Mixing times of a Burnside process Markov chain on set partitions

The Chinese Restaurant Process may be considered to be a Markov chain which generates permutations on n elements proportionally to absorption probabilities θ|π|, θ>0, where |π| is the number of cycles of permutation π. We prove a theorem which provides a way of finding Markov chains, restricted to directed graphs called arborescences, and with given absorption probabilities. We find transition probabilities for the Chinese Restaurant Process arborescence with variable absorption probabilities. The method is applied to an arborescence constructing set partitions, resulting in an analogue of the Chinese Restaurant Process for set partitions. We also apply our method to an arborescence for the Feller Coupling Process. We show how to modify the Chinese Restaurant Process, its set partition analogue, and the Feller Coupling Process to generate derangements and set partitions having no blocks of size one.

Markov processes, Bernoulli schemes, and Ising model

A randomisation of the Berele insertion algorithm is proposed, where the insertion of a letter to a symplectic Young tableau leads to a distribution over the set of symplectic Young tableaux. Berele’s algorithm provides a bijection between words from an alphabet and a symplectic Young tableau along with a recording oscillating tableau. The randomised version of the algorithm is achieved by introducing a parameter $0< q <1$. The classic Berele algorithm corresponds to letting the parameter $q\to 0$. The new version provides a probabilistic framework that allows to prove Littlewood-type identities for a $q$-deformation of the symplectic Schur functions. These functions correspond to multilevel extensions of the continuous $q$-Hermite polynomials. Finally, we show that when both the original and the $q$-modified insertion algorithms are applied to a random word then the shape of the symplectic Young tableau evolves as a Markov chain on the set of partitions.

A probabilistic analysis of a discrete-time evolution in recombination

The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of harmonic analysis on the infinite symmetric group. They are a special and distinguished case of Okounkov's Schur measures. It is known that any Schur measure determines a determinantal point process on the 1-dimensional lattice. In the particular case of z-measures, the correlation kernel of this process, called the discrete hypergeometric kernel, has especially nice properties. The aim of the paper is to derive the discrete hypergeometric kernel by a new method, based on a relationship between the z-measures and the Meixner orthogonal polynomial ensemble. The present paper can be viewed as an introduction to another our paper where the same approach is applied to studying a dynamical model related to the z-measures (Markov processes on partitions, Prob. Theory Rel. Fields 135 (2006), 84-152; arXiv: math-ph/0409075).

We studyk-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integerk= 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, fork&amp;gt; 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeablek-divisible partitions that are consistent under random deletion. We further introduce the notion ofMarkovian partition structures, which are ensembles of exchangeable Markov chains onk-divisible partitions that are consistent under a random process ofMarkovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k &gt; 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

We consider refined versions of Markov chains related to juggling introduced by Warrington. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities. The normalization factor in one case can be explicitly written as a homogeneous symmetric polynomial. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in bounded time.

The goal of this work is to formally abstract a Markov process evolving over a general state space as a finite-state Markov chain, with the objective of precisely approximating the state probability distribution of the Markov process in time. The approach uses a partition of the state space and is based on the computation of the average transition probability between partition sets. In the case of unbounded state spaces, a procedure for precisely truncating the state space within a compact set is provided, together with an error bound that depends on the asymptotic properties of the transition kernel of the Markov process. In the case of compact state spaces, the work provides error bounds that depend on the diameters of the partitions, and as such the errors can be tuned. The method is applied to the problem of computing probabilistic invariance of the model under study, and the result is compared to an alternative approach in the literature.

We introduce a family of Markov processes on set partitions with a bounded number of blocks, called Lipschitz partition processes. We construct these processes explicitly by a Poisson point process on the space of Lipschitz continuous maps on partitions. By this construction, the Markovian consistency property is readily satisfied; that is, the finite restrictions of any Lipschitz partition process comprise a compatible collection of finite state space Markov chains. We further characterize the class of exchangeable Lipschitz partition processes by a novel set-valued matrix operation.

On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a λ-polyharmonic function is a complex function f on the vertex set which satisfies (λ ⋅ I − P)nf(x) = 0 at each interior vertex. Here, P may be the normalised adjacency matrix, but more generally, we consider the transition matrix P of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these “global” polyharmonic functions, we turn to solving the Riquier problem, where n boundary functions are preassigned and a corresponding “tower” of n successive Dirichlet type problems is solved. The resulting unique solution will be polyharmonic only at those points which have distance at least n from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by Cohen, Colonna, Gowrisankaran and Singman, and more recently, by Picardello and Woess.

Nowakowska's (1978) refinement of the classical model of group participation is subjected to empirical test, using data from Tsai's (1977) previous study. The two-group model performs marginally better with respect to the within-group distribution of participation, but Nowakowska's stipulation of a Markov chain process to describe between-group transitions is shown to be fundamentally inaccurate for the two-group partitioning of the data set.

On a finite graph with a chosen partition of the vertex set into interior and boundary vertices, a $\lambda$-polyharmonic function is a complex function $f$ on the vertex set which satisfies $(\lambda \cdot I - P)^n f(x) = 0$ at each interior vertex. Here, $P$ may be the normalised adjaceny matrix, but more generally, we consider the transition matrix $P$ of an arbitrary Markov chain to which the (oriented) graph structure is adapted. After describing these `global' polyharmonic functions, we turn to solving the Riquier problem, where $n$ boundary functions are preassigned and a corresponding `tower' of $n$ successive Dirichlet type problems are solved. The resulting unique solution will be polyharmonic only at those points which have distance at least $n$ from the boundary. Finally, we compare these results with those concerning infinite trees with the end boundary, as studied by Cohen, Colonnna, Gowrisankaran and Singman, and more recently, by Picardello and Woess.

Lumping evolutionary game dynamics on networks

We construct a diffusion process in the infinite-dimensional simplex consisting of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The process is constructed as a limit of a certain sequence of Markov chains. The state space of the nth chain is the set of all strict partitions of n (that is, partitions with distinct parts). As n →∞, these random walks converge to a continuous-time strong Markov process in the infinite-dimensional simplex. The process has continuous sample paths. The main result about the limit process is an expression for its pre-generator as a formal second-order differential operator in a polynomial algebra. Bibliography: 29 titles.