Projections of the Aldous chain on binary trees: Intertwining and consistency

Abstract

Consider the Aldous Markov chain on the space of rooted binary trees with n labeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix 1 ≤ k < n and project the leaf mass onto the subtree spanned by the first k leaves. This yields a binary tree with edge weights that we call a "decorated k-tree with total mass n." We introduce label swapping dynamics for the Aldous chain so that, when it runs in stationarity, the decorated k-trees evolve as Markov chains themselves, and are projectively consistent over k. The construction of projectively consistent chains is a crucial step in the construction of the Aldous diffusion on continuum trees by the present authors, which is the n → ∞ continuum analogue of the Aldous chain and will be taken up elsewhere.

The leftmost column of ordered Chinese Restaurant Process up-down chains: intertwining and convergence

“…In this paper we begin to study nontrivial aspects of the order structure of these diffusions. Motivated by [6,Theorem 2 and Theorem 19] and [5,Theorem 5], which consider similar properties in closely related tree-valued processes, we consider the evolution of the left-most maximal open interval of X (α,0) in running in its (α, 0)-Poisson-Dirichlet interval partition stationarity distribution. Recall that the (α, 0)-Poisson-Dirichlet interval partition is the distribution of {t ∈ (0, 1) : V 1−t > 0} where V t is a (2 − 2α)-dimensional Bessel process started from 0.…”

Diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains

“…The diffusions we construct are indexed by the parameters (α, θ), with θ ≥ 0 and 0 < α < 1, and are natural ordered analogues of the EKP(α, θ) diffusions, which are a two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model [3] constructed by Petrov [17]. There has been significant interest in the EKP(α, θ) diffusions, including studying sample path properties [5,9], giving biological interpretations to the parameters [2,9], and constructing associated Fleming-Viot processes [6,10] Ordered analogues of the EKP(α, θ) diffusions have recently been studied in [8,11]. In those papers, the constructions took place in the continuum based on a general method for constructing open set-valued processes using marked Lévy processes [7].…”

“…Up-down chains on compositions and more generally on similarly graded sets like C have been studied in a variety of contexts [1,9,12,13,14,17,18], often in connection with their nice algebraic and combinatorial properties.…”

A down-up chain with persistent labels on multifurcating trees

“…In the exchangeable case where α = 1 2 , the relabelling in step (iv) is unnecessary and one can therefore adjust (v) accordingly and just insert the deleted leaf again, which yields the uniform chain from [11].…”

“…In [9,11] a modified version of the Aldous chain, the α-chain, was introduced, where the up-step is governed by Ford's α-model. For α = 1 2 , Ford's α-model is the same as Rémy's growth process, and so the leaf labels are exchangeable, but this is not true for 0 < α = 1 2 < 1, so a more complicated down-step was introduced: (i) Down-step (selection): Select leaf i uniformly at random.…”