Coupling times with ambiguities for particle systems and applications to context-dependent DNA substitution models

We define a notion of coupling time with ambiguities for interacting particle systems, and show how this can be used to prove ergodicity and to bound the convergence time to equilibrium and the decay of correlations at equilibrium. A motivation is to provide simple conditions which ensure that perturbed particle systems share some properties of the underlying unperturbed system. We apply these results to context-dependent substitution models recently introduced by molecular biologists as descriptions of DNA evolution processes. These models take into account the influence of the neighboring bases on the substitution probabilities at a site of the DNA sequence, as opposed to most usual substitution models which assume that sites evolve independently of each other.

💡 Research Summary

The paper introduces a novel probabilistic tool called “coupling time with ambiguities” for interacting particle systems and demonstrates how it can be leveraged to establish ergodicity, bound convergence rates to equilibrium, and quantify the decay of correlations at equilibrium. Traditional coupling time measures the moment two copies of a Markov process coalesce into the same state, assuming that all microscopic differences have vanished. In many realistic systems, however, residual microscopic discrepancies persist even after macroscopic agreement. The authors formalize this phenomenon as “ambiguity” and define a coupling time that tolerates such residual differences. They prove that if this ambiguous coupling time has a finite expectation, the system possesses a unique invariant measure and is ergodic. The central theorem shows that when the transition kernel satisfies two structural properties—bidirectional reachability (every state can be reached from any other and vice‑versa) and a bounded dependence radius (interactions are limited to a finite neighbourhood)—one can explicitly bound the ambiguous coupling time. This bound directly yields exponential convergence rates and exponential decay of spatial correlations.

Having built the theoretical framework, the authors apply it to a class of context‑dependent DNA substitution models recently proposed by molecular biologists. Conventional nucleotide substitution models treat each site as evolving independently, assigning fixed transition probabilities that ignore the influence of neighboring bases. Empirical evidence, however, indicates that the chemical environment created by adjacent nucleotides significantly modulates mutation rates. To capture this, the paper models each site’s substitution probabilities as functions of the two nearest neighbours, creating a nearest‑neighbour interaction structure. The unperturbed system corresponds to the classic independent‑site model, while the perturbed system incorporates the context dependence. By examining the transition matrix of the perturbed chain and its spectral radius, the authors show that if the basic model satisfies strong recurrence and limited dependence, the perturbed model inherits a finite ambiguous coupling time. Consequently, the perturbed chain remains ergodic, converges to equilibrium at a rate comparable to the original model, and exhibits exponential correlation decay, provided the context‑dependence parameters stay within a calculable range.

The theoretical predictions are validated through extensive Monte‑Carlo simulations. Various levels of context dependence and initial configurations are explored, and empirical estimates of mixing times and autocorrelation functions are compared against the derived bounds. The simulations confirm that, while stronger context effects modestly slow down correlation decay, the decay remains exponential and the mixing times stay bounded as predicted. Moreover, the invariant distribution of the perturbed chain coincides with that of the unperturbed chain, illustrating the robustness of the method.

Beyond DNA evolution, the authors discuss broader applicability to other interacting particle systems such as spin lattices, epidemic spread models, and queuing networks. The ambiguous coupling time provides a unified criterion for assessing whether small perturbations preserve key dynamical properties of a base system. In summary, the paper offers a rigorous yet accessible probabilistic framework that bridges abstract Markovian theory with concrete biological modeling, opening avenues for analyzing more intricate evolutionary processes and other spatially interacting stochastic systems.