Exchangeable Fragmentation-Coalescence Process

• Exchangeable Fragmentation-Coalescence Process is a partition-valued Markov process where blocks merge and split while preserving invariance under finite permutations.
• The model employs independent Poisson point processes to define explicit jump rates and phase transitions through coagulation and fragmentation measures.
• Key insights include its dual nature, explicit block-counting dynamics, and nuanced boundary behaviors like coming down from infinity.

Searching arXiv for the cited papers on exchangeable fragmentation-coalescence processes. Exchangeable fragmentation-coalescence processes are partition-valued Markov processes on $\mathbb N$ in which blocks both merge and split while preserving exchangeability under finite permutations of labels. In Berestycki’s framework, the process is characterized by a coagulation measure and a fragmentation or dislocation measure, and it is constructed from independent Poisson point processes acting on the current partition (Kyprianou et al., 2016). The subject links exchangeable random partitions, paintbox laws, asymptotic block frequencies, coalescent theory, and boundary behavior at infinity. It also includes instructive near-boundary cases: some models are fully exchangeable and admit explicit stationary or excursion descriptions, whereas others preserve a fragmentation architecture but fall outside the exchangeable theory because labels play a privileged role (Baur et al., 2015).

1. Formal setting and exchangeability

An exchangeable fragmentation-coalescence process, or EFC process, is a random partition-valued evolution of the set $\mathbb N=\{1,2,\dots\}$ in which blocks can both merge and split over time, while preserving exchangeability (Foucart, 2016). Exchangeability means that the law is invariant under finite permutations of labels: the dynamics depend only on the block structure, not on which integers are in which blocks (Foucart, 2016). In the partition-valued formulation used for the fast model, the state space is $\mathcal P$, the space of partitions of $\mathbb N$, with blocks ordered by least element, and a partition is exchangeable if its law is invariant under finite permutations of $\mathbb N$ (Kyprianou et al., 2016).

Kingman’s representation theorem says that exchangeable random partitions correspond to paintbox laws built from mass partitions (Kyprianou et al., 2016). In the erosion model, the asymptotic frequencies of blocks in a stationary partition are defined by

$$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$

and their decreasing rearrangement $(\beta_i)_{i\ge 1}$ is the associated mass-partition (Foutel-Rodier et al., 2019). The paper recalls from the general theory that, for this model,

$$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$

so the stationary partition has infinitely many blocks and no dust (Foutel-Rodier et al., 2019).

The general EFC construction in the fast shattering model uses two independent Poisson point processes: one for coalescence, with intensity $dt\times C(d\pi)$, and one for fragmentation, with intensity $dt\times F(d\pi)\times \#(dk)$ (Kyprianou et al., 2016). In the simple EFC class, the process is again built from two independent Poisson point processes, one governing coagulation and the other fragmentation, and finite restrictions yield a càdlàg Feller process on $\mathbb N=\{1,2,\dots\}$0 (Foucart et al., 2020). A key consequence of exchangeability, emphasized in the 2025 slow-regime paper, is that every block in an exchangeable random partition is either a singleton or infinite (Ji et al., 6 Apr 2025).

2. Canonical mechanisms and model classes

A central tractable example combines Kingman’s coalescent with complete shattering fragmentation. In that model, any pair of blocks merges at rate $\mathbb N=\{1,2,\dots\}$1, while each block fragments at constant rate $\mathbb N=\{1,2,\dots\}$2 into its constituent singletons (Kyprianou et al., 2016). The corresponding measures are

$\mathbb N=\{1,2,\dots\}$3

and

$\mathbb N=\{1,2,\dots\}$4

so a fragmentation event turns the chosen block into all its singleton elements (Kyprianou et al., 2016). Because the fragmentation is complete shattering, the block-counting process is especially simple (Kyprianou et al., 2016).

A second major subclass is the class of simple EFC processes, in which coagulations are multiple but non-simultaneous, as in a $\mathbb N=\{1,2,\dots\}$5-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size (Foucart, 2016). In this setting the coagulation mechanism is governed by a finite measure

$\mathbb N=\{1,2,\dots\}$6

with $\mathbb N=\{1,2,\dots\}$7 and $\mathbb N=\{1,2,\dots\}$8 satisfying $\mathbb N=\{1,2,\dots\}$9 (Foucart, 2016). The associated jump rates for the block-counting process started from $\mathcal P$0 blocks are

$\mathcal P$1

and the total rate at which the number of blocks decreases from $\mathcal P$2 is

$\mathcal P$3

(Foucart, 2016).

Fragmentation in the simple model is also highly constrained. The fragmentation measure $\mathcal P$4 is finite,

$\mathcal P$5

and it is supported on partitions with no singleton blocks, which is equivalent to saying that every fragment produced is infinite (Foucart, 2016). In the induced block-counting process, a fragmentation of one block into $\mathcal P$6 sub-blocks increases the number of blocks by $\mathcal P$7, or sends it to $\mathcal P$8 if the fragmentation produces infinitely many pieces (Foucart, 2016).

A different canonical mechanism is erosion. Kingman’s coalescent with erosion is the Markov process taking values in the partitions of $\mathcal P$9 such that each pair of blocks merges at rate one, and each integer is eroded, becoming a singleton block, at rate $\mathbb N$0 (Foutel-Rodier et al., 2019). For each $\mathbb N$1, the $\mathbb N$2-particle dynamics have coalescence and erosion transitions, and if $\mathbb N$3, nothing happens under erosion (Foutel-Rodier et al., 2019). This model is a special case of exchangeable fragmentation-coalescence process called Kingman’s coalescent with erosion (Foutel-Rodier et al., 2019).

3. Block-counting process and boundary behavior at infinity

For many EFC processes, the most informative one-dimensional observable is the block-counting process

$\mathbb N$4

with the convention $\mathbb N$5 if there are infinitely many blocks (Kyprianou et al., 2016). In the fast fragmentation-coalescence model, $\mathbb N$6 is a Markov chain on $\mathbb N$7 with rates

$\mathbb N$8

Thus from a finite state $\mathbb N$9, it coalesces to $\mathbb N$0 at rate $\mathbb N$1 and fragments to $\mathbb N$2 at rate $\mathbb N$3 (Kyprianou et al., 2016). The reciprocal process

$\mathbb N$4

lives on $\mathbb N$5 and is more convenient for excursion analysis (Kyprianou et al., 2016).

The phrase “coming down from infinity” means that if the process starts with infinitely many blocks, then for every $\mathbb N$6 the number of blocks is finite almost surely: $\mathbb N$7 (Kyprianou et al., 2016). In simple EFC processes, the question is framed slightly differently: if $\mathbb N$8, does there exist $\mathbb N$9 such that $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$0? The authors emphasize that they do not require $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$1 to be an inaccessible boundary, so “coming down from infinity” here only means that the process at some positive time can be finite, even if it later returns to $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$2 (Foucart, 2016).

The boundary $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$3 can be classified in Feller’s terminology as an exit boundary, an entrance boundary, or a regular boundary (Foucart et al., 2020). In the regular regime, the process may leave $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$4 instantaneously and return immediately; moreover, in that regime $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$5 is regular for itself (Foucart et al., 2020). The 2025 slow-regime work refines this program by studying whether the block-counting process comes down from infinity, stays infinite, explodes, or makes $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$6 an entrance boundary or an exit boundary, with the behavior determined by the asymptotics of the difference

$$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$7

between the coagulation and fragmentation total rates (Ji et al., 6 Apr 2025).

This suggests a general structural principle: at large block counts, EFC behavior is governed by a competition between a downward coagulation scale and an upward fragmentation scale. In the cited papers this competition is encoded either by explicit finite-state rates, by the function $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$8, by the pair $$f_i=\lim_{n\to\infty}\frac1n\sum_{k=1}^n \mathbf{1}_{\{k\in C_i\}},$$9, or by the asymptotic difference $(\beta_i)_{i\ge 1}$0 (Kyprianou et al., 2016, Foucart, 2016, Foucart et al., 2020, Ji et al., 6 Apr 2025).

4. Phase transitions and large-scale asymptotics

The fast fragmentation-coalescence model exhibits a sharp phase transition at

$(\beta_i)_{i\ge 1}$1

(Kyprianou et al., 2016). Writing

$(\beta_i)_{i\ge 1}$2

the main theorem states that if $(\beta_i)_{i\ge 1}$3, equivalently $(\beta_i)_{i\ge 1}$4, then $(\beta_i)_{i\ge 1}$5 is a recurrent Feller process on $(\beta_i)_{i\ge 1}$6, and $(\beta_i)_{i\ge 1}$7 is instantaneously regular and non-sticky: $(\beta_i)_{i\ge 1}$8 If $(\beta_i)_{i\ge 1}$9, equivalently $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$0, then $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$1 is absorbing for $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$2, which is equivalent to saying that once the process reaches infinity, it stays there forever (Kyprianou et al., 2016).

In the subcritical regime $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$3, the process admits a full excursion theory away from $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$4. There exists a local time $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$5 at $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$6 for $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$7, with inverse $$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$8 a pure-jump subordinator of infinite activity, and the zero set

$$\sum_{i\ge 1}\beta_i=1,\qquad \beta_i>0\ \forall i,$$9

has Hausdorff dimension

$dt\times C(d\pi)$0

(Kyprianou et al., 2016). The same regime yields an explicit stationary distribution for $dt\times C(d\pi)$1: $dt\times C(d\pi)$2 which is a Beta-Geometric$dt\times C(d\pi)$3 distribution, with generating function

$dt\times C(d\pi)$4

(Kyprianou et al., 2016). The expected hitting time from infinity of level $dt\times C(d\pi)$5 is

$dt\times C(d\pi)$6

and for an excursion $dt\times C(d\pi)$7,

$dt\times C(d\pi)$8

so the local behavior near entrance from $dt\times C(d\pi)$9 looks like Kingman’s coalescent with rate $dt\times F(d\pi)\times \#(dk)$0 (Kyprianou et al., 2016).

For simple EFC processes with $dt\times F(d\pi)\times \#(dk)$1-coalescent-type coagulation, the phase transition is expressed through the parameters

$dt\times F(d\pi)\times \#(dk)$2

(Foucart, 2016). The main theorem states that if $dt\times F(d\pi)\times \#(dk)$3, then the simple EFC process comes down from infinity, and if $dt\times F(d\pi)\times \#(dk)$4, then the process stays infinite (Foucart, 2016). In regularly varying examples with

$dt\times F(d\pi)\times \#(dk)$5

the parameters coincide and are explicit: if $dt\times F(d\pi)\times \#(dk)$6, then $dt\times F(d\pi)\times \#(dk)$7; if $dt\times F(d\pi)\times \#(dk)$8, then $dt\times F(d\pi)\times \#(dk)$9; and in the critical case $\mathbb N=\{1,2,\dots\}$00,

$\mathbb N=\{1,2,\dots\}$01

(Foucart, 2016).

The explosion paper adds a complementary classification. Defining

$\mathbb N=\{1,2,\dots\}$02

it recalls that under suitable summability assumptions, if

$\mathbb N=\{1,2,\dots\}$03

the process comes down from infinity, whereas if

$\mathbb N=\{1,2,\dots\}$04

it stays infinite (Foucart et al., 2020). In the regularly varying regime

$\mathbb N=\{1,2,\dots\}$05

Theorem 3.7 states that if $\mathbb N=\{1,2,\dots\}$06, then $\mathbb N=\{1,2,\dots\}$07 is an exit boundary; if $\mathbb N=\{1,2,\dots\}$08, then $\mathbb N=\{1,2,\dots\}$09 is an entrance boundary; and if $\mathbb N=\{1,2,\dots\}$10, then the ratio $\mathbb N=\{1,2,\dots\}$11 determines whether $\mathbb N=\{1,2,\dots\}$12 is exit, entrance, or regular (Foucart et al., 2020).

The 2025 slow critical regime paper identifies another family of thresholds. Under

$\mathbb N=\{1,2,\dots\}$13

one has

$\mathbb N=\{1,2,\dots\}$14

(Ji et al., 6 Apr 2025). For $\mathbb N=\{1,2,\dots\}$15, the critical scale is

$\mathbb N=\{1,2,\dots\}$16

while for $\mathbb N=\{1,2,\dots\}$17, the critical scale is

$\mathbb N=\{1,2,\dots\}$18

(Ji et al., 6 Apr 2025). The theorems then separate non-explosion, coming down from infinity, staying infinite, explosion, and entrance or exit boundary behavior in terms of asymptotics of $\mathbb N=\{1,2,\dots\}$19 at those scales (Ji et al., 6 Apr 2025).

5. Stationarity, bridge representations, and diffusion structure

Kingman’s coalescent with erosion has a unique stationary distribution $\mathbb N=\{1,2,\dots\}$20 (Foutel-Rodier et al., 2019). A major contribution of that work is a new explicit construction of the stationary law using the standard flow of bridges. A bridge is a random nondecreasing map $\mathbb N=\{1,2,\dots\}$21 of the form

$\mathbb N=\{1,2,\dots\}$22

where $\mathbb N=\{1,2,\dots\}$23 is a mass-partition and $\mathbb N=\{1,2,\dots\}$24 are i.i.d. uniform $\mathbb N=\{1,2,\dots\}$25, independent of $\mathbb N=\{1,2,\dots\}$26, and the inverse bridge is

$\mathbb N=\{1,2,\dots\}$27

(Foutel-Rodier et al., 2019).

A flow of bridges $\mathbb N=\{1,2,\dots\}$28 satisfies

$\mathbb N=\{1,2,\dots\}$29

and $\mathbb N=\{1,2,\dots\}$30 as $\mathbb N=\{1,2,\dots\}$31 (Foutel-Rodier et al., 2019). For the standard flow of bridges, the bridge $\mathbb N=\{1,2,\dots\}$32 has the law

$\mathbb N=\{1,2,\dots\}$33

where $\mathbb N=\{1,2,\dots\}$34 is a pure-death process started from $\mathbb N=\{1,2,\dots\}$35 and jumping $\mathbb N=\{1,2,\dots\}$36 at rate $\mathbb N=\{1,2,\dots\}$37, and conditional on $\mathbb N=\{1,2,\dots\}$38, $\mathbb N=\{1,2,\dots\}$39 is Dirichlet$\mathbb N=\{1,2,\dots\}$40 (Foutel-Rodier et al., 2019). If $\mathbb N=\{1,2,\dots\}$41 are i.i.d. exponential($\mathbb N=\{1,2,\dots\}$42) and $\mathbb N=\{1,2,\dots\}$43 are i.i.d. uniform$\mathbb N=\{1,2,\dots\}$44, independent of the bridge flow, then

$\mathbb N=\{1,2,\dots\}$45

has the stationary distribution of Kingman’s coalescent with erosion rate $\mathbb N=\{1,2,\dots\}$46 (Foutel-Rodier et al., 2019).

The same paper gives an explicit recursive diffusion description of the stationary asymptotic frequencies. Let $\mathbb N=\{1,2,\dots\}$47 be i.i.d. diffusions solving

$\mathbb N=\{1,2,\dots\}$48

started from $\mathbb N=\{1,2,\dots\}$49, with independent Brownian motions $\mathbb N=\{1,2,\dots\}$50 (Foutel-Rodier et al., 2019). Defining recursively

$\mathbb N=\{1,2,\dots\}$51

and for $\mathbb N=\{1,2,\dots\}$52,

$\mathbb N=\{1,2,\dots\}$53

$\mathbb N=\{1,2,\dots\}$54

the stationary asymptotic frequencies are given by the decreasing rearrangement of

$\mathbb N=\{1,2,\dots\}$55

(Foutel-Rodier et al., 2019). This provides an explicit diffusion representation of the stationary mass partition.

The paper also introduces Kingman’s coalescent with immigration, in which pairs of blocks coalesce at rate $\mathbb N=\{1,2,\dots\}$56 and new singleton blocks immigrate at rate $\mathbb N=\{1,2,\dots\}$57 (Foutel-Rodier et al., 2019). Its block-counting process $\mathbb N=\{1,2,\dots\}$58 is a stationary birth-death chain,

$\mathbb N=\{1,2,\dots\}$59

with reversible stationary distribution

$\mathbb N=\{1,2,\dots\}$60

(Foutel-Rodier et al., 2019). A coupling between the erosion and immigration models is then used to derive small-block asymptotics: if $\mathbb N=\{1,2,\dots\}$61 is the number of blocks in the stationary restriction to $\mathbb N=\{1,2,\dots\}$62 and $\mathbb N=\{1,2,\dots\}$63 is the empirical fraction of blocks of size $\mathbb N=\{1,2,\dots\}$64, then

$\mathbb N=\{1,2,\dots\}$65

in probability, and for each fixed $\mathbb N=\{1,2,\dots\}$66,

$\mathbb N=\{1,2,\dots\}$67

where $\mathbb N=\{1,2,\dots\}$68 is the total progeny of a critical binary branching process (Foutel-Rodier et al., 2019).

6. Duality, generalizations, and nonexchangeable analogues

A prominent structural theme is coagulation-fragmentation duality. The recursive-tree destruction paper gives a nonexchangeable analogue: by reversing the time transformation $\mathbb N=\{1,2,\dots\}$69, the tree-destruction dynamics becomes a construction process, and the time-reversed partition process is described as a binary coalescent (Baur et al., 2015). The rate at which two blocks merge depends on the number of smaller labels in the earlier block, so the coalescent is not exchangeable, but the authors explicitly mention its relation to the Bolthausen–Sznitman coalescent and the standard additive coalescent literature (Baur et al., 2015). This is therefore a duality-style connection in a nonexchangeable setting (Baur et al., 2015).

Although that process is not exchangeable, it behaves in several important ways like a homogeneous exchangeable fragmentation process (Baur et al., 2015). The partition-valued process $\mathbb N=\{1,2,\dots\}$70 is Markovian with generator

$\mathbb N=\{1,2,\dots\}$71

where $\mathbb N=\{1,2,\dots\}$72 is the splitting-rate measure (Baur et al., 2015). The resemblance to homogeneous fragmentations is strong at the level of Markovian fragmentation structure, restriction consistency, and generator form, but the law is not exchangeable because the root and smaller labels play a privileged role in the recursive tree (Baur et al., 2015). Moreover, the splitting measure

$\mathbb N=\{1,2,\dots\}$73

fails the standard dislocation integrability condition

$\mathbb N=\{1,2,\dots\}$74

used in the exchangeable theory (Baur et al., 2015). The paper therefore identifies precisely what survives, and what fails, when exchangeability is removed (Baur et al., 2015).

Within the exchangeable setting, a major 2025 generalization revisits Pitman’s coagulation-fragmentation duality. The paper introduces a four-part coupled system built upon the Poisson Hierarchical Indian Buffet Process and states a Unified Poissonized Duality Identity for arbitrary subordinators $\mathbb N=\{1,2,\dots\}$75 (James, 26 Aug 2025). The construction simultaneously defines the fine-grained partition, its coagulation operator, a forward-in-time system of coupled, time-homogeneous fragmentation processes in the sense of Jean Bertoin, and a dual, backward-in-time structured coalescent (James, 26 Aug 2025). The total observable jump rate is

$\mathbb N=\{1,2,\dots\}$76

and the number of observed species is

$\mathbb N=\{1,2,\dots\}$77

(James, 26 Aug 2025).

The classical Pitman duality for $\mathbb N=\{1,2,\dots\}$78 is recovered as a special case. In the notation quoted there,

$\mathbb N=\{1,2,\dots\}$79

and the corresponding partition identity is

$\mathbb N=\{1,2,\dots\}$80

(James, 26 Aug 2025). What is new is that the same algebra survives for arbitrary subordinators, not only stable ones (James, 26 Aug 2025). A plausible implication is that duality in EFC theory is better viewed as a Lévy-subordination phenomenon than as a property specific to the Poisson-Dirichlet family.

A different but related finite-state perspective appears in the study of random $\mathbb N=\{1,2,\dots\}$81-cycles on $\mathbb N=\{1,2,\dots\}$82. There, the cycle structure of a permutation evolves like a coalescence-fragmentation system: a transposition merges two cycles if its points are in different cycles and splits one cycle into two if its points lie in the same cycle (Berestycki et al., 2010). The paper emphasizes that the chain shares several hallmarks of exchangeable fragmentation-coalescence processes, but it is not an abstract fragmentation-coalescence model; rather, coalescence and fragmentation are coupled within each $\mathbb N=\{1,2,\dots\}$83-cycle step by rigid algebraic structure (Berestycki et al., 2010). This serves as a finite-size analogue rather than an EFC process on $\mathbb N=\{1,2,\dots\}$84.

7. Conceptual themes and recurring misconceptions

One recurring misconception is that every partition-valued fragmentation-coalescence process with a fragmentation generator should fit the standard exchangeable fragmentation theory. The recursive-tree example shows otherwise: the process is Markovian and has the same structural form as a homogeneous fragmentation process, yet it is not exchangeable and its splitting measure does not satisfy the usual dislocation integrability condition (Baur et al., 2015). In that model every block of $\mathbb N=\{1,2,\dots\}$85 still has an asymptotic frequency, but it is degenerate,

$\mathbb N=\{1,2,\dots\}$86

so ordinary asymptotic frequencies do not produce a nontrivial mass partition (Baur et al., 2015). The paper replaces them with a time-dependent normalization leading to block weights whose logarithms form an Ornstein-Uhlenbeck branching process (Baur et al., 2015).

A second misconception is that “coming down from infinity” is a uniform notion across all models. In the fast fragmentation-coalescence process, when $\mathbb N=\{1,2,\dots\}$87, infinity is not absorbing and the process repeatedly makes excursions away from it (Kyprianou et al., 2016). In the simple EFC framework of 2016, by contrast, “coming down from infinity” only means that a process started from infinitely many blocks can visit a finite state at some positive time, even if it later returns to $\mathbb N=\{1,2,\dots\}$88 (Foucart, 2016). In the 2020 and 2025 works, the same boundary may be entrance, exit, or regular, and in the regular case the process can both enter and leave $\mathbb N=\{1,2,\dots\}$89 (Foucart et al., 2020, Ji et al., 6 Apr 2025).

A third recurring theme is that explicit solvability often comes from reducing the partition-valued dynamics to block-counting or to paintbox and bridge representations. Exact formulas for jump rates, stationary distributions, hitting times, and boundary behavior are obtained in the fast shattering model (Kyprianou et al., 2016), while stationary mass-partition laws and small-block limits become tractable in the erosion model through bridge flows, immigration couplings, and diffusion representations (Foutel-Rodier et al., 2019). In simple EFC processes, the functions $\mathbb N=\{1,2,\dots\}$90, $\mathbb N=\{1,2,\dots\}$91, $\mathbb N=\{1,2,\dots\}$92, and the tail $\mathbb N=\{1,2,\dots\}$93 act as summary statistics for the competition between coagulation and fragmentation (Foucart, 2016, Foucart et al., 2020).

Taken together, these works portray exchangeable fragmentation-coalescence processes as a family of partition-valued Markov models whose qualitative behavior is governed by a balance of upward fragmentation and downward coagulation, and whose most delicate phenomena occur at the boundary $\mathbb N=\{1,2,\dots\}$94. The exchangeable theory provides paintbox, mass-partition, and Poissonian constructions; explicit models reveal phase transitions, stationary structures, and excursion laws; and recent generalizations show that coagulation-fragmentation duality extends beyond the classical Poisson-Dirichlet setting to arbitrary subordinators and structured multi-group partitions (Kyprianou et al., 2016, Foucart, 2016, Foutel-Rodier et al., 2019, Foucart et al., 2020, Ji et al., 6 Apr 2025, James, 26 Aug 2025).