Doob–Martin compactification of a Markov chain for growing random words sequentially

Abstract

We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the nth word is uniformly distributed over the set of words of length 2n in which n letters are a and n letters are b: at each step an a and a b are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain and thereby delineate all the ways in which the Markov chain can be conditioned to behave at large times. Writing N(u) for the number of letters a (equivalently, b) in the finite word u, we show that a sequence (un)n∈ℕ of finite words converges to a point in the boundary if, for an arbitrary word ν, there is convergence as n tends to infinity of the probability that the selection of N(ν) letters a and N(ν) letters b uniformly at random from un and maintaining their relative order results in ν. We exhibit a bijective correspondence between the points in the boundary and ergodic random total orders on the set {a1, b1, a2, b2, …} that have distributions which are separately invariant under finite permutations of the indices of the a's and those of the b's. We establish a further bijective correspondence between the set of such random total orders and the set of pairs (μ, ν) of diffuse probability measures on [0,1] such that ½(μ + ν) is Lebesgue measure: the restriction of the random total order to {a1, b1,…, an, bn} is obtained by taking X1,…, Xn (resp. Y1,… ,Yn) i.i.d. with common distribution μ (resp. ν), letting (Z1,…, Z2n) be {X1, Y1,…, Xn, Yn} in increasing order, and declaring that the kth smallest element in the restricted total order is ai (resp. bj) if Zk = Xi (resp. Zk = Yj).

Exchangeable interval hypergraphs and limits of ordered discrete structures

“…Martin boundaries and limits of ordered discrete structures. We will give a very short definition of Martin boundary that is adapted best to our already used choice of symbols and refer the reader to [CE,EGW,Ge17,Ve] for more details. We introduce this concept for interval systems first and then relate this to Martin boundaries associated with Schröder trees and finally with binary trees.…”

A boundary theory approach to de Finetti's theorem

“…Further, we restrict ourselves to what we call combinatorial Markov chains. This class is sufficiently rich to provide a framework for many sequentially growing random discrete structures, such as permutations [VK81], compositions and partitions [Gne97], various binary tree models [EGW12, EGW16,EW16], graphs [Grü15], words [CE16], and many others, with applications to the representation theory of the infinite symmetric group, population genetics, and the analysis of algorithms. Both lists are far from complete.…”

“…Among the above, [Gne97] is an interesting example for the use of boundary theory in the context of the analysis of exchangeable random structures. Recently, in the other direction, exchangeability results have employed to determine the boundary for various combinatorial Markov chains; see [CE16,EGW16,EW16].…”