Cutting Random Recursive Trees, and the Bolthausen–sznitman Coalescent

Cutting random recursive trees, and the Bolthausen–Sznitman coalescent Christina Goldschmidt, University of Warwick (joint work with James Martin) The Bolthausen–Sznitman coalescent was introduced in the context of spin glasses in [1]. These days, it is usually thought of as a special case of a more general class of coalescent processes introduced by Pitman [5] and Sagitov [7] and usually referred to as the Λ-coalescents. These are Markov processes taking values in the space P∞ of partitions of N, or the space Pn of partitions of {1, 2, . . . , n}, where the blocks of the partition represent particles which gradually coalesce over the course of time. The dynamics of the Bolthausen–Sznitman coalescent on Pn are very simple. Suppose that we have an initial state π ∈ Pn consisting of b blocks. Then any k of them coalesce at rate λb,k = (k − 2)!(b− k)! (b− 1)! , 2 ≤ k ≤ b ≤ n, regardless of block sizes or which integers the blocks contain. Since the statespace Pn is finite, the distribution of the coalescent is entirely specified by its initial distribution and these transition rates. A random partition Π of [n] is exchangeable if P(Π = π) = P(Π = σ(π)) for any π ∈ Pn and any permutation σ of [n]. A random partition of N is exchangeable if its restriction to [n] is exchangeable in the above sense for all n ≥ 1. The above dynamics preserve the property of exchangeability: if the initial state of the Bolthausen–Sznitman coalescent is exchangeable, then the state remains exchangeable for all times. Moreover, the rates are such that the restriction of the coalescent evolving in Pn+1 to [n] evolves exactly as the coalescent evolving in Pn; in other words, we have consistency for each n ≥ 1. This means that we can define the coalescent evolving in P∞ simply as a projective limit. Let (Π(t), t ≥ 0) be the Bolthausen–Sznitman coalescent in P∞. An important consequence of the exchangeability of Π(t) for all t ≥ 0 is that its blocks possess asymptotic frequencies i.e. if B is a block of Π(t) then