Coalescence in Markov chains

A Markov chain $X^i$ on a finite state space $S$ has transition matrix $P$ and initial state $i$. We may run the chains $(X^i: i\in S)$ in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are $|S|$ trajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number $k(μ)$ of coalescence classes of the process, and what is the set $K(P)$ of such numbers $k(μ)$, as the coupling $μ$ of the chains ranges over couplings that are consistent with $P$? We continue earlier work of the authors (‘Non-coupling from the past’, $\textit{In and Out of Equilibrium 3}$, Springer, 2021) on these two fundamental questions, which have special importance for the ‘coupling from the past’ algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented, and also of non-block measure with similar properties.

💡 Research Summary

The paper investigates the simultaneous evolution of a family of Markov chains that start from every possible state in a finite state space S, under the rule that any two chains coalesce permanently as soon as they occupy the same state. Such a construction is called a “grand coupling” and is described by a probability measure μ on the set F_S of functions f : S→S that is consistent with the given transition matrix P (i.e., μ assigns to each pair (i,j) the transition probability p_{i,j}). The set of all consistent measures is denoted L_P.

The authors first introduce the independence coupling μ_ind, obtained by drawing each step’s function independently according to the product of the row‑wise transition probabilities. They prove (Theorem 2.3) that μ_ind is the unique grand coupling precisely when every row of P contains at most one entry in the open interval (0,1). If at least two rows contain such non‑deterministic entries, then |L_P|≥2 and multiple grand couplings exist. This result is proved by analysing the directed graph of P and showing that the presence of more than one “non‑trivial” edge permits alternative joint distributions of the functions.

Next, the paper defines an equivalence relation ∼ on S by i∼j iff the two chains ever meet (i.e., ∃t : X_i^t = X_j^t). The equivalence classes are called coalescence classes, and their number is denoted k(μ). The set of all possible numbers of coalescence classes over all grand couplings is K(P) = {k(μ): μ∈L_P}. Two central questions are posed (Question 3.3): (1) can K(P) be determined for a given P, and (2) which couplings achieve k(μ)=1 (i.e., full coalescence)?

Theorem 3.13 shows that the independence coupling μ_ind minimizes k(μ) among all grand couplings: for any μ∈L_P, k(μ_ind) ≤ k(μ). Moreover, k(μ_ind)=1 if and only if P is aperiodic. This follows from Doeblin’s theorem, which guarantees that in an aperiodic, irreducible finite Markov chain any two trajectories meet almost surely in finite time. Consequently, the forward coalescence time T and the backward coalescence time C (used in Coupling From The Past, CFTP) have the same distribution, and the set K(P) is the same for forward and backward processes, although the mechanisms differ when coalescence classes are nondeterministic.

The paper then introduces a special class of couplings called “block measures”. These arise when the chain is lumpable: the state space can be partitioned into blocks such that the aggregated process on the blocks is itself a Markov chain. A block measure couples the original chains by first coupling the lumped chain (often via an independence coupling) and then refining the coupling inside each block. Theorem 5.3 provides sufficient conditions for a block measure to exist, essentially requiring that the transition probabilities be compatible with the lumping. Conversely, Theorem 6.1 shows that for certain highly symmetric transition matrices (e.g., all rows equal, each entry 1/|S|) no non‑trivial block measure exists, illustrating that block measures are not universal.

The authors also study bounds on the maximal possible number of coalescence classes, k_max(P) = max_{μ∈L_P} k(μ). Theorem 3.5 gives simple probabilistic upper bounds: if the stationary distribution π of P places more than 1/m mass on some state, then k_max < m; similarly, if every row assigns more than 1/m probability to a particular state, then k_max < m. These results follow from density arguments: with high stationary mass, many chains will visit that state frequently, forcing coalescence among them. Remark 3.6 refines the second condition to allow exclusion of the diagonal entry.

Section 7 treats random transition matrices. By drawing each entry q_{i,j} independently from a continuous distribution on (0,1) and normalising rows, one obtains a “typical” matrix P that almost surely has all entries in (0,1). Consequently, with probability one, |L_P|≥2 and the set K(P) contains more than one element. This observation underlines that in practical applications most transition matrices admit multiple grand couplings, which has implications for the design and analysis of CFTP algorithms.

Overall, the paper provides a systematic framework for understanding how different couplings of a finite‑state Markov chain affect the number of eventual coalescence classes. It establishes that the independence coupling is optimal for minimizing coalescence classes, characterises when multiple grand couplings exist, introduces block measures as a structured way to build non‑trivial couplings, and gives concrete bounds on the maximal number of coalescence classes based on stationary probabilities and row‑wise transition weights. These insights deepen the theoretical foundations of exact sampling methods such as CFTP and open several avenues for further research, including the full characterisation of K(P) for arbitrary P and the exploration of avoidance couplings in the simultaneous‑update setting.