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We consider a Markov chain that iteratively generates a sequence of randomfinite words in such a way that the $n^{\mathrm{th}}$ word is uniformlydistributed 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 inuniformly at random among the letters of the current word. We obtain a concretecharacterization of the Doob-Martin boundary of this Markov chain. Writing$N(u)$ for the number of letters $a$ (equivalently, $b$) in the finite word$u$, we show that a sequence $(u_n)_{n \in \mathbb{N}}$ of finite wordsconverges to a point in the boundary if, for an arbitrary word $v$, there isconvergence as $n$ tends to infinity of the probability that the selection of$N(v)$ letters $a$ and $N(v)$ letters $b$ uniformly at random from $u_n$ andmaintaining their relative order results in $v$. We exhibit a bijectivecorrespondence between the points in the boundary and ergodic random totalorders on the set $\{a_1, b_1, a_2, b_2, \ldots \}$ that have distributionswhich are separately invariant under finite permutations of the indices of the$a'$s and those of the $b'$s. We establish a further bijective correspondencebetween the set of such random total orders and the set of pairs $(\mu,\nu)$ ofdiffuse probability measures on $[0,1]$ such that $\frac{1}{2}(\mu+\nu)$ isLebesgue measure: the restriction of the random total order to $\{a_1, b_1,\ldots, a_n, b_n\}$ is obtained by taking $X_1, \ldots, X_n$ (resp. $Y_1,\ldots, Y_n$) i.i.d. with common distribution $\mu$ (resp. $\nu$), letting$(Z_1, \ldots, Z_{2n})$ be $\{X_1, Y_1, \ldots, X_n, Y_n\}$ in increasingorder, and declaring that the $k^{\mathrm{th}}$ smallest element in therestricted total order is $a_i$ (resp. $b_j$) if $Z_k = X_i$ (resp. $Z_k =Y_j$).

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