The spatial $\Lambda$-coalescent

This paper extends the notion of the $\la$-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $\Lambda$-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $\Lambda$-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $\Lambda$-coalescents on large tori in $d\ge 3$ dimensions. Our results generalize and strengthen those of Greven et al. (2005), who studied the spatial Kingman coalescent in this context.


Introduction
The Λ-coalescent, sometimes also called the coalescent with multiple collisions, is a Markov process Π whose state space is the set of partitions of the positive integers. The standard Λ-coalescent Π starts at the partition of the positive integers into singletons, and its restriction to [n] := {1, . . . , n}, denoted by Π n , is the Λ-coalescent starting with n initial partition elements. The measure Λ, which is a finite measure on [0, 1], dictates the rate of coalescence events, as well as how many of the (exchangeable) partition elements, which we will also refer to as blocks, may coalesce into one at any such event. The Λ-coalescent was introduced by Pitman [20], and also studied by Schweinsberg [23]. It was obtained as a limit of genealogical trees in a Moran-like model by Sagitov [22].
The well-known Kingman coalescent [17] corresponds to the Λ-coalescent with Λ(dx) = δ 0 (dx), the unit atomic measure at 0. For this coalescent, each pair of current partition elements coalesces at unit rate, independently from other pairs. Papers [1] and [13] are devoted to stochastic coalescents where again only pairs of partitions are allowed to coalesce, but the coalescence rate is not uniform over all pairs. The survey [1] gives many pointers to the literature. The Λ-coalescent generalizes the Kingman coalescent in the sense that now any number of partition elements may merge into one at a coalescence event, but the rate of coalescence for any k-tuple of partition elements depends still only on k. The first example of such a Λ-coalescent (other than the Kingman coalescent) was studied by Bolthausen and Sznitman [9], who were interested in the special case where Λ(dx) is Lebesgue measure on [0, 1] in connection with spin glasses. Bertoin and Le Gall [6] observed a correspondence of this particular coalescent to the genealogy of continuous state branching processes (CSBP). More recently, Birkner et al. [8] extended this correspondence to stable CSBP's to Λ-coalescents, where Λ is given by a Beta-distribution. Berestycki et al. [5] use this correspondence to study fine small time properties of the corresponding coalescents.
A further generalization of the Λ-coalescents, known as the coalescents with simultaneous multiple collisions, was originally studied by Möhle and Sagitov [18] and Schweinsberg [24]. Further connections to bridge processes and generalized Fleming-Viot processes were discovered by Bertoin and Le Gall [7], and to asymptotics of genealogies during selective sweeps, by Durrett and Schweinsberg [11].
Our first goal, in Section 2, is to extend the notion of the Λ-coalescent to the spatial setting. Here, partition elements migrate in a geographical space and may only coalesce while sharing the same location. Earlier works on variants of spatial coalescents, sometimes also referred to as structured coalescents, have all assumed Kingman coalescent-like behavior, and include Notohara (1990) [19], Herbots (1997) [16], and more recently Barton et al. [2] in the case of finite initial configurations, and Greven et al. [14] with infinite initial states. A related model has been studied by Zähle et al. [25] on two-dimensional tori.
In most of this paper we assume that Λ is a finite measure on [0, 1] without an atom at 0 or at 1, such that Λ([0, 1]) > 0. At the end of Section 2 we comment on how atoms at 0 or 1 would change the behavior of the coalescent.
Define for 2 ≤ k ≤ b, k, b integers, The parameter λ b,k ≥ 0 is the rate at which k blocks coalesce when the current configuration has b blocks. Extend the definition by setting λ b,k = 0 for b = 1 or b = 0, k ∈ N. Define in addition and Note that λ b is the total rate of coalescence when the configuration has b blocks, and that γ b is the total rate of decrease in the number of blocks when the configuration has b blocks. From the above definitions, one may already observe (see also proof of Theorem 1) that the Λ-coalescent can be derived from a Poisson point process on R + ×[0, 1] (R + := [0, ∞)) with intensity measure dtx −2 dΛ(x) : If (t, x) is an atom of this Poisson point process, then at time t, we mark each block independently with probability x, and subsequently merge all marked blocks into one. Now consider a finite graph G, and denote by |G| the number of its vertices. Call the vertices of G sites. Consider a process started from a finite configuration of n blocks on sites in G where we allow only two types of transitions, referred to as coalescence and migration respectively: (i) at each site blocks coalesce according to the Λ-coalescent, (ii) the location process of each block is an independent continuous Markov chain on G with jump rate 1 and transition probabilities p(g i , g j ), g i , g j ∈ G.
The original Λ-coalescent of [20] and [23] corresponds to the setting where |G| = 1, so migrations are impossible. The spatial Λ-coalescent started from a finite configuration {(1, i 1 ), . . . , (n, i n )} is a well-defined strong Markov process (chain) with state space being the set of all partitions of [n] = {1, . . . , n} labeled by their location in G. This will be stated precisely in Theorem 1 of Section 2 which is devoted to the construction of spatial Λ-coalescents Π ℓ with general (possibly infinite) initial states. After constructing the general spatial Λ-coalescent, we turn to characterizing those that come down from infinity in Section 3. Schweinsberg [23] shows that if holds, then the (non-spatial) Λ-coalescent started with infinitely many blocks at time 0 immediately comes down from infinity, that is, the number of its blocks at all times t > 0 is finite with probability 1; otherwise, the Λ-coalescent stays infinite forever, meaning that it contains infinitely many blocks at all times t > 0 with probability 1. The goal of Section 3 is to show that the spatial Λ-coalescent inherits this property of either coming down from infinity or staying infinite, from its nonspatial counterpart. More precisely, let (Π ℓ (t)) t≥0 be the Λ-coalescent constructed in Theorem 1, and denote by #Π(t) its size at time t, i.e. the total number of blocks in Π ℓ (t), with any label. In Lemma 8 and Proposition 11 we show that condition (4) implies P [#Π(t) < ∞, ∀t > 0] = 1, even if the initial configuration Π(0) contains infinitely many blocks. In this case we say that the spatial Λ-coalescent comes down from infinity. In Proposition 11 we also show via a coupling to the non-spatial coalescent that if (4) does not hold, provided #Π(0) = ∞ and Λ has no atom at 1, then P [#Π(t) = ∞, ∀t > 0] = 1. In this case we say that the spatial Λ-coalescent stays infinite. We note here that the statement of Lemma 8 (saying that sup n E[T n ] < ∞, where T n is the time until there are on average two blocks per site if there are initially n blocks per site) extends to the spatial coalescent for which the migration mechanism may be more general, for example non-exponential or depending on the coalescence mechanism.
In Section 4 we continue the study of the time T n . In particular, in Theorem 12 we obtain an upper bound on its expectation that is not only uniform in n but also, somewhat surprisingly, in the structure (size) of G. In this case, we say that the coalescent comes down from infinity uniformly. The argument of Theorem 12 relies on the independence of the coalescence and migration mechanisms.
Our final goal, in Section 4, is to study space-time asymptotic properties of Λ-coalescents that come down from infinity uniformly on large finite tori at time scales on the order of the volume. In [14], this asymptotic behavior was studied for the spatial Kingman coalescent where Λ = γδ 0 for some γ > 0. It is interesting that on appropriate space-time scales, the scaling limit is again (as in [14]) the Kingman coalescent, with only its starting configuration depending on the specific properties of the underlying Λ-coalescent. We obtain functional limit theorems for the partition structure and for the number of partitions, in Theorems 13 and 19 respectively.

Construction of the coalescent
The construction of the spatial coalescent on an appropriate state space follows quite standard steps. The construction below is inspired by those in Evans and Pitman [13], Pitman [20], and Berestycki [4].
Let P be the set of partitions on N, which can be identified with the set of equivalence relations on N. Any π ∈ P can be represented uniquely by π = (A 1 , A 2 , A 3 , . . . ) where A j ⊂ N for j ≥ 1 are called the the blocks of π, indexed according to the increasing ordering of the set {min A j : j ≥ 1} that contains the smallest element of each block. So in particular min A n−1 < min A n , for any n ≥ 2. Likewise, we define for any n ∈ N, P n as the set of partitions of [n], and for π ∈ P n we have π = (A 1 , A 2 , . . . , A n ) in an analogous way. We will write A ∈ π if A ⊂ N is a block of π, and If the number of blocks of π, denoted by #π, is finite, then set A j = ∅ for all i > #π.
For concreteness in the rest of the paper, let |G| = υ for υ a positive integer and let the vertices of G be {g 1 , . . . , g υ }. The spatial coalescent takes values in the set P ℓ of partitions on N, indexed as described above, and labelled by G, so Similarly, the coalescent started from n blocks takes values in P ℓ n := {(A j , ζ j ) : A j ∈ π, ζ j ∈ G, π ∈ P n , 1 ≤ j ≤ n}. Here, the ζ j ∈ G is the label (or location) of A j ∈ π, j ≥ 1.
For any element π ∈ P ℓ or π ∈ P ℓ n with n ≥ m define π| m ∈ P ℓ m as the labeled partition induced by π on P ℓ m . We equip P ℓ with the metric d(π, π ′ ) = sup and likewise P ℓ n with the metric It is easy to see that (P ℓ n , d n ) and (P ℓ , d) are both compact metric spaces, and that d(π, π ′ ) = sup n d n (π| n , π ′ | n ).
In the rest of the paper, whenever Π ℓ is a spatial coalescent process, we denote by Π the partition (without the labels of the blocks) of Π ℓ , and by (#Π(t)) t≥0 the corresponding total number of blocks process. Thus #Π(t) is the number (finite or infinite) of blocks in Π(t), or equivalently, in Π ℓ (t).
With the above notation we are finally able to construct the spatial Λ-coalescent started from potentially infinitely many blocks, as stated in the following theorem. Recall the migration mechanism stated in the introduction: each block performs an independent continuous Markov chain on G with jump rate 1 and transition kernel p(·, ·).
Theorem 1 Assume that Λ has no atom at 0. Let G be a finite graph with vertex set {g 1 , . . . , g υ }. Then, for each π ∈ P ℓ , there exists a càdlàg Feller and strong Markov process Π ℓ on P ℓ , called the spatial Λ-coalescent, such that Π ℓ (0) = π and (i) blocks with the same label coalesce according to a (non-spatial) Λ-coalescent, (ii) each block of label g i ∈ G changes its label to g j ∈ G at rate p(g i , g j ) as mentioned in introduction.
This process also satisfies and its law is characterized by (iii) and the initial configuration π.
Proof. In order to define a càdlàg Markov process Π ℓ with values in P ℓ such that Π ℓ n := Π ℓ | n is a spatial coalescent starting at Π ℓ (0)| n ∈ P ℓ n for any Π ℓ (0) ∈ P ℓ , we will make use of suitably chosen Poisson point processes.
Let δ n denote the Kronecker delta measure with unit atom at n. Let M be another independent Poisson point process on the same probability space Ω with values in R + × N × G υ and intensity measure given by dt ∞ k=1 δ k (dm)P υ (ds 1 , . . . , ds υ ), where P υ is the joint law of υ independent G-valued random variables S 1 , . . . , S υ , such that P (S g i = g j ) = p(g i , g j ), g i , g j ∈ G.
Using the above random objects define a spatial Λ-coalescent with n initial blocks, Π ℓ n , on Ω for each n ≥ 1 as follows: At any atom (t, x, ξ) of N i , all blocks A j (t−) with ζ j (t−) = g i and ξ j = 1 coalesce together into a new labeled block ( j,ξ j =1,ζ j (t−)=i A j (t−), g i ); at any atom (t, m, (s 1 , . . . , s υ )) of M we set ζ m (t) = s ζm(t−) provided m ≤ #Π n (t−), otherwise nothing changes. For all other t ≥ 0 we set Π ℓ n (t) = Π ℓ n (t−). Note that coalescence causes immediate reindexing (or reordering) of blocks that have neither participated in coalescence nor in migration, and that this reindexing operation decreases each index by a non-negative amount.
Since the sum of the above defined jump rates of Π ℓ n is finite it follows immediately that Π ℓ n is a well defined càdlàg Markov process on Ω for each n ≥ 1 therefore inducing a càdlàg Markov process Π ℓ on P ℓ . It is important to note that each Π ℓ n so constructed is a Λ-coalescent started from Π ℓ (0)| n . Since Π ℓ n+1 (0)| n = Π ℓ n (0) and since clearly the consistency condition (6) is preserved under each transition of Π ℓ n+1 in the construction (this is not always a transition for Π ℓ n ), we have Π ℓ n+1 (t)| n = Π ℓ n (t) for all t ≥ 0. Therefore, (Π ℓ (t)) t≥0 constructed by Π ℓ (t)| n := Π ℓ n (t), n ≥ 1, t ≥ 0 is well-defined. It follows that Π ℓ is a càdlàg Markov process with values in P ℓ , which clearly satisfies properties (i)-(iii), and uniqueness in distribution follows similarly.
In order to verify that the semigroup T t ϕ(π) := E[ϕ(Π ℓ (t))|Π ℓ (0) = π] is a Feller-Dynkin semigroup it now suffices to check the following two properties (see [21] III (6.5)-(6.7)): (i) For any continuous (bounded) real valued function ϕ on P ℓ and all π ∈ P ℓ we have lim and (ii) for any continuous (bounded) real valued function ϕ on P ℓ and all t > 0, π → T t ϕ(π) is continuous (and bounded). Note that (i) is an immediate consequence of the right-continuity of the paths and continuity with respect to (5). One can easily argue for (ii): if Π ℓ,k is the spatial coalescent started from π k and Π ℓ is the spatial coalescent started from π such that lim k→∞ π k = π ∈ P ℓ , then, due to the definition of the metric (5) on P ℓ , there exists for all k ∈ N an m = m(k) such that π k | m = π| m , with the property m(k) → ∞ as k → ∞. This implies that one can construct a coupling of Π ℓ,k and Π ℓ (using the same Poisson point processes for all) such that Π ℓ, for all t ≥ 0 and, since m(k) → ∞, we conclude that the second property holds due to the continuity of ϕ. Given that T t is a Feller-Dynkin semigroup the strong Markov property holds.
✷ Remark. A variation of the above construction could be repeated for the cases where Λ has an atom at 0. This would correspond to superimposing Kingman coalescent type transitions on top of the Poisson process induced coalescent events. One easily observes that all such coalescents come down from infinity. Also note that an atom of Λ at 1 implies complete collapse in finite time, even if the coalescent corresponding to the measure Λ(· ∩ [0, 1)) stays infinite. See [20] for further discussion of atoms. ⋄ Remark. We stated Theorem 1 for |G| < ∞. The case |G| = ∞ needs a little more work if we also want to be able to start with an infinite configuration π ∈ P ℓ . However, for π ∈ P ℓ n , a finite starting configuration, the Poisson point process construction in the proof of the theorem immediately yields the desired process. This fact will be useful in Section 5 where we consider G = Z d . ⋄

Coming down from infinity
In this section, we first obtain estimates on the coalescence rates and the rates of decrease in the number of blocks, both in the non-spatial and the spatial setting. Several of these estimates will be applied to showing that the spatial Λ-coalescent comes down from infinity if and only if (4) holds.
It is easy to see, using definitions (1)- (3), that The following lemma is listing some facts, which are based on (7) and some simple computations.

Lemma 2
We have the following estimates: , and that the term in the parentheses equals bx 2 . Combined with (7), this gives the initial statement of the lemma. The first inequality λ b+1 ≥ λ b is immediate. The second inequality follows again from (7), by integrating the following inequality with respect to Λ which is easy to check, for example, via the Binomial Theorem.
(ii) The stated property of the sequence γ was already noted and used by Schweinsberg, cf. [23] Lemma 3. For completeness we include a brief argument: From (7) The following two lemmas and a corollary are auxiliary results, often implicitly observed in [20] or [23], and are of interest to anyone studying fine properties of Λ-coalescents. Fix a ∈ (0, 1). Let Λ a be the restriction of Λ to [0, a], namely using Λ a as the underlying measure instead of Λ.
(ii) There exists an a < 1 and Remark. For any fixed Λ let Then it is easy to see that γ b /η b → 1 as b → ∞, so statements (ii) and (iii) above extend to the corresponding η b and η a b . ⋄ Proof. For each a ∈ (0, 1), the first inequalities in both (i) and (ii) are trivial consequences of Λ a being the restriction of Λ, the identities in (7), and the fact that The second inequality in (i) is easy as well, Now it is easy to see by . Part (iii) follows immediately from the argument for (ii), and the following fact (already noticed by Pitman [20], Lemma 25), In particular, (4) must imply that the left hand side in (10) is infinite.
✷ Let the symbol ≍ stand for "asymptotically equivalent behavior" in the sense that a m ≍ b m (as m → ∞) if there exist two finite positive constants c, C such that Proof. (i) To show the first claim, use expression (7) to get for b ≥ 2, Then note that and also that . This in turn implies that dΛ(x) can be ignored in the asymptotics, and the remaining term appears in the asymptotic expression for λ b . (ii) Since γ b ≍ η b , see the above remark, it suffices to show the second statement for η b instead. As in (7), and since it is easy to see that the claim on the asymptotics of γ b (i.e., η b ) follows. ✷ , the conclusion follows by Lemma 2(i), (7), (11) and the fact that λ b → ∞.
(ii) Perhaps the easiest way to see that γ b → ∞ whenever Λ([0, 1]) > 0 is by using the identity (10). The statement then follows immediately from Lemma 2(i), Lemma 4(ii), and the fact that There exists a finite number ρ ≥ 1 such that for any Λ, and all b, m Proof. In this lemma we consider the identities (7) for all real b ≥ 1. It suffices to show that and now one can take ρ > log 2 c + log c log 2 to get the statement of the lemma. Define the function Due to representation (7) for λ b it then suffices to study and show sup 1], and that (by expanding the binomial terms and noting x/(1 − x) ≤ 4 log 8 whenever x < 2 b log 8 and b ≥ 10) ✷ Now we turn to the spatial setting. Recall that the vertex set of G is {g 1 , . . . , g υ }.
Denote by λ(b 1 , b 2 , . . . , b υ ) the total rate of coalescence for the configuration with b i blocks at site g i , Similarly, let Denote by ⌊x⌋ the integer part of the real number x and let ⌈x⌉ := −⌊−x⌋.
The following two lemmas will be useful for the proof of the characterization result given in Proposition 11.
In order to verify the first inequality we observe that for x ∈ [0, 1], since one can simply check that equality holds for x = 0 and that for all x ∈ [0, 1]. The first inequality in (i) now follows from this and from (7), since for by Jensen's Inequality since the function y → a y is convex for every a > 0. Therefore, (14) is bounded below by If β is an integer then the last expression is just υγ β . Now note that the function β → βx − 1 + (1 − x) β is increasing (for β ≥ 1) and this implies the second inequality in (i).
(ii) Use Lemma 6 to conclude The second inequality of (ii) is a simple consequence of the fact that there exists a 1 ≤ j ≤ υ such that b j ≥ ⌈ υ i=1 b i /υ⌉ and Lemma 2(i). ✷ Now consider the coalescent (Π ℓ nυ (t)) t≥0 such that its initial configuration Π ℓ nυ (0) has n blocks at each site of G. Let Proof. The argument is an adaptation of the argument by Schweinsberg [23], Lemma 6, to our situation. In fact we will even use similar notation. For n ∈ N define R 0 := 0 and stopping times (with respect to the filtration generated by Π ℓ nυ ) given by In words, R i is the time of the ith coalescence as long as the number of blocks before this coalescence exceeds 2υ, otherwise R i is set equal to the previous coalescence time. Since there are no more than 2υ blocks left after (n − 2)υ coalescence events, note that Of course, it is also possible that T n = R i for i < (n − 2)υ, but the above identity holds almost surely as R (n−2)υ = R i in this case. Let and note that there exists some finite random number ξ i such that . . are the successive times of migration jumps of various blocks from site to site in between the i − 1th and ith coalescence time. Let B i (t) be the number of blocks located at site g i ∈ G at time t. Since the total number of blocks does not change at the jump times T i j for j = 1, . . . , ξ i we have due to Lemma 7 (ii) that Be(R i−1 )/υ⌉ . This implies (by coupling of exponentials in a straightforward way) that Also note that for all i with Π nυ (R i−1 ) > 2υ, where the first equality is a direct consequence of definitions (2) and (3), and the fact that J i is the decrease in the number of blocks at the ith coalescence time R i . The middle inequality is due to Lemma 7 (i) and (ii). From (17) and (18) and the fact that L i = 0 if J i = 0 we get the important relation for i ≥ 1. Now where we have used Lemma 9 below. ✷ Lemma 9 For a fixed υ, let m, n be positive integers such that m ∈ [nυ, (n + 1)υ). For any k ≥ 1 and j 1 , . . . , Proof. Statement (20) can be proved for each fixed υ by induction in n. The base cases n = 2 with m > 2υ and k i=1 j i = m − 1 explain the extra summands 2υ/γ 2 . Here one also uses the fact that (γ b ) ∞ b=2 is an increasing sequence (cf. Lemma 2 (ii)). ✷ Let us now recall the construction in Theorem 1, and define T (2) n := T n from definition (16), and furthermore define Note that by monotone convergence T (k) n ] can be shown as in the proof of Lemma 8. The second claim above now follows by relation (10) and the observation following it. ✷

Corollary 10 If condition (4) holds then for each
We can now establish the following analogues to Proposition 23 of Pitman [20] and Proposition 5 of Schweinsberg [23] in the spatial setting.

Proposition 11 Assume that Λ has no atom at 1. Then the Λ-coalescent either comes down from infinity or it stays infinite. Furthermore, it stays infinite if and only if E[T ∞ ] = ∞.
Proof. Define T := inf{t ≥ 0 : #Π(t) < ∞}. The first statement could be shown following Pitman [20] Proposition 23 by observing that P [0 < T < ∞] > 0 leads to a contradiction. We choose a different approach, based on Corollary 10 and coupling with non-spatial coalescents.
Suppose that (4) holds. Then E[T ∞ ] < ∞, by Lemma 8, implying T ∞ < ∞ almost surely. Also note that the Λ-coalescent comes down from infinity due to Corollary 10, since for any t > 0, and any k ≥ 2, So assume that #Π(0) = ∞, i.e. that there exists at least one site g in G such that Π ℓ (0) contains infinitely many blocks with label g. Then the spatial coalescent Π ℓ is stochastically bounded below by a coalescing systemΠ ℓ , in which any block that attempts to migrate is assigned to a "cemetery site" ∂ instead. More precisely, the evolution of the processΠ ℓ at each site is independent from the evolution at any other site, and its transition mechanism is specified by: (i) blocks coalesce according to a Λ-coalescent, (ii') each block vanishes (moves to ∂) at rate 1.
By adapting the construction of Π ℓ in Theorem 1, one can easily construct a coupling (Π ℓ (t),Π ℓ (t)) t≥0 on the same probability space, so that at each time t, and for each site g of G, the number of blocks in Π ℓ (t) located at g is larger than (or equal to) the number of blocks inΠ ℓ (t) located at g. We will show that in any given time interval [0, t], at each site of G that initially contained infinitely many blocks, there are infinitely many blocks remaining inΠ ℓ (even though there are infinitely many blocks that do vanish to ∂ by time t). Therefore, ∞ = #Π(t) ≤ #Π(t) so that Π ℓ stays infinite.
To show that P [#Π(t) = ∞] = 1 for each t > 0, it will be convenient to construct a coupling ofΠ ℓ (t) with a new random object Π 1 (t). Since there is no interaction between the sites of G inΠ, it suffices to consider the nonspatial case where |G| = 1. Introduce an auxiliary family (X j ) j≥1 of independent exponential random variables with parameter 1. Take a (non-spatial) Λ-coalescent (Π 1 (s)) s∈[0,t] such that Π 1 (0) =Π(0), and in addition augment the state space for Π 1 to accommodate a mark for each block. Initially all blocks start with an empty mark. At any s ≤ t, any block A ∈ Π 1 (s) is marked by ∂ if {X min A ≤ s}. In this way, if an already marked block A coalesces with a family A 1 , A 2 , . . . of blocks, such that min A ≤ min j (min A j ), the new block A∪ ∪ j A j automatically inherits the mark ∂. Note as well that if a marked block A coalesces with at least one unmarked block containing a smaller element than min A, the new block will be unmarked.
The number # u Π 1 (t) of all unmarked blocks in Π 1 (t) is stochastically smaller than the number #Π(t). To see this, note the difference betweenΠ(t) and Π 1 (t): a marked block in Π 1 (s) is not removed from the population immediately (unlike inΠ) so it may coalesce (and "gather") additional blocks with higher indexed elements during [s, t] resulting in a smaller number of unmarked partition elements in Π 1 (0) than inΠ(t).

Remark.
It is intuitively clear that in the case in which the Λ-coalescent Π ℓ stays infinite, there are infinitely many blocks in Π ℓ at all positive times at all sites, a proof of this fact is left to an interested reader. ⋄

Uniform asymptotics
Note that the upper bound in Lemma 8 and Corollary 10 neither depends on the structure of G nor on the underlying migration mechanism. After a careful look at the proofs the reader will see that in fact the same estimates would hold with an arbitrary migration mechanism, even if it is not independent from the coalescent mechanism. In this section we will use the fact that each block changes its label (i.e. migrates) at rate 1, independently from the coalescent mechanism. Recall the setting of Lemma 8 and Corollary 10.
Theorem 12 If (4) holds, then there exists a constant c uniform in Λ, υ, the structure of G, and the transition kernel of the migration mechanism, such that and moreover Proof. We use the same notation as in the proof of Lemma 8, but this time the calculations are finer. First, fix an i ≥ 1 (note the subscripts i are omitted in a number of places below for notational convenience). Recall the jump times T i j and configurations Π nυ (T i j ) with ) for all j ∈ N 0 . Recall that ξ i := max{k : T i k < R i } is the number of migration events in between (i − 1)st and ith coalescence time. Note that the quantities λ j are relevant for our process only if j ≤ ξ i .
Using the first line of (18) and Lemma 7 (i) as well as further conditioning on (λ l ) l∈N 0 we obtain For the next computation define an auxiliary i.i.d. sequence (X j ) j≥0 of exponential random variables with parameter a, as well as a sequence (Y j ) j≥0 of independent random variables where each Y j has an exponential ( λ j ) distribution. Note that W j := X j ∧ Y j are exponential random variables with rate a + λ j that are independent from Z j = 1 {X j >Y j } .
Observe that conditioned on ((λ j ) j∈N 0 , Π nυ (R i−1 )) the X j correspond to the waiting time until the next migration and the Y j to the waiting time until coalescence as long as Comparing now the terms in (22) and (23) we find that where we gained a factor of υ ρ+1 in the denominator with respect to the analogous relation (19) in the proof of Lemma 8. The rest of the proof proceeds now as the proof of Lemma 8 and Corollary 10 and hence we obtain ✷ Definition. We will say that the Λ-coalescent comes down from infinity uniformly if lim k→∞ sup n ET (k) n = 0. ⋄ In particular, by Proposition 11 and Theorem 12 any coalescent with independent Markovian migration mechanism that comes down from infinity also comes down from infinity uniformly.
is of special interest in [8]. As already noted in [23], for α ∈ (0, 1] this (non-spatial) coalescent stays infinite, and for α ∈ (1, 2) it comes down from infinity. By the previous theorem the spatial Beta(2 − α, α)-coalescent comes down from infinity uniformly. An interesting consequence follows by the results of the next section. ⋄

Asymptotics on large tori
In this section we further restrict the setting in the following way: • the migration corresponds to a random walk on the torus, meaning that the kernel p(x, y), x, y ∈ G is given as p(x, y) • the Λ-coalescent comes down from infinity (uniformly), i.e., condition (4) holds.
We are concerned here with convergence of the Λ-coalescent partition structure on T N , if time is rescaled by the volume (2N + 1) d of T N , to that of a time-changed nonspatial Kingman coalescent as N → ∞. The main results are presented in Theorem 13 and Theorem 19: Theorem 13 states convergence of the partition structure in a functional sense for arbitrary finite initial configurations. Theorem 19 states convergence of the number of partition elements in a functional sense if the initial number of partition elements is infinite.
We write P N,ℓ if we want to emphasize that the partitions are labeled by T N . Let Π N,ℓ π and Π N,ℓ denote the Λ-coalescent started from a partition π ∈ P N,ℓ , and the Λ-coalescent started from any partition that contains infinitely many equivalence classes labeled by (located at) each site of T N , respectively. In order to determine the large space-time asymptotics for Π N,ℓ , at time scales on the order of the volume (2N + 1) d of T N , we imitate a "bootstrapping" argument from [14].
Remark. Observe that in [14], only the singular Λ = δ 0 case was studied in this context. However, the structure of the argument concerning large space-time asymptotics carries over due to the cascading property for general (spatial) Λ-coalescents, in particular due to the fact that any two partition elements π 1 , π 2 ∈ Π N,ℓ (0) coalesce at rate while they are at the same site, and that they do not coalesce otherwise. ⋄ We will need the following notation: for a marked partition π ∈ P ℓ n (or π ∈ P ℓ ), and two real numbers if and only if all the mutual distances for pairs of different partition elements of π are contained in [a, b].
The following theorem states that, viewed on the right timescale t(2N + 1) d , and after some initial collapse of a finite starting configuration, the partitions of the Λ-coalescent on the tori T N with N large behave like those of a (non-spatial) time-changed Kingman coalescent. To make this statement more precise, we introduce the following notation.
Let G = ∞ k=0p k (0) wherep k denotes the k-step transition probability of ap random walk. Note that this random walk is transient on Z d , so that G < ∞. Let Π Z d ,ℓ π be the Λ-coalescent on G = Z d with migration given by the random walk kernelp, started from partition π with #π < ∞. The transience ofp also implies existence of non-trivial limit partitions in the sense that if #π ≥ 2 then #Π Z d π (∞) ≥ 2 with positive probability. We define K π as the non-spatial Kingman coalescent started in the partition π ∈ P or π ∈ P n . This means that K π is the Λ K -coalescent for Λ K = δ 0 and |G| = 1 with initial configuration K π (0) = π.
Denote by D(R + , E) the càdlàg paths on R + with values in some metric space E, and equip the space D(R + , E) with the usual Skorokhod topology. Also let " ⇒ " indicate convergence in distribution. Set Recall that Π N,ℓ starts from a configuration containing infinitely many blocks, namely the partition Π(0) = {{j} : j ∈ N}. The theorem below concerns the behavior of only finitely many blocks. Recall that Π N,ℓ (0)| n is the restriction of the labeled partition Π N,ℓ (0) to [n]. In the theorem below we use the abbreviation Π N,ℓ n := Π N,ℓ Π N,ℓ (0) | n . Again, Π N n is the process of partitions corresponding to Π N,ℓ n .
Theorem 13 Assume that for each fixed n ≥ 1 and all large N we have Π N,ℓ (0)| n = Π N +1,ℓ (0)| n . Then for each n, we obtain as N → ∞, the following convergence of the (unlabeled) partition processes: where convergence is with respect to the Skorokhod topology on D(R + , P n ), and both Π N,ℓ n and Π Z d ,ℓ n are started from the same initial configuration Π N,ℓ (0)| n ∈ P ℓ n .
Remark. The statement is a generalization of Proposition 7.2 in [14], which deals with the case of spatial Kingman coalescents, rather than Λ-coalescents and only states convergence of the marginals. Nevertheless, the first part of the argument is analogous, and we will change it only slightly in preparation for Proposition 18 and Theorem 19. ⋄ As the first step we will state a result for the case in which the initial configuration is sparse on the torus, so that no coalescence involving more than two particles may be seen in the limit. The general case, stated in Theorem 13, will then follow easily.

Proposition 14
Let a N → ∞ be such that a N /N → 0. Fix n ∈ N, and let π N,ℓ ∈ P N,ℓ be such that #π N,ℓ = n ≥ 2, π N ∈ [[a N , √ dN ]], and such that its corresponding (unlabeled) partition π N equals a constant partition π 0 ∈ P for all N . Then as N → ∞, we have the following convergence in distribution of the (unlabeled) partition processes: , where the convergence is in the space D(R + , P).
Proof. To simplify the notation we refer to the ith block of π 0 as {i}, for i = 1, . . . n. In order to show the convergence on the space D(R + , P) we will prove that the joint distribution of inter-coalescence times converges, when appropriately rescaled, to the joint distribution of inter-coalescence times of K(κ ·), and that, at each coalescence time, any pair of remaining blocks is equally likely to coalesce next, see also [15] for a similar argument.
We set τ N 0 = 0. Since there are at most n − 1 coalescence times in general, we then define recursively stopping times for k = 1, . . . , n − 1, Let us first observe that for n = 2 uniformly in t ∈ [0, T ] for any T < ∞, by Lemma 7.3 in [14]. Indeed, as remarked at the beginning of this section, the spatial Λ-coalescent restricted to two-particles is identical in law to the spatial λ 2,2 δ 0 (·)-coalescent from [14]. Let U k be independent exponential random variables with parameters κ n−(k−1) 2 for k < n − 1. We wish to show the convergence in distribution of the random vector as N → ∞. The statement is clear by (28) if n = 2. In order to show (29) for n ≥ 2, the first step is to see that, we may exclude the possibility of coalescence of more than two particles at any given time with probability tending to 1 as N → ∞. Let τ N (i, j) be the time of the coalescence which merges the block A (i) containing i and the block A (j) containing j, and for each i denote by ζ (i) the label associated with the block A (i) . Then, we have for any 0 < T < ∞, and any distinct i, j, k ∈ [n], uniformly over all partitions π N,ℓ ∈ [[a N , √ dN ]], as N → ∞. The statement (30) is analogous to (3.7) in Cox [10], and follows with exactly the same calculation. Likewise, a statement analogous to (3.8) in [10] holds, saying that uniformly over all π N,ℓ ∈ [[a N , as N → ∞ for i, j, k, l ∈ [n] distinct. Now fix T < ∞, ǫ > 0, and let n > 2. Relation (30) implies that for N large enough, and together with (31) it implies that A simple induction (using the strong Markov property and uniformity of (30) and (31) in t ∈ [0, T ]) yields the following statement: for each k < n − 1, and any fixed ε > 0, if N is large enough then for k ≤ n − 2. From this we get that, for any fixed ε > 0, if N is large enough, and Moreover, on the event we have as in (3.1) of [10] that where ε N depends on N only, and where ε N → 0, as N → ∞.
In order to arrive at (29), we show that σ N k is asymptotically independent of σ N k−1 , . . . , σ N 1 for all k = 2, . . . , n − 1. So consider for any fixed 0 ≤ t 1 , . . . , t k , where k i=1 t i < T, the event In particular, on this event we have that τ N i < T (2N + 1) d is satisfied for i = 1, . . . , k. We obtain By iterating the argument we obtain asymptotic independence. This in turn implies that (#Π N π N,ℓ (t(2N + 1) d ) t≥0 ⇒ (#K π 0 (κt)) t≥0 in the Skorokhod topology, since by (33), as N → ∞, so that the convergence of the jump times τ N k in n − n−1 k=1 1 {τ N k <t} implies convergence in the Skorokhod topology, see for example Proposition 6.5 in Chapter 3 of [12].
Finally, (2.8) in [10] states that for thep random walk on T N , This implies that the positions of partition elements in Π N,ℓ are approximately uniformly and independently distributed on the torus. Due to (29), with probability tending to 1 as N → ∞, we also have Therefore, at time τ N k+1 , each pair of partition elements of #Π N π N,ℓ (τ N k ) is approximately equally likely to coalesce, as is the case in the Kingman coalescent. This completes the proof of convergence on the space D(R + , P). Since for N large enough, max{ζ : (A, ζ) ∈ Π N,ℓ (0)| n } ≤ N 2 and since the coalescent has at most n blocks independently performing random walks, we immediately obtain lim N →∞ P τ N > N 3/2 = 1.
In particular, we have To see that the blocks remaining at time N 3/2 are at a mutual distance of N 3/4 / log N with high probability, more precisely that if suffices to observe that again by the functional CLT, where X 1 and X 2 are two independentp-random walks on Z d started at X 1 (0) = X 2 (0) = 0. Due to (37), and the fact that Now (36) and (38) imply ✷ We will also show a uniform convergence to the Kingman coalescent, on the same time scale, in the sense of the number of blocks, cf. Theorem 19 below. One starts with a bound on the mean number of partition elements left in the coalescent Π N at a fixed time, say 1. The following useful monotonicity property carries over from the spatial Kingman coalescent setting to the spatial Λ-coalescent setting: Suppose that the partition elements of Π N,ℓ (0) are initially divided into classes Π N,1,ℓ (0), Π N,2,ℓ (0), . . . (in any prescribed deterministic way) and let (∪ j Π N,j,ℓ (t)) t≥0 denote the united Λ-coalescent where only elements of the same class are allowed to coalesce.

Lemma 15
For each t > 0, Proof. We can couple Π N,ℓ and ∪ j Π N,j,ℓ , using the same Poisson point process (from the construction of Π N,ℓ ) for all the Λ-coalescents corresponding to different classes. It then follows that Π N (t) is a coarser partition than ∪ j Π N,j (t) for each t, almost surely. This gives the inequality, #Π N (t) ≤ j Π N,j (t), and in particular the bound in expectation holds.
✷ The following lemma is taken from [14] and is similar to Theorem 1 in [3] and the proposition in Section 4 of [10].

Lemma 16
There is a finite constant c d such that uniformly in N ∈ N, and in the sequences (Π N (0)) N ∈N satisfying #Π N (0) ≥ (2N + 1) d , Proof. All we need to do is translate the notation and explain the small differences in the argument. Our λ 2,2 is γ in [14]. The migration walkp is from the same class as in [14]. There are only two statements in the argument of [14], Lemmas 7.4 and 7.5 that depend on the structure of the underlying coalescent. One is relation (7.50) at the beginning of the argument of Lemma 7.4. Take A 0 ∈ Π N (0) and note that, similar to (7.44) in [14], leading to (7.46) of Lemma 7.4 in [14], and therefore to relation (7.50) since the remaining calculations concern the behavior of two partition elements (not the joint behavior of several partition elements). The other statement concerns (7.58) in the proof of Lemma 7.5: here, the torus is cut up into boxes and (7.58) states that the expected number of blocks is bounded by the expected number of blocks in a coalescent in which only blocks that start in the same initial box may coalesce. This holds in our setting due to Lemma 15. Given (7.50) and (7.58), the remaining arguments are the same as those in the proof of Lemmas 7.4 and 7.5 of [14].
✷ The next lemma says that the number of the partition elements at time ε(2N + 1) d is tight in N .
Due to Theorem 12, for k ∈ N, Due to (4) (more precisely observation (10)), the right hand side converges to zero as k → ∞. Therefore, we may choose M 0 ≥ 1 large enough so that c( ∞ b=M 0 1 Remark. Note that on A N M we may have #Π N (1) ≥ (2N +1) d and we can apply Lemma 16 directly, otherwise couple the coalescent (Π N (t), t ≥ 1) with another coalescentΠ N (t), t ≥ 1) such thatΠ N almost surely dominates Π N (t) at all times, at all sites, and such that #Π N (0) = (2N + 1) d , and apply Lemma 16 toΠ N . ⋄ It follows that By conditioning on whether A N M or its complement occurs, using (40) Since 2c d M ǫ < ǫ ′ 2 we arrive at for M ≥ M 0 , which gives the statement of the lemma with M 0 = (M 0 ) 2 . ✷ As a consequence, we obtain the following asymptotics for the number of partitions in Π N , a spatial Λ-coalescent started from a partition having infinitely many equivalence classes labeled by (located at) each site of T N . Proposition 18 Let (K(t)) t≥0 be the (non-spatial) Kingman coalescent started from the partition K(0) = {{i}, i ∈ N}, and let κ be defined in (27). Then, for each fixed t > 0, we have #Π N (t(2N + 1) d ) ⇒ #K(κt), as N → ∞, where the above convergence is in distribution.
(46) ✷ An even stronger form of convergence is true. It holds in any setting where Proposition 18 and Theorem 13 hold, in particular in the setting of [14], although there it does not appear explicitly. Its analogue is important for the diffusive clustering analysis in the two-dimensional setting of [15].
Proof. As a consequence of (44) we have for any fixed a > 0, Together with the convergence of marginals in Proposition 18, and Theorem 13, this yields the current statement. ✷