Mixing Time for a Random Walk on Rooted Trees

Fulman, Jason

doi:10.37236/228

Cited by 7 publications

(4 citation statements)

References 26 publications

(55 reference statements)

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“…However, as Fulman notes [16], despite these similarities, the Aldous chain is not naturally amenable to these down-up methods. One of the impediments when working with either the Aldous chain or the uniform chain of Definition 1 is that the same label that is removed in the down-move is reinserted in the up-move.…”

Section: Theorem 1 the Unique Invariant Distribution For The Uniform Chain Is The Uniform Distribution On T [N]mentioning

confidence: 99%

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

Forman

Pal

Rizzolo

et al. 2020

Random Struct Algorithms

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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.

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Section: Theorem 1 the Unique Invariant Distribution For The Uniform Chain Is The Uniform Distribution On T [N]mentioning

confidence: 99%

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

Forman

Pal

Rizzolo

et al. 2020

Random Struct Algorithms

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“…We note that, as a result of [SY89, Th. 1] (repeated as Theorem 5.2 below), our leaf-removal step also occurs in the chain of [Ful09], where it is followed by a leaf-attachment step that is absent here.…”

Section: A Chain On Organisational Structuresmentioning

confidence: 99%

Markov Chains from Descent Operators on Combinatorial Hopf Algebras

Pang

2016

Preprint

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We develop a general theory for Markov chains whose transition probabilities are the coefficients of descent operators on combinatorial Hopf algebras. These model the breaking-thenrecombining of combinational objects. Examples include the various card-shuffles of Diaconis, Fill and Pitman, Fulman's restriction-then-induction chains on the representations of the symmetric group, and a plethora of new chains on trees, partitions and permutations. The eigenvalues of these chains can be calculated in a uniform manner using Hopf algebra structure theory, and there is a simple expression for their stationary distributions. For an important subclass of chains analogous to the top-to-random shuffle, we derive a full right eigenbasis, from which follow exact expressions for expectations of certain statistics of interest. This greatly generalises the coproduct-then-product chains previously studied in joint work with Persi Diaconis and Arun Ram.

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“…Up-down chains on compositions, and more generally, on graded sets, have been studied in a variety of contexts [2,6,9,10,11,15,16], often in connection with their nice algebraic and combinatorial properties.…”

Section: Introductionmentioning

confidence: 99%

The leftmost column of ordered Chinese restaurant process up-down chains: intertwining and convergence

Rivera-Lopez

Rizzolo

2022

Electron. J. Probab.

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Recently there has been significant interest in constructing ordered analogues of Petrov's two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model. One method for constructing these processes goes through taking an appropriate diffusive limit of Markov chains on integer compositions called ordered Chinese Restaurant Process up-down chains. The resulting processes are diffusions whose state space is the set of open subsets of the open unit interval. In this paper we begin to study nontrivial aspects of the order structure of these diffusions. In particular, for a certain choice of parameters, we take the diffusive limit of the size of the first component of ordered Chinese Restaurant Process up-down chains and describe the generator of the limiting process. We then relate this to the size of the leftmost maximal open subset of the open-set valued diffusions. This is challenging because the function taking an open set to the size of its leftmost maximal open subset is discontinuous. Our methods are based on establishing intertwining relations between the processes we study.

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Mixing Time for a Random Walk on Rooted Trees

Cited by 7 publications

References 26 publications

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

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

Markov Chains from Descent Operators on Combinatorial Hopf Algebras

The leftmost column of ordered Chinese restaurant process up-down chains: intertwining and convergence

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