Split merge dynamics for expanding intervals and point processes on the real line

Abstract: We study sequences of partitions of a non decreasing sequence I n of intervals into subintervals, starting from the trivial partition, in which each partition is obtained from the one before by splitting its subintervals in two, according to a given rule, and then merging pairs of subintervals at the break points of the old partition. The nth partition then comprises n+1 subintervals with n break points. When I n = [0, 1] is constant, the empirical distribution of these points was shown to converge weakly to a singular probability supported in {0, 1} in a previous article. When the length of the intervals is regularly varying with a positive index, we show in this article that the limit can be absolutely continuous. In the last part we extend the split merge dynamics to partitions of R. In this case we characterize invariant distributions and show that special instances of split merge dynamics for expanding intervals converge to these invariant measures vaguely in distribution.

• The paper introduces a novel split-merge Markov process framework for expanding intervals that yields distinct limiting empirical measures under varying expansion regimes.
• It employs probabilistic tools, including renewal theory and random walks, to rigorously characterize convergence and invariant stationary measures.
• The results bridge fragmentation theory with spatial processes, offering insights applicable to hierarchical clustering and random recursive partitioning.

Split-Merge Dynamics for Expanding Intervals and Point Processes on the Real Line

Introduction and Framework

The paper "Split merge dynamics for expanding intervals and point processes on the real line" (2604.19220) introduces and systematically analyzes a class of split-merge Markov processes defined on growing sequences of intervals and their induced partitions. These processes are characterized by iterative fragmentation, where intervals are recursively split at specified or random proportions, combined with an erasure at previous breakpoints. This results in a sequence of partitions that dynamically evolve as the endpoints of the base interval expand, accommodating deterministic or random expansion regimes.

The authors generalize prior work where such dynamics were studied on a fixed interval, e.g., $[0,1]$, showing that, under constant length, breakpoints asymptotically concentrate at the boundary—yielding singular limiting distributions. By introducing regular, slow, or fast varying regimes for interval expansion, they reveal a spectrum of empirical distributions, including regimes where absolutely continuous limits emerge. Furthermore, the dynamics are extended to point process-valued Markov chains on the real line, identifying invariant (stationary) measures and characterizing convergence in the vague topology.

Split-Merge Partition Dynamics

The construction is based on a recursive formula for partition points $a_{n,k}$ within intervals $[a_{n,0}, a_{n,n+1}]$. Each partition is generated by splitting the previous intervals according to proportions $p_{n,k}$, potentially random, and then merging per an erasure scheme. The main recursion,

$$a_{n,k} = p_{n,k} a_{n-1, k-1} + (1 - p_{n,k}) a_{n-1, k},$$

encapsulates both deterministic and random (stratified and fully random) protocols. The process creates Markov chains in the space of interval partitions, which under randomization encode nontrivial genealogical and spatial dependencies.

Three cases for expansion sequence $l_n = a_{n, n+1} - a_{n, 0}$ are treated:

1. Slow variation ($c = 0$): $l_n$ increases very slowly with $n$.
2. Regular variation ($0 < c < \infty$): $a_{n,k}$0 grows polynomially; in this regime, novel features arise.
3. Fast increments ($a_{n,k}$1): $a_{n,k}$2 rapidly outpaces $a_{n,k}$3.

The analysis leverages probabilistic representations of the breakpoints via auxiliary Markov processes and embedded random walks, facilitating distributional and limit theorems for empirical measures.

Limiting Empirical Distributions

A core contribution is the identification of the limiting empirical distribution of the normalized breakpoints under various expansion regimes. For a sequence of partitions with $a_{n,k}$4 breakpoints, the empirical measure of (normalized) breakpoints is

$a_{n,k}$5

where $a_{n,k}$6 is an affine map rescaling $a_{n,k}$7 to $a_{n,k}$8.

The limit law for $a_{n,k}$9 exhibits a dichotomy:

• For fixed or slowly varying interval ($[a_{n,0}, a_{n,n+1}]$0), all mass concentrates on the endpoints, in line with earlier findings; the limit is a convex combination of Dirac measures at $[a_{n,0}, a_{n,n+1}]$1 and $[a_{n,0}, a_{n,n+1}]$2.
• For regular variation ($[a_{n,0}, a_{n,n+1}]$3), the empirical distribution converges weakly to an absolutely continuous law, typically a mixture of beta distributions whose parameters depend on $[a_{n,0}, a_{n,n+1}]$4, the mean split proportion $[a_{n,0}, a_{n,n+1}]$5, and an additional parameter $[a_{n,0}, a_{n,n+1}]$6 describing the increment regime. Notably, for $[a_{n,0}, a_{n,n+1}]$7 and uniform randomization, the limit is Lebesgue measure.
• For fast increments ($[a_{n,0}, a_{n,n+1}]$8), the limit is the Dirac mass at the right endpoint.

These results (Theorem~\ref{thm:cv_p} in the paper) are supported by explicit concentration bounds, probabilistic coupling arguments, and law of large numbers for sums of Bernoulli variables parametrizing the split-merge steps.

Detailed Regimes and Special Cases

When the expansion parameters and splitting proportions are fixed (homogeneous regime), regular spacing of breakpoints is established, and their local densities can be finely described. Perturbations from symmetry (nonuniform $[a_{n,0}, a_{n,n+1}]$9) induce asymmetry in breakpoint spacing, quantifiable via explicit recurrences and generating functions.

In the fully random and exponentially expanding case (Section~\ref{sec:rand-expan}), the process is initialized with random boundary points distributed as sums of independent exponentials (i.e., Gamma variables), and all splits are uniformly random. It is shown that subinterval lengths within partitions are i.i.d.\ $p_{n,k}$0-distributed in every iteration. Upon normalization, the empirical measure of breakpoints converges to the Lebesgue measure on $p_{n,k}$1, and the process admits Donsker-type fluctuation results, where the empirical process converges to a Brownian bridge.

Dynamics on Point Processes and Invariant Laws

The split-merge framework is extended to partitions of the entire real line, corresponding to point processes. Here, the system is recast as a Markov process on integer-valued measures via interval endpoints $p_{n,k}$2. Through analysis of the splitting and merging transformation, the authors construct explicit stationary point processes invariant under the dynamics: renewal processes with i.i.d.\ $p_{n,k}$3 increments, which are equivalently obtainable from thinning and shifting Poisson processes. These stationary processes are shown to be the (unique) attractive limit for evolved random measures under the split-merge operation.

This convergence is established in the vague topology for point processes, using techniques from the theory of Markov chains on general state spaces, ergodicity, and coupling with stationary renewal processes. Geometric ergodicity is obtained via Lyapunov functions and small set conditions.

Implications and Future Research

This paper establishes a rigorous mathematical framework for a class of stochastic dynamics relevant for fragmentation, spatial random growth, and interacting partition processes. The results supply exact limit theorems for the distribution of breakpoints under natural regularity and randomization assumptions, demonstrating when singular measures give rise to absolutely continuous analogues under interval expansion.

The characterization of invariant measures for point process-valued Markov chains links fragmentation theory with the ergodic theory of renewal and Poisson processes. The results open avenues to analyze more complicated fragmentation-coalescence systems, nonuniform spatial dynamics, or models with memory or delayed interactions.

From an applied standpoint, these models and limit theorems could find recognition in random spatial algorithms, random recursive partitioning methods, and models for hierarchical clustering and tree growth with randomization and expansion. The explicit limiting distributions can serve as benchmarks for algorithmic randomized partitioning in high-dimensional computational settings.

Possible future developments include the analysis of the central regime (breakpoints in the median region), convergence rates and mixing times for the Markov chains, and scaling limits under more general non-polynomial expansion laws or dependent splitting mechanisms.

Conclusion

The paper provides a comprehensive and technical account of split-merge interval dynamics under expanding environments, demonstrating a transition from singular to absolutely continuous limiting empirical distributions and identifying invariant point processes for their real-line analogues. Its results generalize earlier work, give new explicit limit laws, and contribute to the probabilistic understanding of partition structures in evolving domains. The mathematical tools developed—ranging from renewal theory to Markov process stability—facilitate broader applications and extensions, marking the work as a substantive contribution to the theory of stochastic partition dynamics.