Fisher‐KPP equation, but the stochasticity of the model creates new difficulties.


Definition of the model
In this article, we study the Bolker-Pacala-Dieckmann-Law (BPDL) model, which we also refer to as branching random walk with non-local competition (BRWNLC). The BRWNLC can be regarded as a Markov process (ξ t ) t≥0 taking values in N Z d 0 , with the interpretation that ξ t (x) is the number of particles at site x at time t, for t ≥ 0 and x ∈ Z d . Starting from an initial configuration ξ 0 consisting of a finite number of particles, the model evolves as follows: • Particles branch (or reproduce) at (constant) rate 1.
That is, for each x ∈ Z d , the transition ξ t → ξ t + δ x occurs with rate ξ t (x).
We denote by p : Z d → [0, 1] a finite range jump kernel, i.e. p(x) ≥ 0 ∀x ∈ Z d , x∈Z d p(x) = 1 and for some R 1 < ∞, p(x) = 0 for all x ∈ Z d with x ≥ R 1 . (Here, and throughout the article, we write · to denote the 2 or Euclidean norm.) Furthermore, we assume that the support of p, i.e. the set {x ∈ Z d : p(x) > 0}, is a spanning set of the vector space R d .
Then for each x ∈ Z d and y ∈ Z d , the transition ξ t → ξ t − δ x + δ x+y occurs with rate γξ t (x)p(y).
• Particles compete with each other.
The competition kernel is denoted by Λ : Z d → [0, ∞), and we assume that x∈Z d Λ(x) ∈ (0, ∞). We also assume the existence of λ > 0, fixed throughout the article, such that (1.1) More assumptions regarding the decay of Λ appear below.
A particle at x gets killed with rate i.e. for each x ∈ Z d , the transition ξ t → ξ t − δ x occurs with rate K t (x)ξ t (x).
For ξ ∈ N Z d 0 , write P ξ for the probability measure under which (ξ t ) t≥0 is a BRWNLC with ξ 0 = ξ. A precise construction of the model is given in Section 2.
We further define (1. 3) The parameter N should be interpreted as the local population density or carrying capacity (we do not need to assume that N is an integer). We will be interested in the regime where N is large. One might think of a sequence of competition kernels Λ N = 1 N Λ 1 for some fixed Λ 1 with x∈Z d Λ 1 (x) = 1; however, our results hold in greater generality.

Main results
We state two theorems. We will need an extra condition on the exponential decay of the competition kernel Λ. For this, define R 2 (κ) = R 2 (κ; Λ, N ) := inf{r ∈ N : Λ(x) ≤ e −κ x N −1 ∀x ∈ Z d with x ≥ r} ∈ (0, ∞]. (1.4) The theorems below will be stated uniformly over all competition kernels such that R 2 (κ) ≤ R for some R. Note that if we consider a sequence of competition kernels Λ N = 1 N Λ 1 for some fixed Λ 1 , then R 2 (κ; Λ N , N ) = R 2 (κ; Λ 1 , 1) for all N and we can therefore choose R = R 2 (κ; Λ 1 , 1) to be equal to this value.
The first theorem concerns the global survival of the BRWNLC when N is sufficiently large: Theorem 1.1 (Global survival). There exists κ 0 = κ 0 (γ, p) > 0 such that the following holds. For δ > 0 and R > 0, there exists N 0 = N 0 (δ, γ, p, λ, R) > 0 such that if N ≥ N 0 and R 2 (κ 0 ) ≤ R then for any ξ ∈ N Z d 0 consisting of a finite number of particles with ξ ≡ 0, The next theorem describes the asymptotic spread of the BRWNLC under slightly more restrictive assumptions. Let (X t ) t≥0 denote a continuous-time random walk started at 0 with jump rate γ and jump kernel p. The Cramér transform of X 1 is expressed for u ∈ R d as E e u,X 1 = exp γ x∈Z d p(x) e u,x − 1 < ∞, (1.5) since the jump kernel p is of finite range by assumption. The rate function is expressed for v ∈ R d as (1.6) By the finiteness of the Cramér transform, I is a good convex rate function, i.e. it is convex and all sub-level sets {I ≤ a}, a ∈ R, are compact [DZ93, Lemma 2.2.31]. In particular, the set is compact and convex.
Recall the definition of the Hausdorff distance between two sets X, Y ⊆ R d : where for X ⊆ R d and > 0, (1.8) Theorem 1.2 (Shape theorem). For δ > 0, there exists κ = κ(δ, γ, p) such that for R > 0, there exists N 0 = N 0 (δ, γ, p, λ, R) > 0 such that if N ≥ N 0 and R 2 (κ) ≤ R then We remark that an analogue of Theorem 1.2 has been proven for branching random walk (BRW) without competition by Biggins [Big78]. Theorem 1.2 thus shows that in the limit of large population density, the spreading speed of the BRWNLC is asymptotically the same as in BRW without competition.

Discussion and comparison with the literature
Previous works on the BPDL model. The BPDL model, studied initially by Bolker and Pacala [BP97] as well as Law and Dieckmann [LD02], is a popular individual-based model in population dynamics. It has been studied under various guises in the mathematics, ecology and physics literature. Questions of interest concern global and local survival, asymptotic spread, equilibrium states, and the description of ancestral lineages. The methods used to study the model include the following: 1. moment equations and approximation by scaling limits, see e.g. [ 3. comparison with particle systems and percolation models, see e.g. [Eth04,BPZ07,BD07,BEM07].
The first method is a powerful tool allowing in particular the derivation of precise numerical estimates of various quantities of interest [CSF + 19] but, to our knowledge, has not yet been used to rigorously study the asymptotic behaviour of the process as time goes to infinity. The second method, duality, is a powerful tool, in particular for studying ancestral lineages and the equilibrium distribution, but it is restricted to certain special cases. The third method is well suited for treating questions concerning survival, asymptotic spread and ergodicity. The main technique is to compare the model with simpler models, such as oriented percolation, by means of a renormalization procedure. This method was introduced by Bramson and Durrett for analysing the contact process [BD88] and has been applied in many contexts since. However, in order to make such a comparison work, which is typically done through coupling arguments, the particle system should satisfy a property called monotonicity. This property states in particular that two copies (ξ t ) t≥0 and (ξ t ) t≥0 of the system, starting from two initial configurations ξ 0 and ξ 0 with ξ 0 (x) ≥ ξ 0 (x) for all x ∈ Z d , can be coupled in such a way that ξ t (x) ≥ ξ t (x) for all t ≥ 0 and x ∈ Z d . Unfortunately, the BPDL model is monotone if and only if the competition is on-site only, i.e. if Λ = Λ(0)δ 0 .
In order to get around this problem, several authors have introduced additional assumptions, in particular on the jump kernel p and the competition kernel Λ. For example, Etheridge [Eth04] assumes (for a variant of the model) a condition analogous to p ≥ cΛ for some c > 0, which allows her to obtain a certain monotonicity for a truncated version of the model. Birkner and Depperschmidt [BD07] assume that the model evolves in discrete time and that both p and Λ have finite range (and that Λ is a small perturbation of Λ(0)δ 0 ). While discrete time induces additional complications due to large jumps in the numbers of particles and the chaotic behaviour of the logistic map, it ensures that the dependence between the particles is of finite range only, which allows them to apply a comparison with so-called k-dependent oriented percolation.
In the current article, we allow for Λ to be of arbitrary, even infinite range, and we work in continuous time. This makes the particle system non-monotone and of infinite-range dependence. Instead of comparing the process to an oriented percolation, we implement a contour argument tailored to our process. Much work is devoted to dealing with the infinite-range dependence in space. In order not to be burdened with the dependence in time, we have introduced Assumption (1.1), which effectively allows us to treat the system as being of finite-range dependence in time. For more details of the proof, see Section 1.4.
The hydrodynamic limit. A natural approach to studying the BRWNLC in the large population limit would be to take its hydrodynamic limit, i.e. to consider the limit of (ξ t /N ) t≥0 as N → ∞, and consider the BRWNLC as a perturbation of its limit. This is indeed the underlying idea of the first method mentioned above. The hydrodynamic limit of a related model has been rigorously shown to be a certain evolution equation with a quadratic non-linearity due to the competition term [FM04]. Maybe surprisingly, little is known about the long-time behaviour of solutions to this evolution equation. One might expect it to behave similarly to its continuousspace analogue, known as the non-local Fisher-KPP equation. While the classical Fisher-KPP equation has been extensively studied since the 30's [Fis37,KPP37], with many celebrated results such as Bramson's logarithmic correction to the front position [Bra83], its non-local counterpart has spurred interest only in recent years, see e.g. [HR14,Pen18,BHR20]. The non-local Fisher-KPP equation displays intriguing behavior, such as the existence of non-constant steady states if the competition kernel is "sufficiently non-local" [HR14]. More importantly for our purposes, the study of the non-local Fisher-KPP equation greatly suffers from the lack of a parabolic maximum principle, which is the basic technical tool for the study of semi-linear parabolic partial differential equations such as the Fisher-KPP equation. Indeed, the parabolic maximum principle is crucially used in order to compare the solution to the equation to simpler functions, chosen to be super-or subsolutions to the equation. It is the analytic analogue of the probabilistic concept of monotonicity mentioned above -the lack of a parabolic maximum principle for the non-local Fisher-KPP equation is therefore a heritage of the non-monotonicity of the BRWNLC.
To circumvent this problem, the authors of [HR14,Pen18,BHR20] rely on other techniques, such as: • Focusing on the regions where the solution is small, and comparing it to the solution of the linearised equation.
• Bootstrapping: starting from "crude" global bounds, and using these bounds to obtain bounds on the regularity of the solution (for example through a certain Harnack-type inequality [BHR20]), and obtaining improved lower bounds from crude upper bounds and vice versa (for example using a Feynman-Kac formula [Pen18]).
Our proof is partly influenced by these ideas, but the stochasticity of the model adds additional difficulties. For example, the lack of deterministic global bounds requires us to handle situations where the particle density is much larger than usual, which a priori might lead to extinction in neighbouring (or more distant) regions. See Section 1.4 for more details.
Other branching particle systems with competition. We finish this section with a (biased) review of some other branching particle systems with local or non-local competition. Closely related to the BPDL model is the branching Brownian motion with decay of mass, where the competition between particles leads to a decay of their mass rather than a reduction in their numbers [ABP17]. While this system admits the non-local Fisher-KPP equation as its hydrodynamic limit [ABBP19], it differs from the BPDL model in that the population density (in terms of particle numbers) is not prescribed by a parameter N , but grows with time.
A popular model of branching random walk with competition is the so-called N -BRW, which is a one-dimensional branching random walk in which after each branching step only the N particles at the maximal positions are kept. This model was introduced by Brunet and Derrida [BD97] as a toy model to study finite-N corrections to the speed of travelling wave fronts, and has seen significant interest since in both the physics and the mathematics literature, see e.g. [BDMM06,BG10,Mai16]. The hydrodynamic limit of a closely related model, the N -BBM, has recently been shown to be a certain free boundary problem [DFPSL17,BBP19]. The genealogy of the N -BBM (and N -BRW with light-tailed jump distribution) is conjectured to converge to the Bolthausen-Sznitman coalescent on the time-scale (log N ) 3 , a fact that has been proven for certain related models [BDMM07,BBS13].
One can define a one-dimensional branching Brownian motion with local competition, based on the intersection local time of two Brownian paths. This system is dual to a certain stochastic partial differential equation, the Fisher-KPP equation with Wright-Fisher noise [Shi88]. This duality is used in [BMS] to study the "coming down from infinity" property of the process.
Maillard, Raoul and Tourniaire [MRT21] study a BRW with local competition in an environment which is space-and time-heterogeneous over a macroscopic scale 1/ . In contrast to the homogeneous case, the authors find that the spreading speed, even in the limit of large population size N → ∞, may differ from its hydrodynamic limit, i.e. the limits N → ∞ and → 0 do not commute in general.
Finally, branching particle systems with long-range dispersal and competition have been considered as well in the literature, see e.g. [BM14, BDKT20, PRT22].

Proof outline
Renormalization grid. In order to prove our results, we will take a large constant T and define a renormalization grid of edges of the form e = ((x, kT ), (x + y, (k + 1)T ) for some x, y ∈ Z d and k ∈ N 0 , with x in a suitable one-dimensional lattice and y ∈ {y − , y + } where y ± = O(T ). For each edge e, we will say that the edge is 'closed' if an event C e occurs; this event will be defined in such a way that on the event (C e ) c , if there are at least J = N 1/3 particles at some site near x at time kT then there will be at least J particles at some site near x + y at time (k + 1)T . (We could have chosen any J such that 1 J N 1/2 , but for definiteness, we set J = N 1/3 throughout the remainder of the article.) See Section 3 for precise definitions. The main intermediate step in the proof of Theorems 1.1 and 1.2 will be to show that for any b > 0, if N is large enough then for any n ∈ N and for a suitable collection of edges e 1 , . . . , e n and a suitable initial configuration ξ, (1.9) see Proposition 3.2 below. Using (1.9), we will be able to establish our results using contour arguments.
Dependence in time. We now outline the proof of (1.9). We first deal with the dependence in time of the events C e i . Suppose we are given a subset of edges e i 1 , . . . , e i l of the form ((x j , kT ), (x j + y j , (k + 1)T )) for some x j , y j ∈ Z d and some k ≥ 1, i.e. the time-coordinates are the same for all edges in the subset. We will be able to bound the probability of the event l j=1 C e i j independently of the process up to time (k − 1)T . In other words, we will be able to bound the probability in (1.9) as if the events C e 1 , . . . , C en were 2-dependent in time. The key to this is Proposition 4.1, which allows us to bound the configuration at time kT conditioned on time (k − 1)T , uniformly over all configurations at time (k − 1)T , by a random configuration (not depending on N ) multiplied by N . One may call this a "coming down from infinity" property of the particle system. 1 Underlying this is the fact that the solution to the logistic equation comes down from infinity in finite time. Assumption (1.1) is crucially used here: if there are aN particles at a site x, then since Λ(0) ≥ λN −1 , particles at x are killed by other particles at the same site x at rate Λ(0)(aN ) 2 ≥ λa 2 N , whereas new particles are born at rate aN . We point out that we do not use duality in order to prove Proposition 4.1.
Dependence in space. Assume from now on that the start time for each edge e i is the same kT for each i. We can also assume that none of the edges are too close together (by removing at most a constant proportion of the edges). Recall that a particle at site x at time t is killed at rate K t (x). Let > 0 be a small constant, and for an edge e = ((x 0 , kT ), (x 0 + y 0 , (k + 1)T )), let T e denote a tube from (x 0 , kT ) to (x 0 + y 0 , (k + 1)T ) with radius T , i.e.
1 The term "coming down from infinity" is often used in coalescent theory and in the study of one-dimensional diffusions and birth-and-death processes, and has a precise meaning in these settings, see e.g. [Pit99, CCL + 09, BMR16]. For a particle system, one could define it to mean that the system can be defined for an infinite initial configuration as a unique limit when starting from an increasing sequence of finite initial configurations approaching the initial configuration in a certain sense. See Theorem 2 in Hutzenthaler and Wakolbinger [HW07], where this is shown for a certain system of interacting Feller diffusions with logistic growth. The BRWNLC is in general not monotone, and we do not prove the existence of a unique limit, but we do prove a form of local boundedness uniformly in the initial configuration, which is why we still use the term in a loose sense.
We now consider three possible cases for the values of K t (x) in the tube T e . Let c > 0 be a small constant and let C > 0 be a large constant.
Case 1: K t (x) ≤ c for all (x, t) ∈ T e . In this case, the killing rate inside the tube is very small, so for suitable y 0 , we can use large deviation results to show that if there are J particles at a site near x 0 at time kT , they are likely to have more than J descendants which stay inside the tube T e , are not killed by competition, and are at a site near x 0 + y 0 at time (k + 1)T .
Case 2: K t (x) ≤ C for all (x, t) close to T e , but K t * (x * ) > c for some (x * , t * ) ∈ T e . In this case, at time t * there will be some site y * fairly close to x * at which there are at least N 1/2 particles. (Indeed, if ξ t * (y) < N 1/2 for all y fairly close to x * , then due to the decay of the competition kernel Λ, we must have that ξ t * (y)/N is extremely large for some y further from x * ; we will rule out this possibility using the same argument as for Case 3 below.) Since N 1/2 J, and since the killing rate of particles near T e is at most C, we can show that these N 1/2 particles are likely to have at least J surviving descendants at a site near x 0 + y 0 at time (k + 1)T .
Case 3: K t * (x * ) > C for some (x * , t * ) near T e . We will show that this is unlikely to occur if C is sufficiently large, using the upper bound on the particle configuration at time kT (conditioned on time (k − 1)T ) mentioned above.
The heuristics for the three cases above suggest that the event C e is unlikely to occur. In order to prove (1.9) using these heuristics, we will construct the process (ξ t ) t≥0 using independent families of decorated BRW trees; each particle in the BRW trees will be assigned an independent Exp(1) random variable that will determine the time at which it may be killed by competition (as a function of the killing rate on its trajectory).
Let C > 0 be a large constant. In our construction, at time kT , at most C N particles at each site x will be coloured blue, and the remaining particles at the site will be coloured red. (We will have a stochastic upper bound on the number of red particles, using our bound on the particle configuration at time kT .) The descendants of each particle will be constructed using the independent decorated BRW trees. Using the heuristic in Case 1 above, we can define a 'bad event' B e depending only on the BRW trees for blue particles near x 0 such that on the event (B e ) c , if Case 1 happens then there are at least J blue particles near x 0 + y 0 at time (k + 1)T and so C e cannot occur; moreover, the bad event B e has low probability.
In our construction, we will also define stopping times τ (z) for each site z, given by the first time after time kT at which there are at least N 1/2 red and blue particles at z. At a time τ (z), we turn N 1/2 red and blue particles at z into yellow particles, and their descendants will be constructed using another independent family of BRW trees. Using the heuristic in Case 2 above, we can define a 'bad event' Y e depending only on the BRW trees for yellow particles that appear near the tube T e such that on the event (Y e ) c , if Case 2 happens then there are at least J yellow particles near x 0 + y 0 at time (k + 1)T , which means that C e cannot occur; moreover, the bad event Y e has low probability.
Finally, we need to control the possibility that Case 3 occurs, by controlling the number and spatial spread of blue, red and yellow particles. We will define bad events P z,r for a site z and a 'radius' r ∈ N 0 , which heuristically say that too many descendants of particles at z spread to some distance r ≤ r from z at some time in [kT, (k + 1)T ]. The events will be defined in such a way that for z ∈ Z d , on the event ∩ z∈Z d (P z, z−z ) c we have K t (z ) ≤ C for all t ∈ [kT, (k + 1)T ]. The events P z,r will be independent for different values of z, and we will be able to show that P z,r has low probability for each r, and moreover, for a small constant a > 0, for large r, P (P z,r ) ≤ e −ar log(r+1) .
(1.10) (The proof of (1.10) will rely on the assumption that the jump kernel p has finite range.) A bad event P z,r can only affect edges within distance roughly r of z, and there are at most O(r) such edges (because for each edge e = ((x, kT ), (x + y, (k + 1)T )), x is in a one-dimensional lattice and y ∈ {y − , y + }). Therefore, since the bad events B e i and Y e i are independent for different (not too close together) edges e i , and since C e i can only occur if either B e i or Y e i occurs, or P z,r occurs for some z within distance roughly r of the edge e i , we will be able to show that for some constant a > 0, where B r = {x ∈ Z d : x < r} and the constant C > 0 can be chosen sufficiently large that (1.9) follows. Note that the bound (1.10) is sharp for this argument to work, since otherwise the sum on the right hand side of (1.11) diverges. In particular, this means that we are currently not able to weaken the assumption that the jump kernel has finite range.

Overview of the article
The remainder of the article is organised as follows. In Section 2, we construct the BRWNLC from a collection of BRW trees decorated with "resiliences". Section 3 contains the contour argument used to prove the main results, relying on Proposition 3.2. Section 4 contains the proof of Proposition 3.2. Finally, Section 5 contains certain estimates for sums of independent random variables, which are the building blocks of the proof of Proposition 3.2.
As mentioned in Section 1.1, we write · for the 2 or Euclidean norm. For x ∈ R d and r ≥ 0, we let B r (x) := {z ∈ Z d : x − z < r}. (1.12) (1.13)

Branching random walks
The branching random walks (BRW) we consider here are continuous-time BRW on Z d , started with a single particle at 0, where each particle branches into two child particles at rate 1, located at the position of their parent, and furthermore, each particle jumps at rate γ according to the jump kernel p. (When a particle branches, we say the parent particle 'dies' and the two child particles are 'born'.) For such a branching random walk, we introduce notation as follows: • Particles are given labels from the set of labels U = ∪ ∞ n=0 {1, 2} n according to Ulam-Harris labelling, i.e. the label 12 corresponds to the second child of the first child of the initial particle (ordering of the children is arbitrary). We write u ≺ v for u, v ∈ U if v is a descendant of u (including u itself) in the tree U.
• For t ≥ 0, let N t ⊂ U denote the set of labels of particles alive at time t.
• Let (X t (u), u ∈ N t ) denote the locations in Z d of the particles at time t.
• For u ∈ U, for t < α(u), we write X t (u) to denote the location of the ancestor of particle u at time t.
The existence and formal construction of the process is standard and can easily be obtained by recurrence over the generations, see e.g. [Jag89,HH09]. This way, the trajectory of each particle between branch points is a continuous-time random walk with jump kernel p and jump rate γ until a finite time given by the life length of the particle, and the trajectories, independent over all particles, are "glued together" at the branch points. Equivalently, one could construct the BRW following a more modern approach using random trees: Start with a Yule tree, i.e. a binary tree where each edge has a length given by an Exp(1)-distributed random variable. Consider a Poisson process on the tree with intensity measure equal to γ times the length measure of the tree. Now define a random process indexed by the tree that jumps at the times given by the Poisson process according to the jump kernel p. See Section 3 in Duquesne and Winkel [DW07] for a construction of the Yule tree as a random metric space and the notion of Poisson processes on trees.
The many-to-one lemma for branching Markov processes allows us to calculate additive functionals of the branching Markov process in terms of a single Markov process in a potential. See for example Section 8 in Hardy and Harris [HH09] for a modern presentation using change of measure techniques. Here we will use the following version, which is a special case of Corollary 8.6 in that article.
Lemma 2.1 (Many-to-one lemma [HH09]). Let F be a non-negative path functional. Let (X t ) t≥0 be a continuous-time random walk started at 0 with jump kernel p and jump rate γ. Then The following result will also be needed. Note that |N t |, the number of particles at time t, is a Yule process, i.e. a continuous-time Galton-Watson process with binary offspring distribution. This implies the following classical result:

Resilience and BRW with non-local competition
Let a BRW tree be defined as in the previous section. Let (ρ(u), u ∈ U) be i.i.d. Exp(1) random variables. We call ρ(u) the resilience of the particle u. We denote by the tuple representing the BRW tree with resiliences. Such a tree will be used to encode the evolution of the descendants of a given particle in the process (ξ t ) t≥0 , represented by the root of the tree. For u ∈ U, a BRW tree with resiliences T = ((N t ) t≥0 , (X t (u), t ≥ 0, u ∈ N t ), (α(u), u ∈ U), (β(u), u ∈ U), (ρ(u), u ∈ U)), a function κ = (κ t (y)) t≥0,y∈Z d and a location x ∈ Z d , define with the convention that inf ∅ = ∞. This quantity will be interpreted as the length of time between the birth of the particle u and its death by competition (if this occurs before its "death by branching"), given that κ t (y) is the killing rate experienced by particles at position y at time t and that the root particle of the BRW tree T is at position x at time 0.
The process (ξ t ) t≥0 can now be defined as a deterministic function of a collection of BRW trees with resiliences. Take a collection T 1 , . . . , T n of i.i.d. copies of T , representing the descendants of n particles at positions x 1 , . . . , x n at time 0.
Define K t (x) = ξ t * Λ(x). Now let . . , n}, u ∈ U and denote by (i * , u * ) the minimiser of this quantity. The time σ 1 is interpreted as the first time a particle is killed by competition. We set ξ t = ξ t for t < σ 1 .
We then iterate this process, but ignoring the particle u * in T i * and its descendants from time We continue like this to define ξ t at all times. Then (ξ t (x), x ∈ Z d ) t≥0 is a BRWNLC with ξ 0 (x) = n i=1 1 x i =x . We write (F t ) t≥0 for the natural filtration of the process (ξ t ) t≥0 .

Proof of Theorems 1.1 and 1.2 (contour arguments)
Recall the definition of the rate function I in (1.6). Define its domain by The following lemma gathers a few properties of I and dom(I). Recall the definition of µ in (1.3).
Lemma 3.1. The rate function I satisfies the following properties: 1. The closure of dom(I) is the closed cone generated by the convex hull of the support of the jump kernel p; in other words, it is the set of linear combinations with non-negative coefficients of points x ∈ Z d such that p(x) > 0.
Proof. We provide a proof for completeness. Recall that (X t ) t≥0 is a continuous-time random walk started at 0 with jump rate γ and jump kernel p. By Cramér's theorem, I is the good convex rate function in the large deviation principle for X n /n [DZ93, Theorem 2.2.30] and therefore the closure of its domain equals the closed convex hull of the support of X 1 , see e.g. [Lan73,Pet18], see also [Big78, Lemma 1]. Now, X 1 is the sum of a Poisson(γ) number of i.i.d. random variables with law p, and therefore the convex hull of its support is easily shown to be the closed cone generated by the convex hull of the support of p, see for example Section 6 in [Big78]. This proves the first point. For the second point, recall from our assumptions on p that the set S(p, γ) := {γx : x ∈ Z d , p(x) > 0} is a spanning set of the vector space R d . Hence, since dom(I) includes this set as well as the origin 0 ∈ R d , its affine hull is the whole space R d . It is therefore a convex set of dimension d with non-empty interior dom(I) • (see e.g. [Roc70, Section 2] for basic properties of convex sets). Now we have that µ, which is a convex combination of points in S(p, γ), is contained in the relative interior of the convex hull of S(p, γ) [Roc70, Theorem 6.9], and hence in the relative interior of dom(I), by the first part of the lemma and the fact that the relative interior of a convex set is contained in the relative interior of the cone it generates [Roc70, Renormalization grid. Recall the definition of I 1 in (1.7). For v ∈ R d \{0}, let a v = sup{a ≥ 0 : I(µ + av) ≤ 1} = sup{a ≥ 0 : µ + av ∈ I 1 }. (3.1) Note that a v > 0, because by Lemma 3.1 we have that µ ∈ dom(I) • , I(µ) = 0 and I is continuous on dom(I) • . Furthermore, a v < ∞, because I 1 is compact. Furthermore, we have µ + a v v ∈ I 1 , since I 1 is compact, and hence closed. The function v → 1/a v is also called the gauge function of the convex set I 1 − µ [Roc70, Section 4].
Recall the definition of x for x ∈ R d from (1.13), and recall the definition of B r (x) from (1.12). Take R , T > 1 large positive constants to be fixed later, set J = N 1/3 and x y x y Figure 1: Schematic representation of the renormalization grid defined in Section 3. The arrows represent the edges e of the grid. Open edges are drawn with solid lines, closed edges with dotted lines. An edge from (x, kT ) to (y, (k + 1)T ) is closed if there are at least J particles at some site x ∈ B R (x) at time kT but there are less than J particles at every site y ∈ B R (y) at time (k + 1)T .
take v 0 ∈ R d \{0} and a − , a + ≥ 0. We introduce a renormalization grid, see Figure 1 for a schematic illustration. Define sets of vertices by letting Then define the sets of directed edges for k ∈ N 0 , For an edge e = ((x, s), (y, t)) ∈ E T,a − ,a + ,v 0 , define the event (At first reading, the reader can safely ignore the third event in the definition of C e and concentrate on the first two events; the third event will only be relevant at the end of the proof of Theorem 1.2.) We say that e is closed if C e occurs and e is open otherwise. We want to show by a contour argument (also known as a Peierls argument) that there exists an infinite cluster of open edges with high probability when starting from a "good" initial configuration. The key to this argument is the following proposition.
We will prove Proposition 3.2 in Sections 4-5; in the remainder of this section, we use Proposition 3.2 and the construction in Section 2 to prove Theorem 1.1 and Theorem 1.2.
Proof of Theorem 1.1. Suppose N ≥ 1 (and so J = N 1/3 ≥ 1). Define the stopping times For ξ ∈ N Z d 0 consisting of a finite number of particles with ξ ≡ 0, by conditioning on F τ 1 and applying the strong Markov property, where the last line follows by translational symmetry. Then by conditioning on F τ 2 and applying the strong Markov property again, We now bound the second term on the right hand side of (3.5). Construct the BRWNLC process (ξ t ) t≥0 with initial configuration ξ 0 = δ 0 as in Section 2, using the BRW tree with resiliences T = (

and define the event
We claim that if τ 2 > 3 2 log J then A c occurs. Indeed, suppose (aiming for a contradiction) that A occurs and τ 2 > 3 2 log J. Then for x ∈ Z d and t ∈ [0, 3 2 log J], we have by the definition of τ 2 in (3.3) and since z∈Z d Λ(z) = N −1 . Therefore, by our construction in Section 2, and since ρ(u) > J N min By the definition of the event A, it follows that which implies that τ 2 ≤ 3 2 log J and gives us a contradiction, proving that the claim holds. We now establish an upper bound on P (A c ).
t ) t≥0 denote a Poisson process with rate γ. By the many-to-one lemma (Lemma 2.1), and since the jump kernel p is supported on B R 1 (0), where the last line follows by Markov's inequality. By the many-to-one lemma again, and since Hence by (3.7), (3.8) and (3.9) and a union bound, for N sufficiently large, since J = N 1/3 . By (3.4), (3.5) and (3.10), the proof now reduces to showing that there exists κ 0 > 0 such that for δ > 0 and R > 0, if N is sufficiently large and R 2 (κ 0 ) ≤ R, then for any ξ ∈ N Z d 0 consisting of a finite number of particles with ξ (x) ≤ J ∀x ∈ Z d and ξ (x * ) = J for some x * ∈ Z d , (3.11) By translational symmetry, we may assume that x * = 0. We will now prove (3.11) under this assumption using Proposition 3.2 and a contour argument.
Define sets of directed edges in the lattice Z 2 by letting . Take κ > 0 as in Proposition 3.2; fix R > 0 and suppose R 2 (κ) ≤ R. Take T > 1 sufficiently large that Proposition 3.2 holds and (µ + a − v 0 )T = (µ + a + v 0 )T , and take R > 1 as in Proposition 3.2. Take ξ ∈ N Z d 0 consisting of a finite number of particles with ξ (x) ≤ J ∀x ∈ Z d and ξ (0) = J, and take a BRWNLC For a directed edge e ∈ E 0 , we say that the edge e is closed if the event Note that if ξ t ≡ 0 for some t > 0, then by the definition of the event C e in (3.2), and since ξ 0 (0) ≥ J, we must have |A| < ∞.
Define the set of directed edges in the dual lattice by letting Note that each dual edge e * ∈ E * crosses exactly one edge in E; write c(e * ) for this edge. When travelling along e * , either the start vertex or the end vertex of c(e * ) is on the right; call this vertex r(e * ) ∈ Z 2 , and call the vertex on the left Therefore, if |A| < ∞, then by following a path of distinct edges in E * starting with the edge e * 0 := ((− 1 2 , − 1 2 ), (− 1 2 , 1 2 )) such that for each edge e * in the path, r(e * ) ∈ A and l(e * ) / ∈ A, we can see that there exists a cycle Γ of edges in E * containing e * 0 such that for each edge e * ∈ Γ ∩ E * ,se , the edge in E that it crosses is closed (i.e. C P −1 T,a − ,a + ,v 0 (c(e * )) occurs). For δ > 0, take b ∈ (0, 1) sufficiently small that ∞ =4 3 −1 b /2 < δ/2, and then take N sufficiently large that Proposition 3.2 holds with this choice of b. Let G denote the set of cycles of edges in E * containing e * 0 . Note that each Γ ∈ G must have length at least 4, and for ≥ 4, the number of cycles in G with length is at most 3 −1 . Moreover, if Γ ∈ G has length , then |Γ ∩ E * ,se | = /2. Therefore by a union bound, where the second inequality follows by Proposition 3.2 under condition 3. This establishes (3.11), and completes the proof.
Proof of Theorem 1.2. Recall from (1.8) that for X ⊆ R d , the set X denotes the -fattening of X. We begin by proving the upper bound, i.e. we show that for every > 0, we have This follows from classical results on branching random walks: ) be a BRW tree with resiliences as in Section 2. By our construction in Section 2, for t ≥ 0, But for the BRW, it was shown by Biggins [Big78] (see page 79 of that paper) that for every > 0, This proves the upper bound (3.12). We now prove the lower bound, i.e. we want to show that for every > 0, there exists κ > 0 such that for R > 0, if N is sufficiently large and R 2 (κ) ≤ R, we have Using (3.10) in the proof of Theorem 1.1 and the definition of the event A in (3.6), it suffices to show that for every > 0, there exists κ > 0 such that for R > 0, if N is sufficiently large and R 2 (κ) ≤ R, for any initial condition ξ ∈ N Z d 0 consisting of a finite number of particles with ξ(y) ≤ J ∀y ∈ Z d and ξ(0) = J, and any x * ∈ B (log J) 2 (0) and t * ∈ [0, 3 2 log J], Recall the definition of a v in (3.1), and recall that we observed in (1.7) that I 1 is convex and compact. It follows that a v ]} and hence by a covering argument, it is enough to show that for every v 0 ∈ R d with v 0 = 1, every a ∈ (0, a v 0 ) and every δ, > 0, there exists κ > 0 such that for R > 0, if N is sufficiently large and R 2 (κ) ≤ R, for any initial condition ξ as above, But this is done using the exact same contour argument as in the proof of Theorem 1.1, except that we take a − = a− and a + = a+ , for 0 < < sufficiently small that 0 ≤ a − < a + < a v 0 , and take T sufficiently large that d 1/2 T −1 + < . Here, the last event in the definition of the event C e from (3.2) is used to obtain (3.13) for arbitrary t ≥ t 0 instead of only for multiples of T .

Proof of Proposition 3.2
We will use the following notation throughout Sections 4 and 5. For r ≥ 0, let f (r) = exp(− r 9R 1 log(r + 1)) (4.1) and g(r) = 1 ∧ r −6d−2 . (4. 2) The function f will be used to bound the probability of some "bad events" which describe the spread of particles over distances of order r. The function g on the other hand is used to control the number of such particles in these bad events. We will prove Proposition 3.2 using the following two results. The first result gives us a stochastic upper bound on the particle system at time 1 that holds for any finite initial particle configuration. One may phrase this as a "coming down from infinity" property of the process. Its proof heavily relies on the presence of some competition between particles on the same site, i.e. on assumption (1.1).
For ε > 0, let Z ε be a random variable taking values in N 0 , with Proposition 4.1 (Coming down from infinity). There exists K 0 = K 0 (d, γ, R 1 , λ) > 1 such that the following holds. For ε > 0, there exists N 1 > 0 such that if N ≥ N 1 and ξ is a starting configuration consisting of a finite number of particles, then there exists a coupling between (ξ t ) t≥0 and (Z (x) ε ) x∈Z d such that under the coupling, ξ 0 = ξ and Proposition 4.1 will be proved in Section 4.1. The next result gives us an upper bound on the probability that a collection of edges are all closed, which holds if the (random) initial particle configuration is bounded above in the same way as ξ 1 is bounded above in Proposition 4.1.
Proposition 4.2. For v 0 ∈ R d \{0} and 0 ≤ a − < a + < a v 0 , there exists κ = κ(γ, p, v 0 , a − , a + ) > 0 such that the following holds. For T > 1 sufficiently large and R > 0, there exists R > 1 such that for every b ∈ (0, 1), there exist ε > 0 and N 2 > 1 such that the following holds: Let N ≥ N 2 , n ∈ N and e 1 , e 2 , . . . , e n ∈ E T,a − ,a + ,v 0 (0). Suppose R 2 (κ) ≤ R. Suppose ξ is a (random) configuration of a finite number of particles, and suppose there exists a coupling between ξ and (Z (x) ε ) x∈Z d such that under the coupling, where ζ, ζ (x) are as in Proposition 4.1. Then This result will be proved in Section 4.2 below, using the strategy outlined in Section 1.4. We now show how Propositions 4.1 and 4.2 can be used to prove Proposition 3.2.
Proof of Proposition 3.2. For v 0 ∈ R d \{0} and 0 ≤ a − < a + < a v 0 , take κ > 0 and T > 1 such that Proposition 4.2 holds. Take R > 0 and then take R as in Proposition 4.2. Take b ∈ (0, 1), and then take ε > 0 as in Proposition 4.2, and assume R 2 (κ) ≤ R and N is sufficiently large that Propositions 4.1 and 4.2 hold.
The result under condition 1 now follows from Propositions 4.1 and 4.2. Indeed, take e 1 , . . . , e n ∈ E T,a − ,a + ,v 0 (1) with e i = ((x i , T ), (y i , 2T )), let ξ be an arbitrary particle configuration consisting of a finite number of particles, and take a BRWNLC (ξ t ) t≥0 with ξ 0 = ξ. Since ξ T −1 consists of a finite number of particles almost surely, by Proposition 4.1 there exists a coupling between (ξ t ) t≥0 and (Z (x) ε ) x∈Z d such that under the coupling, This implies the result under condition 1.
The result under condition 2 follows directly from Proposition 4. Now take e 1 , . . . , e n ∈ E T,a − ,a + ,v 0 arbitrary, and suppose ξ ∈ N Z d 0 consists of a finite number of particles with ξ(x) ≤ N ∀x ∈ Z d . In order to prove the result under condition 3, we first divide the edges into two sets: We distinguish between two cases: Case 1: |E odd | ≥ n/2. Let k m be the largest odd k such that E odd ∩ E T,a − ,a + ,v 0 (k) = ∅. We then write Using for each conditional probability the bound from Proposition 3.2 under condition 1, with b 2 instead of b, we get Case 2: |E odd | < n/2. This is similar to the last case, using the edges from E even instead. The only difference is that instead of Proposition 3.2 under condition 1, the result under condition 2 has to be applied for the edges in E even ∩ E T,a − ,a + ,v 0 (0), using the assumption that ξ(x) ≤ N ∀x ∈ Z d . This completes the proof.

Proof of Proposition (coming down from infinity)
Take a particle configuration ξ consisting of a finite number of particles. We first introduce a convenient construction of the BRWNLC process on the time interval [0, 1], building on the construction using the BRW trees with resiliences in Section 2.
Take K 1 > 0 a large constant and c 0 ∈ (0, 1) a small constant to be fixed later. We let ξ 0 = ξ, and on the time using (for example) the construction in Section 2. From time 1 − c 0 onwards, the details of the construction will be important and we will split the particles into blue and red particles.
Initially, at time 1 − c 0 , all particles will be blue. The heuristic to have in mind for the construction below is that 'well-behaved' particles remain blue, but if there are too many blue particles that have come from some site x to some site y, these 'badly-behaved' particles at y will be coloured red. Our construction will ensure that by time 1, there will be at most 2K 1 N blue particles at each site, and by establishing an upper bound on the probability of particles from x being coloured red at y (and, in particular, using the decay of the upper bound in terms of x − y ), we will be able to prove Proposition 4.1. We now give the details of the construction.
For x ∈ Z d , y ∈ Z d , k ∈ N and j ∈ N, let T blue,x,k,j and T red,x,y,k,j be i.i.d. copies of the BRW tree with resiliences T (as defined in (2.1)). The corresponding entries will be denoted by N * , X * , α * , β * , ρ * , where * is replaced by the corresponding superscripts. The process (ξ t ) t∈[1−c 0 ,1] will be constructed as a deterministic function of ξ 1−c 0 and these BRW trees with resiliences; the collections of trees (T blue,x,k,j ) x,k,j and (T red,x,y,k,j ) x,y,k,j will encode the behaviour of blue and red particles respectively.
Note that since ξ consists of a finite number of particles, For k ∈ k , we successively construct the particle system on the time intervals (t k , t k−1 ], starting with k = k. The particles will be split into blue and red particles. Whenever a particle branches, both offspring particles inherit the colour of the parent. At time t k = 1 − c 0 , all particles are blue. Blue particles may become red at times t k−1 , for k ∈ k , according to a rule that we will specify below. Red particles and their descendants remain red until time 1. For 0 ≤ k ≤ k and x ∈ Z d , we denote by ξ blue,k (x) the number of blue particles at x at time t k . For 0 ≤ k ≤ k − 1, we need to keep track of the (blue) ancestor at time t k+1 of the particles that turn red at time t k . We therefore denote by ξ red,x,k (y) the number of particles at y ∈ Z d which turn red at time t k and are descendants of a blue particle positioned at x ∈ Z d at time t k+1 .
We start by defining the particle system on the time interval (t k , t k−1 ]. At time t k , all particles are blue. The particle system is then constructed as in Section 2 from the BRW trees with resiliences T blue,x,k,j for x ∈ Z d , j ∈ ξ blue,k (x) , with the root of T blue,x,k,j positioned at x and with time starting at time t k . Recall (4.2) and define a "threshold function" thr(r, k) := c 1 2 k g(r) 1/3 K 1 N for r ≥ 0 and k ∈ N 0 , (4.8) where c 1 ∈ (0, 1) is chosen sufficiently small that At time t k−1 , some particles may be coloured red by the following rule: For each x, y ∈ Z d , if the blue particles at x at time t k have more than thr( x − y , k) descendants at y at time t k−1 , then turn these descendants red at time t k−1 . On the subsequent time intervals, the construction is similar, but takes into account the red particles created in the previous steps. Specifically, take k ∈ {1, 2, . . . , k − 1}. The particles living between times t k and t k−1 will be subsets of the particles in the following BRW trees with resiliences: • The descendants of the blue particles at time t k will come from the BRW trees with resiliences T blue,x,k,j for x ∈ Z d , j ∈ ξ blue,k (x) , with the root of T blue,x,k,j positioned at x at time t k .
• The descendants of particles which have turned red at a time t k , k ≥ k, will come from the BRW trees with resiliences T red,x,y,k ,j for x, y ∈ Z d and j ∈ ξ red,x,k (y) , with the root of T red,x,y,k ,j positioned at y at time t k .
Using these BRW trees with resiliences, the particle system is constructed on the time interval (t k , t k−1 ] by an obvious extension of the construction from Section 2, taking into account the different times at which the particles corresponding to the roots of the trees appear, and removing from the trees T red,x,y,k ,j with k > k those particles that have been killed by competition before time t k (and their descendants). At time t k−1 , some blue particles may be coloured red by the same rule as above: For each x, y ∈ Z d , if the blue particles at x at time t k have more than thr( x − y , k) descendants at y at time t k−1 , then turn these descendants red at time t k−1 . This completes the construction of the particle system up to time t 0 = 1. Note that by the definition of k in (4.5), we have ξ blue,k (x) ≤ 2 k+1 K 1 N ∀x ∈ Z d . Moreover, for 1 ≤ k < k and x ∈ Z d , by our construction, where the second inequality follows by (4.9). Summarising, we have the following deterministic bound on the number of blue particles: (4.10) It remains to control the number of red particles created during the process.
Controlling the creation of red particles. For x, y ∈ Z d , let η (x) (y) denote the number of red particles at y at time 1 whose ancestor turned red at time t k−1 for some 1 ≤ k ≤ k and whose time-t k blue ancestor was at x. In order to control the creation of red particles, we define a family of "bad events" (R x,r ) x∈Z d , r∈N 0 which have the following properties: • If R x,r does not occur, then at time 1 there are at most O(r d g(r) −1 N ) red particles whose ancestor turned red at some time t k−1 and whose time-t k blue ancestor was at x, and at most O(r d−1 g(r)N ) of those particles are outside B r (x) (see Lemma 4.3).
• The probability of R x,r is small and decays like f (r) when r → ∞ (see Lemma 4.4).
• For every x ∈ Z d , the events (R x,r ) r≥0 only depend on the BRW trees with resiliences (T blue,x,k,j ) k,j and (T red,x,y,k,j ) y,k,j . In particular, if x 1 , . . . , x n ∈ Z d are pairwise distinct and r 1 , . . . , r n ∈ N 0 , then R x 1 ,r 1 , . . . , R xn,rn are independent.
The events R x,r will be defined as a union of several other bad events. The formal definitions are given below. Informally, the role of these events is as follows: • (R cr x,k ) x∈Z d ,k∈N : if the event R cr x,k does not occur, then no descendants of blue particles at x at time t k are turned red at time t k−1 (the abbreviation "cr" stands for "creation").
• (R num x,k,r ) x∈Z d ,k,r∈N : if the event R num x,k,r does not occur, then at most g(r) −1/5 2 k+2 K 1 N descendants of blue particles at x at time t k turn red at time t k−1 , and none of the descendants turn red outside B r/2 (x) (the abbreviation "num" stands for "number").
• (R x,y,k,r ) x,y∈Z d ,k,r∈N : if the event R x,y,k,r does not occur and ξ red,x,k (y) ≤ g(r) −1/5 2 k+3 K 1 N , then these ξ red,x,k (y) red particles at y at time t k will have at most r −d+o(1) g(r) −1 N descendants and at most (r ) −d+o(1) g(2r )N of them spread further than r , for every r ≥ r/2.
We now give the formal definition of these events, which the reader might skip at first reading.
We will prove Lemma 4.3 at the end of this section. The second lemma bounds the probability of the bad event R x,r ; we will prove this result in Section 5.
We now show how Lemmas 4.3 and 4.4 can be used to prove Proposition 4.1.
Proof of Proposition 4.1. Take K 0 > 1 sufficiently large that and fix ε > 0. Recall that we let ξ blue,0 (x) denote the number of blue particles at x ∈ Z d at time t 0 = 1. Then for y ∈ Z d , we have For y ∈ Z d , by our construction, and then using (4.8), by (4.9). Note that for x ∈ Z d , the events (R x,r ) r∈N 0 depend only on the BRW trees (T blue,x,k,j ) k,j∈N and (T red,x,y,k,j ) y∈Z d ,k,j∈N . By Lemma 4.4, we have that for N sufficiently large, for x ∈ Z d and r ∈ N 0 , where the last equality follows by (4.3). Therefore we can couple (T blue,x,k,j ) x∈Z d ,k,j∈N and ε ] such that (R x,rx ) c occurs, and so by Lemma 4.3, for y ∈ Z d , where the second line follows since r → g(r) −1 is non-decreasing and g(r) ≤ 1 for r ≥ 0 by (4.2). By (4.19), for y ∈ Z d we can write Then ξ 1 = ζ + x∈Z d ζ (x) , and by (4.21) and (4.18), for x, y ∈ Z d , Moreover, by (4.20) and (4.21), where the second inequality follows by (4.18).
We now finish this section by proving Lemma 4.3. We first prove that on the event (R cr x,k ) c , no descendants of blue particles at x at time t k are turned red at time t k−1 : Lemma 4.5. For x ∈ Z d and k ∈ k , Proof of Lemma 4.5. We first explain the heuristics behind the argument. We differentiate between the particles turning red at some site y = x and those which turn red at x. For y = x, if the event R cr,4 x,k does not occur, it is a direct consequence of its definition that the number of descendants at y at time t k−1 does not exceed the threshold and so none are coloured blue. If y = x, the argument is more subtle and makes use of the competition between particles. First, if the events R cr,2 x,k and R cr,3 x,k both do not occur, then most particles at x will neither branch nor jump before time t k−1 (this will later be shown to be likely even for small k because c 0 has been chosen sufficiently small). Second, if moreover R cr,1 x,k does not occur, then only a small proportion of those particles have an exceedingly large resilience. It will follow that if the number of blue particles at x at time t k exceeds the time-t k−1 threshold, most of these particles get killed by competition, thus leaving only a small fraction of these particles by time t k−1 , much smaller than the threshold. Using again the fact that R cr,2 x,k and R cr,3 x,k do not occur, one can moreover make sure that the particles which have branched or jumped do not suffice to make the total number of particles exceed the threshold. Hence, no particle turns red at x.
We now get to the formal details. Let x ∈ Z d and k ∈ k . Recall that ξ blue,k (x) ≤ 2 k+1 K 1 N by (4.10). The descendants on the time interval [t k , t k−1 ] of the blue particles at x at time t k are encoded by the BRW trees with resiliences (T blue,x,k,j ) j≤ξ blue,k (x) . For j ≤ ξ blue,k (x) and K t k +s (x+X blue,x,k,j s (v)) ds ∀v ≺ u (4.22) for the set of particles at time t k + t from the BRW tree T blue,x,k,j which have not been killed by competition, and whose ancestors were not killed by competition.
By the definition of the event (R cr,1 x,k ) c and by (4.10) we have that which contradicts the assumption that S x,k t k−1 −t k > c 1 2 k−2 K 1 N . Therefore S x,k t k−1 −t k ≤ c 1 2 k−2 K 1 N . Now suppose that the event (R cr x,k ) c occurs. Then since A blue,x,k,j t k−1 −t k ⊆ N blue,x,k,j t k−1 −t k for each j ≤ ξ blue,k (x), and by the definition of S x,k t in (4.23), where the second inequality follows by the definition of the events (R cr,2 x,k ) c and (R cr,3 x,k ) c and since ξ blue,k (x) ≤ 2 k+1 K 1 N by (4.10) and t k−1 − t k ≤ c 0 2 1−k by (4.7). Hence, in particular, the blue particles at x at time t k have less than c 1 2 k K 1 N descendants at x at time t k−1 , and so, since g(0) = 1 and therefore thr(0, k) = c 1 2 k K 1 N , by our construction we have ξ red,x,k−1 (x) = 0.
By the definition of the event (R cr,4 x,k ) c , and since ξ blue,k (x) ≤ 2 k+1 K 1 N by (4.10) and t k−1 − t k ≤ c 0 2 1−k by (4.7), for r ≥ 1, we have In particular, for any y = x, the blue particles at x at time t k have at most thr( x − y , k) descendants at y at time t k−1 . Hence by our construction, ξ red,x,k−1 (y) = 0 ∀y = x. We have now established that on the event (R cr x,k ) c , we have ξ red,x,k−1 (y) = 0 ∀y ∈ Z d . This proves the lemma.
We now finally prove Lemma 4.3.
Proof of Lemma 4.3. Take x ∈ Z d and r ∈ N 0 , and suppose the event (R x,r ) c occurs. We begin by considering the case r = 0; in this case, (R cr x,k ) c occurs for each k ≥ 1, and so ξ red,x,k−1 (y) = 0 ∀y ∈ Z d , k ∈ k , by Lemma 4.5. Therefore η (x) (y) = 0 ∀y ∈ Z d and the statement of the lemma holds.
From now on we assume r ≥ 1. Recall the definition of k r in (4.11). For k > k r , since (R cr x,k ) c occurs we have that ξ red,x,k−1 (y) = 0 ∀y ∈ Z d , by Lemma 4.5. For k ∈ k r , by the definition of the event (R num x,k,r ) c and since ξ blue,k (x) ≤ 2 k+1 K 1 N by (4.10) and t k−1 − t k ≤ c 0 2 1−k by (4.7), Moreover, for y ∈ Z d with x − y ≥ r/2, the blue particles at x at time t k have at most thr( y − x , k) descendants at y at time t k−1 , so by our construction, ξ red,x,k−1 (y) = 0 ∀y / ∈ B r/2 (x).
We have now established that no blue particles at x at a time t k with k > k r have descendants that turn red at time t k−1 , and no blue particles at x at any time t k with k ≥ 1 have descendants that turn red at time t k−1 outside B r/2 (x). Note that for y ∈ B r/2 (x) and y / ∈ B r (x), we have y − y ≥ x − y − x − y > r/2 > x − y , which implies x − y ≤ x − y + y − y < 2 y − y . (4.25) For y / ∈ B r (x), by summing over the possible times t k and locations y at which descendants of blue particles at x may turn red, and then by (4.24) and (4.25), and using that 1 − t 0 = 0 and y / ∈ B r/2 (x) to remove the k = 0 term from the sum, where the third inequality follows by the definition of the events (R x,y,k,r ) c and since 1 − t k ≤ c 0 by (4.6) and x − y ≥ r, and the last inequality follows since x − y ≥ r. Moreover, for any y ∈ Z d , again by summing over possible times t k and locations y at which descendants of blue particles at x may turn red, and then by (4.24) and the definition of (R x,y,k,r ) c for y ∈ B r/2 (x) and 1 ≤ k < k r , where the last line follows since we chose K 1 > 16 (before (4.15)) and so 1 2 K 2 1 + 8K 1 ≤ K 2 1 . The result follows directly from (4.26) and (4.27).

Proof of Proposition 4.2
Take T > 1. We now construct the BRWNLC particle system on the time interval [0, T ], using a construction that is different from the one used in Section 4.1, but still based on the BRW trees with resiliences construction in Section 2. Take a (random) particle configuration ξ consisting of a finite number of particles with ξ = ζ + x∈Z d ζ (x) , where ζ, ζ (x) ∈ N Z d 0 are as in Proposition 4.2 (for some ε > 0 to be specified later). We construct (ξ t ) t∈[0,T ] with ξ 0 = ξ.
The particles will be split into blue, red and yellow particles. The reader is referred to Section 1.4 for a heuristic description of the roles of the blue, red and yellow particles in the construction.
For x, y ∈ Z d and j ∈ N, let T blue,x,j , T red,x,y,j and T yellow,x,j be i.i.d. copies of the BRW tree with resiliences T (defined in (2.1)). The corresponding entries will be denoted by N * , X * , α * , β * , ρ * , where * is replaced by the corresponding superscripts. The process (ξ t ) t∈[0,T ] will be constructed as a deterministic function of ζ, (ζ (x) ) x∈Z d and the BRW trees with resiliences (T blue,x,j ) x∈Z d ,j∈N , (T red,x,y,j ) x,y∈Z d ,j∈N and (T yellow,x,j ) x∈Z d ,j∈N , which will encode the behaviour of the blue, red and yellow particles respectively.
Particles at time 0 are initially coloured either blue or red, and given a label x ∈ Z d , according to the following rule: For each y ∈ Z d , each particle at y that contributes to ζ is coloured blue and given label y, and each particle at y that contributes to ζ (x) for some x ∈ Z d is coloured red, and given label x. We then recolour and relabel some particles according to the following rule: If ξ(y) ≥ J, and if ζ(y) < J, then choose J − ζ(y) red particles at y according to some arbitrary rule, and recolour them blue with label y, so that there are exactly J blue particles at y.
As mentioned above, during the time interval [0, T ], particles will be coloured blue, red or yellow. Whenever a particle branches, both offspring particles inherit the colour and label of the parent particle. For t ∈ [0, T ] and x, y ∈ Z d , we let ξ blue,x t (y) (resp. ξ red,x t (y), ξ yellow,x t (y)) denote the number of blue (resp. red, yellow) particles with label x at location y at time t. Similarly, for t ∈ [0, T ] and y ∈ Z d , we let ξ blue t (y) (resp. ξ red t (y), ξ yellow t (y)) denote the total number of blue (resp. red, yellow) particles at location y at time t.
For every site x ∈ Z d , there may be some time τ (x) ∈ [0, T ) at which blue or red particles at that site are coloured yellow. These are the only times in the time interval (0, T ) at which particles change colour. The particles living between times 0 and T will be subsets of the particles in the following BRW trees with resiliences: • The descendants of the blue particles at x at time 0 will come from the BRW trees with resiliences T blue,x,j for 1 ≤ j ≤ ξ blue 0 (x), with the root of T blue,x,j positioned at x at time 0.
• The descendants of red particles with label x at y at time 0 will come from the BRW trees with resiliences T red,x,y,j for 1 ≤ j ≤ ξ red,x 0 (y), with the root of T red,x,y,j positioned at y at time 0.
• The descendants of particles which turn yellow at x at time τ (x) will come from the BRW trees with resiliences T yellow,x,j , j ≥ 1, with the root of T yellow,x,j positioned at x at time τ (x).
The process (ξ t ) t∈[0,T ] is then constructed from these BRW trees with resiliences as in Section 2, removing the particles that turn yellow at times τ (x) (and their descendants) from their original trees T blue,x ,j or T red,x ,y,j from time τ (x) onwards.
Having explained the idea of the construction, we can now specify the times τ (x), x ∈ Z d . We let with the convention that inf ∅ = ∞. At time τ (x), if τ (x) < T , choose N 1/2 blue or red particles at x according to some arbitrary rule and turn them yellow with label x. This completes the construction of (ξ t ) t∈[0,T ] .
Bad events. We now define some "bad events" for the BRW trees with resiliences T blue,x,j , T red,x,y,j and T yellow,x,j which control the blue, red and yellow particles respectively. We will give informal descriptions of what happens on the complements of these bad events; these descriptions will be made precise in the proofs of Lemmas 4.8 and 4.9 below. The first two events make sure that particles issued from a point x will arrive close to a point y, as long as the killing rate is not "too large" along their trajectory.
Let L > 0 and K 2 > 0 be large constants and c 2 > 0 a small constant to be chosen later. For x, y ∈ Z d , set B x,y = # (j, u) : j ∈ J , u ∈ N blue,x,j T , ρ blue,x,j (v) ≥ c 2 T ∀v ≺ u, On the event (B x,y ) c , if there are at least J blue particles at x at time 0, and if the killing rate is at most c 2 in a tube from (x, 0) to (y, T ) with radius L, and no particles are turned yellow in this tube, then there will be at least J descendants of the blue particles at some site y ∈ B L (y) at time T . Now take 1 > 0 a small constant to be chosen later. For x, y ∈ Z d , set On the event (Y x,y ) c , roughly speaking, if the time τ (x) is triggered, a sufficiently large fraction of yellow particles created at x will quickly move close to y (within a time 1 T ) and then stay in the vicinity of y. More precisely, if τ (x) ≤ (1 − 1 )T and the killing rate is at most K 2 in both a tube from (x, τ (x)) to (y, τ (x) + 1 T ) of radius L and a tube from (y, τ (x) + 1 T ) to (y, T ) of radius L, then there will be at least J descendants of the yellow particles from x at some site y ∈ B L (y) at time T . In order to make sure the killing rate is not "too large", we have to control the growth and spread of the particles. For x ∈ Z d and r ≥ 0, set Since J ≤ N < K 0 N and ζ(x) ≤ K 0 N , by our construction we have ξ blue 0 (x) ≤ K 0 N , and so, on the event (B spread x,r ) c , the blue particles at x at time 0 have fewer than e 3T g(r)K 0 N descendants outside B r/2 (x), and fewer than e 3T g(r) −2 K 0 N descendants in total, at all times in [0, T ].
For x ∈ Z d and r ≥ 0, set On the event (Y spread x,r ) c , any yellow particles created at x at time τ (x) have fewer than e 3T g(r)N 1/2 descendants outside B r/2 (x), and fewer than e 3T g(r) −2 N 1/2 descendants in total, at all times in Finally, the next event controls the spread of the red particles with label x in a ball around x. For x ∈ Z d and r ≥ 0, set For x ∈ Z d , r ≥ 0 and ε > 0, define a bad event involving particles of all colours by letting We will see in the proof of Lemma 4.9 below that on the complement of this event, for any y / ∈ B r/2 (x), blue and red particles with label x contribute at most 2e 3T g(r)K 0 N particles at y at any time in [0, T ]. Moreover, yellow particles with label x contribute at most e 3T g(r)N 1/2 particles at y at any time in [0, T ].
Let (X t ) t≥0 denote a continuous-time random walk started at 0 with jump rate γ and jump kernel p. We need the following result, which will be deduced from a large deviation principle for the process (X t ) t≥0 in Section 5. This result will allow us to bound the probabilities of the bad events B x,y and Y x,y for suitable x and y.
Note that, in particular, 1 ≤ 1/36. Let κ = 4 −1 0 and let t 0 = max(t 0 ( 0 , v 0 , a − ), t 0 ( 0 , v 0 , a + )) as in Lemma 4.6. We now choose T > 1; we will take T sufficiently large that several conditions hold. For t ≥ 0, let D t = max( (µ + a − v 0 )t , (µ + a + v 0 )t ). First, take T sufficiently large that T ≥ t 0 , 0 T ≥ R, 1 D T < 1 3 2 T, 2 T ≤ 1 2 δ 0 T − d 1/2 , 4 2 T + 1 < 1 2 D T , D T > 60R 1 and T ≥ δ −1 0 log(2|B 0 T (0)|) + 1. (4.30) Also take T sufficiently large that (µ + a − v 0 )T = (µ + a + v 0 )T . Moreover, (by considering the cases r > 8R 1 γT e and r ≤ 8R 1 γT e separately) suppose T is sufficiently large that which is possible since g(r) = r −6d−2 for r ≥ 1, by (4.2). Finally, letting and using that 0 κ = 4 by our choice of κ after (4.29), take T sufficiently large that We can now define the remaining constants. Let Take K 2 sufficiently large that and (4.37) Now we can state the two main intermediate lemmas in the proof of Proposition 4.2. The first lemma bounds the probabilities of the bad events; we will prove this result in Section 5. Let Lemma 4.7. For every ε > 0 sufficiently small, there exists N 0 > 0 such that for N ≥ N 0 , the following holds for a ∈ {a − , a + }: The second lemma says that if an edge e in the renormalization grid is closed, i.e. if the event C e defined in (3.2) occurs, then one of a collection of bad events must occur; we will prove this result at the end of this section.
Lemma 4.8. For any ε > 0, there exists N 0 > 0 such that if N ≥ N 0 and R 2 (κ) ≤ R, the following holds. If the coupling in (4.4) holds for a (random) particle configuration ξ ∈ N Z d 0 consisting of a finite number of particles, then under the construction of (ξ t ) t∈[0,T ] with ξ 0 = ξ at the start of Section 4.2, for e = ((x 0 , 0), (y 0 , T )) ∈ E T,a − ,a + ,v 0 (0), Before proving Lemma 4.8, we now show how we can deduce Proposition 4.2 from Lemma 4.7 and Lemma 4.8.
Using (4.46) and (4.47), we get Then by the definition of f in (4.1), by our choice of ε in (4.39). It follows that which was to be proven.
In the remainder of this section, we will prove Lemma 4.8; we will use the following result in the proof.
Lemma 4.9. If N is sufficiently large and R 2 (κ) ≤ R, for any ε > 0 and y ∈ Z d , if the coupling in (4.4) holds for a random particle configuration ξ ∈ N Z d 0 consisting of a finite number of particles, then under the construction of (ξ t ) t∈[0,T ] with ξ 0 = ξ at the start of Section 4.2, if Proof. Recall that for x, y ∈ Z d and t ∈ [0, T ], we write ξ blue,x t (y) (resp. ξ red,x t (y), ξ yellow,x t (y)) to denote the number of blue (resp. red, yellow) particles at y at time t which have label x.
Take x ∈ Z d and r ≥ 0, and suppose the event (P x,r,ε ) c occurs. Note that by our construction, since ζ(x) ≤ K 0 N and J < K 0 N we have ξ blue 0 (x) ≤ K 0 N. Hence for t ∈ [0, T ] and y / ∈ B r/2 (x), by the definitions of the events (B spread Now take y ∈ Z d and suppose the event x∈Z d (P x, x−y ,ε ) c occurs. For t ∈ [0, T ], by (1.2), (4.54) We will split the sum on the right hand side of (4.54) according to whether y / ∈ B x−y /2 (x) or y ∈ B x−y /2 (x). First, by (4.50) and (4.52), for N sufficiently large, where the third inequality follows by the definition of K 2 in (4.36) and since z∈Z d Λ(z) = N −1 .
To bound the other terms in the sum on the right hand side of (4.54), by (4.51) and (4.53), we have Now note that for x, y ∈ Z d with x − y < x − y /2 we have It follows that if x − y < x − y /2 then x ∈ B y −y (y ).
(4.57) Therefore by (4.56), where the second inequality follows since x − y ≤ x − y + y − y < 2 y − y for x ∈ B y −y (y ), and since g(r) −2 is non-decreasing in r, and the last inequality follows by the definition of K 2 in (4.37) and since N Λ(z) ≤ 1 z <R + e −κ z ∀z ∈ Z d by the definition of R 2 (κ) in (1.4). By (4.54) and (4.55), item 1 now follows.
We now prove item 2. Again suppose ∩ x∈Z d (P x, x−y ,ε ) c occurs, and take t ∈ [0, T ]. Then by (4.50) and (4.51), For the first sum on the right hand side of (4.58), we have where the last line follows since z∈Z d Λ(z) = N −1 . For the second sum on the right hand side of (4.58), by (4.57), and then since g(r) −2 is non-decreasing in r and x−y ≤ x−y + y −y < 2 y − y for x ∈ B y −y (y ), we have where the last line follows since N Λ(z) ≤ 1 z <R +e −κ z ∀z ∈ Z d by (1.4). Taking N sufficiently large, item 2 follows from (4.58), (4.59) and (4.60). Finally, for item 3, again suppose ∩ x∈Z d (P x, x−y ,ε ) c occurs; then for t ∈ [0, T ], where the inequality follows from (4.50), (4.51), (4.52) and (4.53). Therefore, using that L ≥ R ≥ R 2 (κ) by (4.30) and (4.35), and (for the second sum) using (4.57) and that x − y < 2 y − y for x ∈ B y −y (y ), where K 3 is defined in (4.33). Hence by (4.34), and since L = 0 T and c 2 = δ 0 T −1 by (4.35), item 3 follows, which completes the proof.
We finish this section by proving Lemma 4.8.
Proof of Lemma 4.8. Suppose N is sufficiently large that Lemma 4.9 holds. Recall from the statement of the lemma that and define H = where conv(S) denotes the convex hull of the set S. Define the event and from now on suppose A c occurs. Moreover, suppose ξ 0 (x 1 ) ≥ J for some x 1 ∈ B R (x 0 ). By our construction, this implies that ξ blue 0 (x 1 ) ≥ J. We will now show that we must have ξ T (y) ≥ J for some y ∈ B R (y 0 ). We consider two cases. See Figure 2 and Figure 3 for a schematic illustration of the strategy in both of these cases.
the event ∩ y∈Z d (P y, y−x ,ε ) c occurs. By the definition of τ (x) in (4.28), we have that In this case, we assume that no yellow particles are created within the hatched space-time region, which implies that the number of blue and red particles is small there. We wish to show that on a good event, denoted by A c in the proof, if at time 0 there are at least J = N 1/3 blue particles at some x 1 ∈ B R (x 0 ), these particles will have at least J descendants at some point in B L (y 0 ) ⊆ B R (y 0 ) at time T . The hatched space-time region contains a tube of radius L around the space-time line from x 1 at time 0 to y 0 at time T . On the good event A c , the blue particles have at least J descendants at some point in B L (y 0 ) at time T whose trajectories stayed inside this tube, provided the competition inside the tube is sufficiently small. To ensure this, the hatched region also contains an additional "buffer" of width L around the tube (the shaded region in the figure). Within the tube, on the good event A c , the competition from particles outside the buffer is bounded by the small constant c 2 /3 by virtue of point 3 from Lemma 4.9. Furthermore, the competition from particles inside the buffer is small because of point 2 from Lemma 4.9 (for the yellow particles) and because the number of blue and red particles is small there by assumption. The total competition felt by the blue particles in the tube is therefore small, as required. In this case, we assume that yellow particles appear at some point (x * , τ (x * )) within the hatched space-time region (i.e. x * ∈ B T −t * T R +2L (x 0 + t * T (y 0 −x 0 )) for some t * ∈ [0, T ] such that τ (x * ) ≤ t * ). The number of these particles is N 1/2 by definition. We wish to show that on a good event, denoted by A c in the proof, these particles will have at least J = N 1/3 descendants at some point in B R (y 0 ) at time T . It is enough to consider the descendants that stay within distance L of a certain space-time polyline. This polyline is chosen so that the tube of distance L around the polyline stays within the spatial region delimited by the dashed lines, denoted by H in the proof. In this region, on the good event A c , the competition is bounded by a (large) constant K 2 , by virtue of point 1 from Lemma 4.9. The good event A c then also ensures that at least J descendants of the N 1/2 yellow particles will travel along this tube without being killed and reach some point in B R (y 0 ) at time T . The choice of the polyline depends on t * . If t * > (1 − 1 )T , where 1 is sufficiently small, then B L (x * ) ⊆ B R (y 0 ) and the polyline is simply the constant x * . If t * ≤ (1 − 1 )T , the polyline moves from x * to y 0 within time 1 T and then stays there.
for N sufficiently large, where the first inequality comes from considering the contributions from yellow particles, red and blue particles in B L (x), and particles outside B L (x) separately, and the second inequality follows by Lemma 4.9 item 2 for the first term, (4.61) (since B L (x) ⊆ B T −t T R +2L (x 0 + t T (y 0 − x 0 ))) for the second term and by Lemma 4.9 item 3 for the third term, and the last inequality since z∈Z d Λ(z) = N −1 .
Recall that we have ξ blue , by (4.62) we have K t (y) < c 2 , and by our assumption for Case 1, we have τ (y) > t. Then by counting blue particles with label x 1 which are not killed by competition or turned into yellow particles before time T , and which are in B L (y 0 ) at time T , we have where the last inequality follows by the definition of the event (B x 1 ,y 0 ) c . Note that by (4.35) and (4.29), we have L ≤ R . Hence, by the pigeonhole principle, there exists y ∈ B L (y 0 ) ⊆ B R (y 0 ) such that ξ T (y) ≥ J.
Case 2: Suppose instead that there exist t * ∈ [0, T ] and x * ∈ B T −t * T R +2L (x 0 + t * T (y 0 − x 0 )) such that τ (x * ) ≤ t * . Choose x * and t * in such a way that τ (x * ) is minimised -this will be useful later. Note that B L (x * ) ⊆ H t * ; in particular, x * ∈ H t * . Let Note that (Y x * ,y * ) c occurs, by the definition of the event A c . Also note that B L (y * ) ⊆ H since B L (x * ) ⊆ H t * and B L (y 0 ) ⊆ H T . Since H is a convex set, it follows that Note that for x ∈ H, we have that ∩ y∈Z d (P y, y−x ,ε ) c occurs, and so by Lemma 4.9 item 1, (4.64) Using (4.64) and (4.63), we get that (4.65) Note that x * + (y * − x * )( T −τ (x * ) 1 T ∧ 1) = y * by the definition of y * . At time τ (x * ), by our construction, N 1/2 red or blue particles at x * are turned yellow and given label x * . By counting the descendants of these particles which are not killed by competition before time T and are in B L (y * ) at time T , we obtain where the second inequality follows by (4.65), and the last inequality follows by the definition of the event (Y x * ,y * ) c . Using the pigeonhole principle again, it follows that ξ T (y) ≥ J for some y ∈ B L (y * ). Now suppose t * ≤ (1 − 1 )T and so y * = y 0 . Then since L ≤ R by (4.35) and (4.29), we have ξ T (y) ≥ J for some y ∈ B L (y 0 ) ⊆ B R (y 0 ). Suppose instead that t * > (1 − 1 )T , and so y * = x * ; then where the last line follows since, by (4.29), 1 < 1/3 and 0 < 1 9 2 , and since 1 D T < 1 3 2 T by (4.30), and R = 2 T , L = 0 T by (4.35). Therefore B L (y * ) ⊆ B R (y 0 ), and again we must have ξ T (y) ≥ J for some y ∈ B R (y 0 ).
To summarise, in both cases, we have ξ T (y) ≥ J for some y ∈ B R (y 0 ). We now claim that we also have that for all t ∈ [0, T ], there exists a particle at time t at some site contained in Note that by (4.29) and (4.35), we have 3L = 3 0 T ≤ 2 T = R , so this is enough to conclude the proof of the lemma. First consider Case 1 above. The blue particles considered in this case stay within distance R + L of x 0 Figure 2), at every intermediate time t ∈ [0, T ], which in particular yields the claim in this case. In Case 2, the same holds for all times t < τ (x * ), with x * as chosen at the start of Case 2. Indeed, inspecting the argument from Case 1, the argument used to derive the bound (4.62) also works for every such t. We can then argue as in Case 1 to count blue particles that are not killed by competition or turned into yellow particles before time τ (x * ). From time τ (x * ) on, inspecting Case 2 we see that the yellow particles involved in this case stay within distance L of the line connecting x * and y * (see also Figure 3). Using (4.63), and the fact that H ⊆ H, this proves the claim and finishes the proof.

Missing proofs of lemmas
In this section, we prove the remaining lemmas from Section 4. In Section 5.1, we first recall a few concentration bounds on binomial and sums of independent geometric random variables, which are used in the following sections. In Section 5.2, we prove Lemma 4.4, which was used in Section 4.1 to prove the "coming down from infinity" property of the BRWNLC (Proposition 4.1). Finally, in Section 5.3, we prove the lemmas missing for Proposition 4.2, namely Lemmas 4.6 and 4.7.

Some concentration inequalities
For n ∈ N and p 0 ∈ [0, 1], let Y (n, p 0 ) be a random variable following the binomial distribution with parameters n and p 0 , i.e., Y (n, p 0 ) ∼ Bin(n, p 0 ). (5.1) The following classical concentration inequalities can be found e.g. in Theorem 2.3 in McDiarmid [McD98]: for all n ∈ N, p 0 ∈ [0, 1] and ε > 0, Let T 1 , T 2 , . . . be i.i.d. copies of the BRW tree with resiliences T as defined in (2.1), with ). The next lemma uses Lemma 2.2 and bounds on sums of independent geometric random variables to give useful bounds on the size of certain sets of particles in the BRW trees. (1 − te) n , (5.5) ∀t ≥ 0 with te < 1, P #{(j, u) : j ∈ n , u ∈ N j t , u = ∅} ≥ m ≤ e (te 2 /(1−te))n−m . (5.6) Proof. Take t ≥ 0. For j ∈ N, let G j = |N j t |, so that the random variable on the left hand side of (5.4) equals n j=1 G j . Now, G 1 , G 2 , . . . are i.i.d. with G 1 ∼ Geom(e −t ), by Lemma 2.2. Note that for s ≥ 0 such that e s (1 − e −t ) < 1, using that e −t ≤ 1. Hence by Markov's inequality (or a Chernoff bound), which is the first inequality (5.4). The second inequality (5.5) readily follows by setting s = 1 and using that 1 − e −t ≤ t.
For the third inequality, we note that the random variable on the left hand side of (5.6) equals n j=1 G j , where G j = G j 1 (G j ≥2) . Using the above expression for E e sG 1 with s = 1, and using that 1 − e −t ≤ t, we have that using furthermore that e −t ≤ 1 and 1 + t ≤ e t ∀t ∈ R for the last inequality. Using Markov's inequality again, we get that which is the third inequality (5.6).

Proof of Lemma 4.4
As in Section 5.1, let T 1 , T 2 , . . . be i.i.d. copies of the BRW tree with resiliences T as defined in (2.1), with , u ∈ U)). We start with a lemma that gives us two bounds on the number of particles in (T i ) i≤n that move a large distance from the origin.
Lemma 5.2. For n ∈ N and m, r, t > 0, For n ∈ N, m, t > 0 and r ≥ 0, if either r > 0 and ne t e − r R 1 log r R 1 γte ≤ m/2, or r ≥ 0 and ne t ≤ m/2, then Proof. Take n ∈ N, m, t > 0 and r ≥ 0. For j ∈ N, let Let (N t ) t≥0 denote a Poisson process with rate γ, and let (X t ) t≥0 denote a continuous-time random walk starting at 0 with jump rate γ and jump kernel p. By the many-to-one lemma (Lemma 2.1), and then since p(x) = 0 for x ≥ R 1 , Suppose first that r ≥ R 1 γt. Then by Markov's inequality, Note that if instead 0 < r < R 1 γt then we have Therefore, for any r > 0, by Markov's inequality and then by (5.9), which completes the proof of (5.7).
In the next three lemmas, we bound the probabilities of the bad events in the definition of R x,r ; this will allow us to prove Lemma 4.4.
Lemma 5.3. For N sufficiently large, for k ∈ N and x ∈ Z d , Proof. Let k ∈ N and x ∈ Z d . Recall that R cr x,k = 4 i=1 R cr,i x,k . We will bound P R cr,i x,k for each i ∈ {1, 2, 3, 4} separately. Case i = 1. Recall from (4.15) that we chose K 1 sufficiently large that 16e − 1 4 λK 1 c 0 c 1 < c 1 . Then since (ρ blue,x,k,j (∅)) j∈N are i.i.d. with distribution Exp(1), and recalling (5.1), where the last line follows by (5.2).
For j, r ∈ N, let Therefore, for r ∈ N, by Markov's inequality, We can bound the expectation on the right hand side by writing Note that if X ∼ Poisson(c) for some c > 0, then for y ≥ c, by Markov's inequality, P (X ≥ y) ≤ e −y log(y/c) E e X log(y/c) = e −y log(y/c) e c(y/c−1) ≤ e y(1−log(y/c)) .
We can now combine Lemmas 5.3, 5.4 and 5.5 to prove Lemma 4.4.

Proof of Lemmas 4.6 and 4.7
We first prove Lemma 4.6. We will see that it is basically a consequence of the following large deviation result, which is a direct consequence of Theorem 1.2 in [dA94] (see also Theorem 13 part 1 in [Bor67]).
Let (X t ) t≥0 be a continuous-time random walk starting at 0 with jump rate γ and jump kernel p. Recall from our assumptions in Section 1.1 that p has finite range, and so in particular E e θ X 1 < ∞ ∀θ > 0. Also, recall from (1.6) that for v ∈ R d , we let I(v) = sup u∈R d v, u − log E e u,X 1 .
Let > 0 and take a finite /2-mesh V of V , i.e. V is a finite subset of V such that for all v ∈ V there exists v * ∈ V with v − v * < /2. By Lemma 5.6, we can take t 0 > 1 sufficiently large that for t ≥ t 0 , for each v ∈ V , P X ts t − sv < 1 2 ∀s ∈ [0, 1] ≥ e −(1−3δ 0 )t .
The following lemma will easily imply item 2 of Lemma 4.7.
It remains to prove item 3 of Lemma 4.7, which will follow easily from the following result.
Proof of Lemma 4.7 item 3. Fix ε > 0; then for x ∈ Z d and r ∈ N 0 , by a union bound, P (P x,r,ε ) ≤ P Z (x) ε > r/4 + P R spread x,r + P B spread x,r + P Y spread x,r .