--- title: Exchangeable Fragmentation-Coalescence Process url: https://www.emergentmind.com/topics/exchangeable-fragmentation-coalescence-process type: topic --- # Exchangeable Fragmentation-Coalescence Process 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 [1602.05241]. 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 [1501.01400]. ## 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 [1605.07039]. 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 [1605.07039]. 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 \(\mathbb N\) [1602.05241]. Kingman’s representation theorem says that exchangeable random partitions correspond to paintbox laws built from mass partitions [1602.05241]. 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 [1907.05845]. 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 [1907.05845]. 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 \(dt\times F(d\pi)\times \#(dk)\) [1602.05241]. 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 \(\mathcal P_\infty\) [2009.11173]. 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 [2504.04327]. ## 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 \(c>0\), while each block fragments at constant rate \(\lambda>0\) into its constituent singletons [1602.05241]. The corresponding measures are \[ C=c\kappa,\qquad \kappa=\sum_{i0\) the number of blocks is finite almost surely: \[ \mathbf P_\infty(N(t)<\infty)=1,\qquad t>0 \] [1602.05241]. In simple EFC processes, the question is framed slightly differently: if \(\#\Pi(0)=\infty\), does there exist \(t>0\) such that \(\#\Pi(t)<\infty\)? The authors emphasize that they do not require \(\infty\) 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 \(\infty\) [1605.07039]. The boundary \(\infty\) can be classified in Feller’s terminology as an exit boundary, an entrance boundary, or a regular boundary [2009.11173]. In the regular regime, the process may leave \(\infty\) instantaneously and return immediately; moreover, in that regime \(\infty\) is regular for itself [2009.11173]. The 2025 slow-regime work refines this program by studying whether the block-counting process comes down from infinity, stays infinite, explodes, or makes \(\infty\) an entrance boundary or an exit boundary, with the behavior determined by the asymptotics of the difference \[ \Phi_A(n)-\Phi_u(n) \] between the coagulation and fragmentation total rates [2504.04327]. 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 \(\Phi(n)\), by the pair \((\phi(n),l(n))\), or by the asymptotic difference \(\Phi_A(n)-\Phi_u(n)\) [1602.05241; 1605.07039; 2009.11173; 2504.04327]. ## 4. Phase transitions and large-scale asymptotics The fast fragmentation-coalescence model exhibits a sharp phase transition at \[ \lambda=\frac c2 \] [1602.05241]. Writing \[ \theta:=\frac{2\lambda}{c}, \] the main theorem states that if \(0<\theta<1\), equivalently \(\lambda1\), then the process stays infinite [1605.07039]. In regularly varying examples with \[ \Phi(n)\sim d\,n^{1+\beta},\qquad \beta\in(0,1), \qquad \bar\mu(n)\sim \frac{\lambda}{n^\alpha},\qquad \alpha>0, \] the parameters coincide and are explicit: if \(\beta<1-\alpha\), then \(\theta_\star=\theta^\star=\infty\); if \(\beta>1-\alpha\), then \(\theta_\star=\theta^\star=0\); and in the critical case \(\beta=1-\alpha\), \[ \theta_\star=\theta^\star=\frac{\lambda}{d(1-\alpha)} \] [1605.07039]. The explosion paper adds a complementary classification. Defining \[ \phi(n):=\sum_{k=2}^n (k-1)\binom{n}{k}\lambda_{n,k}, \qquad l(n):=\sum_{k=1}^n k\,\mu(k), \] it recalls that under suitable summability assumptions, if \[ \theta^*:=\limsup_{n\to\infty}\frac{\phi(n)}{n\,l(n)}<1, \] the process comes down from infinity, whereas if \[ \theta_*:=\liminf_{n\to\infty}\frac{\phi(n)}{n\,l(n)}>1, \] it stays infinite [2009.11173]. In the regularly varying regime \[ \phi(n)\sim d\,n^{1+\beta},\qquad \mu(n)\sim b\,n^{-(1+a)}, \] Theorem 3.7 states that if \(a+\beta<1\), then \(\infty\) is an exit boundary; if \(a+\beta>1\), then \(\infty\) is an entrance boundary; and if \(a+\beta=1\), then the ratio \(b/d\) determines whether \(\infty\) is exit, entrance, or regular [2009.11173]. The 2025 slow critical regime paper identifies another family of thresholds. Under \[ u(n)\sim b(\log n)^a n^{-2},\qquad n\to\infty, \] one has \[ \Phi_u(n)\sim \frac{b}{a+1}\,n(\log n)^{a+1} \] [2504.04327]. For \(01\), the critical scale is \[ n(\log n)^a \] [2504.04327]. The theorems then separate non-explosion, coming down from infinity, staying infinite, explosion, and entrance or exit boundary behavior in terms of asymptotics of \((\Phi_A(n)-\Phi_u(n))\) at those scales [2504.04327]. ## 5. Stationarity, bridge representations, and diffusion structure Kingman’s coalescent with erosion has a unique stationary distribution \(\Pi\) [1907.05845]. 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 \(B:[0,1]\to[0,1]\) of the form \[ B(u)=\Big(1-\sum_{i\ge 1}\beta_i\Big)u+\sum_{i\ge 1}\beta_i\,\mathbf{1}_{\{u\ge V_i\}}, \] where \((\beta_i)\) is a mass-partition and \((V_i)\) are i.i.d. uniform \([0,1]\), independent of \((\beta_i)\), and the inverse bridge is \[ B^{-1}(u)=\inf\{t\in[0,1]:B(t)>u\},\qquad u\in[0,1),\qquad B^{-1}(1)=1 \] [1907.05845]. A flow of bridges \((B_{s,t})_{s\le t}\) satisfies \[ B_{s,u}\circ B_{u,t}=B_{s,t}, \qquad B_{t_1,t_2},\dots,B_{t_{p-1},t_p}\ \text{independent}, \qquad B_{t_1,t_2}\stackrel{d}=B_{0,t_2-t_1}, \] and \(B_{0,t}\to \mathrm{Id}\) as \(t\downarrow 0\) [1907.05845]. For the standard flow of bridges, the bridge \(B_{0,t}\) has the law \[ B_{0,t}(u)=\sum_{i=1}^{N_t}\beta_i\,\mathbf{1}_{\{V_i\le u\}}, \] where \(N_t\) is a pure-death process started from \(\infty\) and jumping \(k\to k-1\) at rate \(k(k-1)/2\), and conditional on \(N_t\), \((\beta_1,\dots,\beta_{N_t})\) is Dirichlet\((1,\dots,1)\) [1907.05845]. If \((T_i)_{i\ge 1}\) are i.i.d. exponential(\(d\)) and \((U_i)_{i\ge 1}\) are i.i.d. uniform\([0,1]\), independent of the bridge flow, then \[ i\sim_\Pi j \iff B^{-1}_{-T_i,0}(U_i)=B^{-1}_{-T_j,0}(U_j) \] has the stationary distribution of Kingman’s coalescent with erosion rate \(d\) [1907.05845]. The same paper gives an explicit recursive diffusion description of the stationary asymptotic frequencies. Let \((Y_i)_{i\ge 1}\) be i.i.d. diffusions solving \[ dY_i=(1-Y_i)\,dt+\sqrt{Y_i(1-Y_i)}\,dW_i, \] started from \(0\), with independent Brownian motions \(W_i\) [1907.05845]. Defining recursively \[ Z_1(t)=Y_1(t),\qquad \tau_1(t)=\int_0^t \frac{1}{1-Z_1(s)}\,ds, \] and for \(i\ge 1\), \[ Z_{i+1}(t)=\bigl(1-Z_1(t)-\cdots-Z_i(t)\bigr)\,Y_{i+1}(\tau_i(t)), \] \[ \tau_{i+1}(t)=\int_0^t \frac{1}{1-Z_1(s)-\cdots-Z_{i+1}(s)}\,ds, \] the stationary asymptotic frequencies are given by the decreasing rearrangement of \[ z_i=\int_0^\infty d\,e^{-dt}\,Z_i(t)\,dt,\qquad i\ge 1 \] [1907.05845]. 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 \(1\) and new singleton blocks immigrate at rate \(d\) [1907.05845]. Its block-counting process \(M_t\) is a stationary birth-death chain, \[ k\to k+1 \text{ at rate } d,\qquad k\to k-1 \text{ at rate } \frac{k(k-1)}{2}, \] with reversible stationary distribution \[ \nu_k \propto \frac{(2d)^k}{k!\,(k-1)!} \] [1907.05845]. A coupling between the erosion and immigration models is then used to derive small-block asymptotics: if \(M^n\) is the number of blocks in the stationary restriction to \([n]\) and \(\mu_k^n\) is the empirical fraction of blocks of size \(k\), then \[ \frac{M^n}{\sqrt n}\longrightarrow \sqrt{2d} \] in probability, and for each fixed \(k\ge 1\), \[ \mu_k^n \longrightarrow \mathbb{P}(J=k), \] where \(J\) is the total progeny of a critical binary branching process [1907.05845]. ## 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 \(t\mapsto -\log(1-e^{-t})\), the tree-destruction dynamics becomes a construction process, and the time-reversed partition process is described as a binary coalescent [1501.01400]. 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 [1501.01400]. This is therefore a duality-style connection in a nonexchangeable setting [1501.01400]. Although that process is not exchangeable, it behaves in several important ways like a homogeneous exchangeable fragmentation process [1501.01400]. The partition-valued process \(\Pi\) is Markovian with generator \[ Gf(\pi)=\int_{\mathcal P} r(d\pi') \sum_i \Big(f(\pi\circ_i \pi')-f(\pi)\Big), \] where \(r\) is the splitting-rate measure [1501.01400]. 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 [1501.01400]. Moreover, the splitting measure \[ r=\sum_{k=2}^{\infty} \mathbb P_k \] fails the standard dislocation integrability condition \[ \int_{\mathcal P}\bigl(1-|\pi_1|\bigr)\,s(d\pi)<\infty \] used in the exchangeable theory [1501.01400]. The paper therefore identifies precisely what survives, and what fails, when exchangeability is removed [1501.01400]. 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 \((\sigma_0,\sigma_1,\dots,\sigma_J)\) [2508.18668]. 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 [2508.18668]. The total observable jump rate is \[ \Psi_0\!\left(\sum_{j=1}^J\psi_j(\gamma_j)\right), \] and the number of observed species is \[ \varphi\sim \mathrm{Poisson}\!\left(\Psi_0\!\left(\sum_{j=1}^J\psi_j(\gamma_j)\right)\right) \] [2508.18668]. The classical Pitman duality for \(\mathrm{PD}(\alpha,\theta)\) is recovered as a special case. In the notation quoted there, \[ G_{\beta,\theta}(y)\overset{d}{=}F_{\alpha,\theta}\!\left(G_{\beta/\alpha,\theta/\alpha}(y)\right), \qquad G_{\alpha,\theta}(y)\overset{d}{=}\sum_{l\ge 1}V_l F_{\alpha,-\beta}^{(l)}(y), \] and the corresponding partition identity is \[ p_{\beta/\alpha,\theta/\alpha}(x_1,\dots,x_r)\,p_{\alpha,\theta}(c_1,\dots,c_K) = \prod_{l=1}^r p_{\alpha,-\beta}(c_{l,1},\dots,c_{l,x_l})\,p_{\beta,\theta}(n_1,\dots,n_r) \] [2508.18668]. What is new is that the same algebra survives for arbitrary subordinators, not only stable ones [2508.18668]. 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 \(k\)-cycles on \(\mathcal S_n\). 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 [1001.1894]. 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 \(k\)-cycle step by rigid algebraic structure [1001.1894]. This serves as a finite-size analogue rather than an EFC process on \(\mathbb N\). ## 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 [1501.01400]. In that model every block of \(\Pi(t)\) still has an asymptotic frequency, but it is degenerate, \[ |B(t)|=\lim_{n\to\infty} n^{-1}\#\{i\le n:i\in B(t)\}=0 \quad \text{a.s.}, \] so ordinary asymptotic frequencies do not produce a nontrivial mass partition [1501.01400]. The paper replaces them with a time-dependent normalization leading to block weights whose logarithms form an Ornstein-Uhlenbeck branching process [1501.01400]. A second misconception is that “coming down from infinity” is a uniform notion across all models. In the fast fragmentation-coalescence process, when \(\lambda