A Spectral decomposition for the Bolthausen-Sznitman coalescent and the Kingman coalescent

We consider both the Bolthausen-Sznitman and the Kingman coalescent restricted to the partitions of $\{1, \ldots, n\}.$ Spectral decompositions of the corresponding generators are derived. As an application we obtain a formula for the Green's functions and a short derivation of the well-known formula for the transition probabilities of the Bolthausen-Sznitman coalescent.


Introduction
An exchangeable coalescent process is a discrete or continuous-time Markov chain that encodes the dynamics of particles grouped into so-called blocks. As time passes, only mergers of some blocks may occur, and the rate at which a merger happens only depends on the current number of blocks, but not, for instance, on their sizes or the specific particles they contain. The theory of exchangeable coalescent processes has its origins in the study of genealogies in population genetics, culminating in the seminal work of Kingman [9]. In the context of population genetics Sagitov [16] and later Sagitov and Möhle [10] derived exchangeable coalescents as limiting genealogies of so-called Cannings models, which are discrete-time models of neutral haploid populations with exchangeable family sizes. This derivation is also implicit in the work of Donnelly and Kurtz [7]. Among exchangeable coalescent processes the so-called Λ-coalescents have received increasing attention in recent years. The latter were introduced independently by Donnelly and Kurtz [7], Pitman [14] and Sagitov [16]. A Λ-coalescent {Π(t), t ≥ 0} is a time-homogeneous exchangeable coalescent process in continuous time with state space P N , the set of partitions of the non-negative integers N := {1, 2, . . .}, that only allows for one merger of blocks at any jump. It can be characterized via its restrictions {Π n (t), t ≥ 0} to [n] := {1, . . . , n} as follows. If at any given time Π n (t) contains b ≥ 2 blocks, then any 2 ≤ k ≤ b of these blocks merge at rate λ b,k := 1 0 x k−2 (1 − x) b−k Λ(dx), where Λ denotes a finite measure on the unit interval. This measure Λ together with the initial state Π(0) uniquely determines Π, hence the name Λ-coalescent.
In this note we consider both the Kingman coalescent Π K = {Π K (t), t ≥ 0} and the Bolthausen-Sznitman coalescent Π BS = {Π BS (t), t ≥ 0}, which are both Λ-coalescents. For convenience we drop the superscripts K and BS when there is no risk of ambiguity. From its introduction Kingman's coalescent has been used in population genetics as a model approximating the genealogy of a sample drawn from a large neutral population of haploid individuals, i.e. in this context the particles are interpreted as individuals in the population. The Bolthausen-Sznitman coalescent was discovered by Bolthausen and Sznitman [4] in the context of the Sherrington-Kirkpatrick model for spin glasses in statistical physics. Goldschmidt and Martin [8] gave a construction of Π BS via a cutting of a random recursive tree. Bertoin and Le Gall [2] derived the Bolthausen-Sznitman coalescent as the genealogy of a continuous-state branching process.
An increasing number of recent works suggest that the class of beta coalescents, which contains the Bolthausen-Sznitman coalescent, should provide better null models than Kingman's coalescent for the genealogy of highly fecund populations, see the introduction in [3]. Neher and Hallatschek [12] argue that the Bolthausen-Sznitman coalescent occurs naturally as the genealogy in models of rapidly adapting asexual populations. Motivated by models of populations undergoing natural selection [6], [5], Berestycki, Berestycki and Schweinsberg [1] showed that the genealogy of a population governed by Branching Brownian Motion with absorption is again the Bolthausen-Sznitman coalescent.
The article is organized as follows. Our main results are spectral decompositions of the generator of the Bolthausen-Sznitman n-coalescent, Theorem 1, respectively of the generator of Kingman's n-coalescent, Theorem 2. As Corollaries we obtain for the Bolthausen-Sznitman coalescent a derivation of the formula for the transition probabilites that goes back to [4], and a formula for its Green's matrix. As a further application we obtain a spectral decomposition of the generator of the block counting process of the Bolthausen-Sznitman, respectively Kingman's coalescent.

Results
Let us introduce some notation. A partition of a set A is a set, π say, of nonempty pairwise disjoint subsets of A whose union is A. The members of π are called the blocks of π. Let #A denote the cardinality of A and let P A denote the set of partitions of A.
2.1. Bolthausen-Sznitman n-coalescent. The Bolthausen-Sznitman n-coalescent Π n,BS = {Π n,BS (t), t ≥ 0} is obtained by choosing Λ to be the uniform measure on [0, 1], that is the corresponding Q-matrix Q := Q n,BS = (q πρ ) π,ρ∈P [n] is given by where π ≺ ρ if and only if ρ is obtained by exactly one merger of blocks of π. In this section we only consider the Bolthausen-Sznitman n-coalescent and therefore write Π instead of Π n,BS . For any two sets A and B ⊆ A and a partition π ∈ P A we call π| B := {C ∩ B : C ∈ π, C ∩ B = ∅} ∈ P B the restriction of π to B. Theorem 1 (Spectral decomposition of the Bolthausen-Sznitman coalescent). Let L = (l πρ ) π,ρ∈P [n] and R = (r πρ ) π,ρ∈P [n] be matrices defined by and where π ≤ ρ if and only if each block of π is contained in a block of ρ. Then a spectral decomposition of Q is given by Q = RDL, where D = (d πρ ) π,ρ∈P [n] is defined by d ππ = −(#π − 1) and d πρ = 0 if π = ρ. In particular, l ππ = r ππ = 1 for any π ∈ P [n] .
The right eigenvectors r πρ may be interpreted as the probability that a random recursive tree on the label set π can be cut down to a tree on the label set ρ, see the paragraph "A connection with random recursive trees" in Section 3.1 below. As an application of this spectral decomposition we derive the well-known formula for the transition probabilities of Π given by Bolthausen and Sznitman in [4], Proposition 1.4.
Thanks to the spectral decomposition, Theorem 1, we obtain a formula for the Green's matrix G = (g πρ ) π,ρ∈P [n] of the Bolthausen-Sznitman n-coalescent defined by g πρ : is the expected total time that Π spends in ρ starting from π. We denote by i j the Stirling permutation numbers or unsigned Stirling numbers of the first kind, which count the number of permutations of a set of i elements with j cycles.
For a set A and j ∈ N let P A,j denote the set of partitions of A into j blocks. Moreover, i j denotes the Stirling partition numbers or Stirling numbers of the second kind, which count the number of partitions into j blocks of a set of i elements.
Remark 2. From Corollary 2 it follows by a technical but straightforward computation that for any π ∈ P [n],i and j ≤ i in agreement with the entries of the Green's matrix of the block counting process of the Bolthausen-Sznitman coalescent provided in the proof of Corollary 1.5 in [11].
As a further application of the spectral decomposition of Q, Theorem 1, we derive a spectral decomposition of the generator Q ′ = (q ′ ij ) i,j∈[n] of the block counting process {N (t), t ≥ 0} of the Bolthausen-Sznitman n-coalescent defined by N (t) := #Π(t). It is well-known that the matrix Q ′ is given by In [11] this spectral decomposition of Q ′ was derived by means of generating functions without recourse to the partition-valued process Π.
be matrices given by Then a spectral decomposition of Q ′ is given by where π ⋖ ρ if and only if π ≤ ρ and #π − #ρ = 1. In other words, the jump chain of Kingman's n-coalescent is the directed simple random walk on the partition lattice P [n] , where at each step the chain jumps into a coarser partition. From now on we only consider Kingman's n-coalescent and therefore write Π instead of Π n,K .
As in the case of the Bolthausen-Sznitman coalescent, consider the block counting process . From Theorem 2 we obtain a spectral decomposition of Q ′ .
j! denotes the unsigned Lah numbers which count the number of partitions into j blocks of a set of i elements, where the elements in each block are ordered. Then

Proofs
In order to prove our results, we need some notions and facts from the theory of lattices, that we collect from [17]. Recall that a partially ordered set (poset for short) (P, ≤) is a set P together with a binary relation ≤ satisfying: (1) For all p ∈ P, p ≤ p (reflexivity).
Recall that two posets P, Q are called isomorphic, in which case we write P ∼ = Q, if there exists an order-preserving bijection φ : P → Q whose inverse is order-preserving. The cartesian product P × Q of two posets P, Q is defined on the set {(p, q) : p ∈ P, q ∈ Q} by letting (p, q) ≤ (p ′ , q ′ ) in P × Q if and only if p ≤ p ′ in P and q ≤ q ′ in Q. For p, q ∈ P an upper bound of p and q is an element r ∈ P such that p, q ≤ r. A least upper bound of p and q is an upper bound s ∈ P of p and q such that for any upper bound r of p and q one has s ≤ r. Clearly, if a least upper bound of two elements p and q exists, it is unique. A greatest lower bound is defined in complete analogy. A lattice L is a poset with the property that any two of its elements have a least upper bound and a greatest lower bound. It is well-known, that P [n] together with the relation ≤ as defined in Theorem 1 is a lattice, the so-called partition lattice. For π, ρ ∈ P [n] with π ≤ ρ we call the set [π, ρ] := {σ ∈ P [n] : π ≤ σ ≤ ρ} an interval. We will make repeated use of the isomorphism cf. Example 3.10.4 in [17]. For more information on posets in general and the partition lattice in particular the reader is referred to [17]. Evidently, we have P [n] = [∆ [n] , {[n]}], where for any set A we let ∆ A := {{a} : a ∈ A} be the partition of A into singletons.
Using the notation we just introduced, the n-Λ-coalescent Π n is a Markov chain with state space P [n] , initial state ∆ [n] and Q-matrix Q n = (q n πρ ) π,ρ∈P [n] given by where There are several extensions of ≤ to a linear order on P [n] . Let us fix such an extension ≤ ex and notice that the following quantities of Q n that we are interested in do not depend on the specific extension chosen. The linear order ≤ ex induces a natural bijection ψ from P [n] to [B n ], where B n denotes the nth Bell number, which is the number #P [n] of partitions of [n], defined inductively by letting ψ(∆ [n] ) = 1 and ψ(π) ≤ ψ(ρ) iff π ≤ ex ρ. Then Q n can be seen as an upper right triangular matrix with entries ordered according to ≤ ex , if we define row/column π to be lower than row/column ρ iff π ≤ ex ρ.
The determinant of Q n is therefore given by the product of its diagonal entries. Hence, the characteristic polynomial of Q n is given by where I n is the identity matrix on P [n] . Hence, for each i ∈ [n], −λ i is an eigenvalue of Q n with algebraic multiplicity #P [n],i = n i . Since Q n is a Q-matrix, (1, . . . , 1) ⊤ ∈ R Bn is an eigenvector corresponding to the eigenvalue 0, and direct inspection yields that (1, 0, . . . , 0) ⊤ ∈ R Bn is also an eigenvector corresponding to the eigenvalue −λ n . From now on we fix an n ∈ N, n ≥ 2, and drop this index in the notation, if there is no risk of confusion.
3.1. Bolthausen-Sznitman n-coalescent. In order to prepare the proof of the spectral decomposition of Q = Q n,BS , Theorem 1, we calculate the left and right eigenvectors of Q. In the sequel we give two proofs for the right eigenvectors of Q that are of rather different flavours. The first proof is completeley self-contained and only makes use of the partition lattice P [n] . Together with the proof of Lemma 5 it might serve as a starting point to find a spectral decomposition for more general coalescents, e.g. beta coalescents. There is a probabilistic interpretation of the right eigenvector of Q in terms of random recursive trees which then motivates our second proof that heavily draws on random recursive trees and their connection to the Bolthausen-Sznitman coalescent as explored in Goldschmidt and Martin [8].
is a right eigenvector of Q with corresponding eigenvalue 1 − #ρ.
Proof. (first proof of Lemma 1) In order to carry out the following calculations, for any two partitions π, ρ ∈ P [n] we need to parameterize the set {σ : π ≺ σ ≤ ρ}. To construct an arbitrary partition σ such that π ≺ σ, we could choose a subset C ⊆ π of at least two blocks of π and merge them in order to obtain σ. In this case #σ = #π − #C + 1. If, additionally, we require σ ≤ ρ, we certainly cannot choose any collection C of blocks in π. Instead, all blocks chosen have to be in π| B for some block B ∈ ρ, in which case #σ| B = #π| B − #C + 1. To summarize, we have Using this parametrization, we obtain and the claim follows.
A connection with random recursive trees. Let us now recall the notion of a random recursive tree in order to prepare our second proof of Lemma 1, where we closely follow Goldschmidt and Martin [8]. We call a tree on n nodes labelled by 1, 2, . . . , n an increasing tree if the root has label 1 and the labels in any path from the root to another node are increasing. nodes, we obtain an increasing tree on n nodes by adding a node labelled n and attaching it by an edge to one of the nodes in the given tree. Starting from the tree that only consists of the root node 1 this gives an explicit construction of all increasing trees on n nodes. Consequently, there are (n − 1)! increasing trees on n nodes. A random recursive tree is a tree chosen uniformly at random from all increasing trees on n nodes. An explicit construction of a random recursive tree on n nodes is the following. Start with the root node labelled 1. If the tree has k nodes, choose one of these nodes uniformly at random and attach to it node k + 1 by an edge. Stop after attaching node n. For any partition π ∈ P [n] an increasing tree on π is a tree with #π nodes that are labelled by the blocks in π such that the labels in any path from the root to another node are increasing with respect to their least element. We denote a random recursive tree on π by T π .
Crucial to the construction of the Bolthausen-Sznitman coalescent via random recursive trees is the following cutting procedure. When given a tree T on π, a cut is performed by picking an edge, removing the subtree above this edge (here we picture trees as they grow in nature: from the root at the bottom to the leaves at the top) and adding the labels of this subtree to the labels of the node below the edge. For a simple tree the cutting procedure is depicted in Figure 2. We denote by cT the tree obtained by cutting T at an edge chosen uniformly at random. A striking result then is Proposition 2.1 in Goldschmidt and Martin [8], which states that after cutting a random recursive tree T π on π at an edge chosen uniformly at random, the new tree cT π is a random recursive tree on the new label set (which is again a partition of [n]).
For π ∈ P [n] define a time-homogeneous continuous-time Markov chain R π := {R π (t), t ≥ 0} with values in the set of random recursive trees as follows. The initial state R π (0) is the random recursive tree T π on π. The process R π evolves according to the following dynamics. Suppose R π is in state T . If T has only one vertex, do nothing. Otherwise, attach to each edge in T an exponential 1 clock, all clocks being independent. When the first clock rings, cut T at the associated edge to obtain the next state of R π . For any tree T on the label set π let p(T ) = π. Proposition 2.2 of Goldschmidt and Martin [8] establishes that the Bolthausen-Sznitman ncoalescent Π is equal in distribution to {p(R ∆n (t)), t ≥ 0}.
Fix two partitions π, ρ ∈ P [n] such that π ≤ ρ. We say that an increasing tree T on π contains ρ iff one can obtain an increasing tree on ρ by successively cutting T . If T contains ρ we write ρ ⊏ T . Let {R π (k), k ≥ 0} denote the jump chain of R π . From the definition of R it follows that a process that is equal in distribution to the jump chain R can be constructed recursively by letting R π (0) := T π , R π (k + 1) := cR π (k), k ≥ 0.
A simple question then is: what is the probability P{ρ ⊏ T π } that a random recursive tree on π contains ρ? Notice first, that an increasing tree T on π contains ρ if for each block B ∈ ρ we can find a node v = v(B) in T such that the labels in the subtree above v coincide with the elements in π| B . In other words, we can construct all increasing trees on π that contain ρ by first constructing an increasing tree T on ρ. Then each node v ∈ T is labelled by some block B ∈ ρ. Now if #π| B > 1, build an increasing tree on π| B and replace v by this tree. This procedure is done for all nodes v ∈ T to obtain an increasing tree on π that contains ρ. Therefore, the number of increasing trees on π containing ρ is On the other hand, the total number of increasing trees on π is (#π − 1)!. Since a random recursive tree on π is a tree chosen uniformly at random from all increasing trees on π, we obtain which is just r πρ . We can now turn to our second proof of Lemma 1 in terms of random recursive trees.
Proof. (second proof of Lemma 1) We rewrite the statement as Let J = {J(k), k ≥ 0} denote the jump chain of Π = Π BS,n . Since P{J(1) = σ|J(0) = π} = q πσ /q π is the probability that J jumps from π to σ, after dividing (14) by q π = #π − 1 we obtain for the left hand side where we used that for any random recursive tree T one has T p(T ) = d T . Conditional on ρ ⊏ T π , there are precisely #π − #ρ among the #π − 1 edges in T π that we may cut in order to obtain a tree cT π that contains ρ. Since, by definition, cT π is obtained by cutting T π at an edge chosen uniformly at random, we have The claim follows.
Proof. Notice that From the interpretation of n i as counting the number of permutations of [n] with i cycles it is clear that n i = π∈P [n],i B∈π (#B − 1)!, cf. equation (1.15) in [15]. Using the isomorphism where in the last step we used x n = n k=1 n k x k , cf. equation (1.16) in [15]. Lemma 3. The matrix L = (l πρ ) π,ρ∈P [n] defined by is the inverse matrix of R, i.e. σ∈P [n] r πσ l σρ = δ πρ .
Proof. (of Theorem 1) The claim follows by Lemmata 1 and 3.
Letting x = e −t in Lemma 2 proves the claim.
Proof. (of Corollary 2) From (17) we have where |k| := B∈ρ k B . Letting x = e −t and integrating out with respect to t we see for The claim follows from (16) and the definition of g πρ .
Proof. (of Corollary 3) Evidently, the rate at which a jump from i to j blocks occurs equals the sum of all rates at which a jump from a partition π ∈ P [n],i to a partition ρ ∈ P [n],j occurs, i.e. q ′ ij = ρ∈P [n],j q πρ , where π ∈ P [n],i is fixed arbitrarily. The quantity ρ∈P [n],j q πρ does not depend on the choice of π ∈ P [n],i , as the following calculation shows. By the spectral decomposition of Q, Theorem 1, we obtain
Lemma 4 (Number of maximal chains). For π, ρ ∈ P [n] with π ≤ ρ we have that Proof. Notice that any maximal chain (π 1 , . . . , π n ) in [∆ [n] , {[n]}] can be constructed as follows. Let π 1 := ∆ [n] , and if π i with i < n is constructed, π i+1 is obtained by merging two blocks in π i , which can be done in #πi For the general case, recall the isomorphism [π, ρ] ∼ = × B∈ρ P π|B . As a consequence, any maximal chain in [π, ρ] can be built by choosing a maximal chain in each factor P π|B and then 'intertwining' these chains, i.e. ordering their elements (excluding the first element -the partition into singletons -in each chain) in any order subject to the restriction that the order of elements of the same chain is preserved. Consequently, we have m(π, ρ) = (#π − #ρ)!
where in the third step we used the identity i k = σ∈P [i],k B∈σ #B! twice. This identity is obvious from the interpretation of i k as the number of partitions into k ordered blocks of a set of i elements, where the elements in each block are ordered.