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

Abstract

We consider both the Bolthausen-Sznitman and the Kingman coalescent restricted to the partitions of {1, . . . , 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. IntroductionAn 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.

On the block counting process and the fixation line of the Bolthausen–Sznitman coalescent

“…Spectral decompositions are of fundamental interest since they lead to diagonal representations of the corresponding operators or matrices which simplify many mathematical calculations and numerical computations significantly. Explicit spectral decompositions for (the block counting process of) the Kingman coalescent and the Bolthausen-Sznitman coalescent are provided in [17] and [22]. We are interested in analog spectral decompositions for the fixation line.…”

Site Frequency Spectrum of the Bolthausen-Sznitman Coalescent

“…Progress has also been made for the finite coalescent even for the general coalescent process. The finite Bolthausen-Sznitman coalescent has been studied through the spectral decomposition of its jump rate matrix described in [11] where the authors used it to derive explicit expressions for the transition probabilities and the Green's matrix of this coalescent, and also the Kingman coalescent. The spectral decomposition of the jump rate matrix of a general coalescent, including coalescents with multiple mergers, is also used in [17] where an expression for the expected Site Frequency Spectrum is given in terms of matrix operations which in the case of the Bolthausen-Sznitman coalescent result in an algorithm requiring on the order of n 2 computations.…”

Site Frequency Spectrum of the Bolthausen-Sznitman Coalescent

“…Progress has also been made for the finite coalescent even for the general coalescent process. The finite Bolthausen-Sznitman coalescent has been studied through the spectral decomposition of its jump rate matrix described in Kukla and Pitters (2015) where the authors used it to derive explicit expressions for the transition probabilities and the Green's matrix of this coalescent, and also the Kingman coalescent. The spectral decomposition of the jump rate matrix of a general coalescent, including coalescents with multiple mergers, is also used in Spence et al (2016) where an expression for the expected site frequency spectrum is given in terms of matrix operations which in the case of the Bolthausen-Sznitman coalescent result in an algorithm requiring on the order of n 2 computations.…”