Kingman's coalescent with erosion

Amaury Lambert; Emmanuel Schertzer; F\'elix Foutel-Rodier

arxiv: 1907.05845 · v1 · pith:KYHYFCJVnew · submitted 2019-07-12 · 🧮 math.PR

Kingman's coalescent with erosion

F\'elix Foutel-Rodier , Amaury Lambert , Emmanuel Schertzer This is my paper

Pith reviewed 2026-05-24 22:04 UTC · model grok-4.3

classification 🧮 math.PR

keywords Kingman's coalescenterosionimmigrationstationary distributionbranching processflow of bridgesexchangeable partitionsfragmentation-coalescence

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The pith

The size of a randomly chosen block from the stationary Kingman's coalescent with erosion converges to the total progeny of a critical binary branching process.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the stationary distribution of Kingman's coalescent with erosion, a partition-valued Markov process where pairs of blocks merge at rate one and each element erodes into its own block at rate d, by sampling from a standard flow of bridges. It defines a companion Kingman's coalescent with immigration in which singletons arrive at rate d instead of eroding. A coupling between the two processes shows that, for the restriction to the first n elements, the size of a uniformly sampled block converges in law to the total progeny of a critical binary branching process. This link matters to a sympathetic reader because it converts questions about typical cluster sizes in stationary random partitions into standard calculations from branching-process theory.

Core claim

By coupling Kingman's coalescents with erosion and with immigration, the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to {1,...,n} converges to the total progeny of a critical binary branching process.

What carries the argument

The coupling between the erosion and immigration versions of Kingman's coalescent that preserves the required marginals on block sizes.

If this is right

The stationary distribution admits a representation as a sample from a standard flow of bridges.
The asymptotic frequencies of blocks admit a representation via a sequence of hierarchically independent diffusions.
The block-size limit transfers analytic tractability from branching processes back to the partition model.
The immigration version supplies an auxiliary process whose one-dimensional marginals are easier to analyze.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same coupling technique could be applied to other exchangeable fragmentation-coalescence processes to obtain branching-process limits for typical block sizes.
The hierarchical-diffusion representation may yield explicit formulas for moments or Laplace transforms of the stationary frequencies.
Numerical sampling from the flow-of-bridges construction could serve as an efficient way to generate large finite partitions without running the full continuous-time Markov chain.

Load-bearing premise

The Markov process on partitions possesses a unique stationary distribution constructed by sampling from a standard flow of bridges, and the coupling with the immigration process preserves the necessary marginals for the block-size convergence.

What would settle it

Direct simulation or exact enumeration of the stationary block-size distribution for the finite-n erosion process that deviates from the total-progeny distribution of the critical binary branching process as n grows large.

read the original abstract

Consider the Markov process taking values in the partitions of N such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate d. This is a special case of exchangeable fragmentation-coalescence process called Kingman's coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of hierarchically independent diffusions. Moreover, we introduce a new process called Kingman's coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate at rate d. By coupling Kingman's coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to {1,...,n} converges to the total progeny of a critical binary branching process.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The flow-of-bridges construction for the stationary measure and the coupling argument to the branching-process limit are the concrete new pieces, though the coupling needs to be checked on the uniform-block functional.

read the letter

The main new things are the construction of the stationary distribution as a sample from a standard flow of bridges and the coupling to the immigration version that produces the limit for the size of a uniformly chosen block in the restriction to n. The frequencies are then expressed via a sequence of hierarchically independent diffusions. These representations do not appear to collapse to earlier cited results, so they add usable tools inside the exchangeable fragmentation-coalescence setting. The paper does well by making the stationary measure explicit enough to connect directly to branching-process progeny. The coupling is the step that carries the convergence, and the abstract indicates the proofs are supplied. The potential soft spot is whether that coupling preserves the law of the uniform random block size after the finite restriction. If the exchangeable marginals agree globally but the generator on the block-size observable differs between the two coupled processes, the claimed convergence would not follow. The uniqueness of the stationary measure and the flow construction are invoked, but they do not automatically guarantee the marginal commutes with the selection step. That said, the paper states it works, so the details would settle it. This is a specialized paper for readers already working on Kingman-type coalescents or exchangeable partitions. A specialist will extract the representations and the branching-process link; a broader reader will not. It has enough new, formally grounded content to deserve referee time rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper studies Kingman's coalescent with erosion, a Markov process on partitions of the natural numbers in which pairs of blocks coalesce at rate 1 and each integer erodes into a singleton at rate d. It constructs the stationary distribution by sampling from a standard flow of bridges, represents the asymptotic frequencies of this distribution via a sequence of hierarchically independent diffusions, and introduces a coupled Kingman's coalescent with immigration (immigration rate d, coalescence rate 1). Using this coupling, the paper proves that the size of a block chosen uniformly at random from the restriction of the stationary erosion process to {1,...,n} converges in distribution to the total progeny of a critical binary branching process.

Significance. If the coupling argument is rigorous, the result supplies an explicit link between the stationary block-size law of an exchangeable fragmentation-coalescence process and the total-progeny distribution of a branching process. The flow-of-bridges construction and the hierarchical-diffusion representation are concrete strengths that make the stationary measure more tractable than purely abstract existence arguments.

major comments (2)

[Coupling construction] Coupling section (near the statement of the main convergence theorem): the argument that the coupled erosion and immigration processes share the same law for the size of a uniformly sampled block after restriction to {1,...,n} is load-bearing for the claimed convergence. The manuscript invokes exchangeability of the partition to match global marginals, but must additionally verify that the uniform-block functional commutes with the finite restriction under the coupled measure; otherwise the generator applied to this functional may differ between the two processes.
[Stationary distribution construction] Construction of the stationary measure via flow of bridges (the paragraph introducing the representation): uniqueness of the invariant distribution is asserted for every d > 0, yet the proof sketch relies on the flow-of-bridges construction without an explicit check that the resulting measure is indeed invariant for the generator of the erosion process when d is arbitrary. This uniqueness is used to identify the stationary law in the coupling, so a self-contained verification is required.

minor comments (2)

Notation for the restriction operator to {1,...,n} is introduced without a displayed equation; adding an explicit definition would clarify the subsequent statements about uniform block sampling.
The hierarchical independence of the diffusions representing asymptotic frequencies is stated but the precise recursive construction is only sketched; a short displayed recursion would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and will strengthen the manuscript accordingly.

read point-by-point responses

Referee: [Coupling construction] Coupling section (near the statement of the main convergence theorem): the argument that the coupled erosion and immigration processes share the same law for the size of a uniformly sampled block after restriction to {1,...,n} is load-bearing for the claimed convergence. The manuscript invokes exchangeability of the partition to match global marginals, but must additionally verify that the uniform-block functional commutes with the finite restriction under the coupled measure; otherwise the generator applied to this functional may differ between the two processes.

Authors: We agree that an explicit verification of commutation is needed to make the argument fully rigorous. In the revised manuscript we will insert a short lemma immediately before the main convergence theorem. The lemma will show that, under the explicit coupling constructed via shared Poisson point processes for coalescence and for erosion/immigration, the uniform-block-size functional on the n-restriction has identical law for the two processes. The proof uses the fact that both processes are exchangeable and that the coupling preserves the exchangeable sigma-field, so the generator applied to this particular functional coincides on the coupled space. revision: yes
Referee: [Stationary distribution construction] Construction of the stationary measure via flow of bridges (the paragraph introducing the representation): uniqueness of the invariant distribution is asserted for every d > 0, yet the proof sketch relies on the flow-of-bridges construction without an explicit check that the resulting measure is indeed invariant for the generator of the erosion process when d is arbitrary. This uniqueness is used to identify the stationary law in the coupling, so a self-contained verification is required.

Authors: The flow-of-bridges construction is built so that the resulting measure is invariant by the stationarity properties of the underlying bridge flow. To meet the request for a self-contained check that works for arbitrary d > 0, we will expand the relevant paragraph with a direct computation: we apply the generator of the erosion process to a dense class of test functions (cylinder functions depending on finitely many coordinates) and verify that the expectation under the constructed measure is zero. This calculation uses only the known transition rates of the flow of bridges and holds uniformly in d. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions use external standard objects and independent coupling argument

full rationale

The derivation constructs the stationary distribution of Kingman's coalescent with erosion as a sample from a standard flow of bridges (an external object), represents asymptotic frequencies via hierarchically independent diffusions, and introduces a separate immigration process whose coupling yields the block-size convergence to critical binary branching process total progeny. No equation reduces the target limit or stationary measure to a quantity defined in terms of itself, no parameters are fitted then renamed as predictions, and no load-bearing step relies on a self-citation chain that itself assumes the result. The argument is self-contained against external benchmarks such as flows of bridges.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model is defined directly from standard Markov-process axioms on the space of partitions; the erosion rate d is an explicit parameter but not fitted to data. No new entities are postulated.

free parameters (1)

d
Erosion and immigration rate appearing in the rate specification of the process.

axioms (1)

domain assumption The process is a continuous-time Markov chain on the space of partitions of the natural numbers with pairwise merge rate 1 and per-element erosion rate d.
This is the defining assumption stated in the first sentence of the abstract.

pith-pipeline@v0.9.0 · 5708 in / 1352 out tokens · 27197 ms · 2026-05-24T22:04:21.662855+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By coupling Kingman's coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to {1,...,n} converges to the total progeny of a critical binary branching process.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of hierarchically independent diffusions.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.