Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize $(α,θ)$} and $(β,θ/α)$, wherein $PD(α, θ)$ is coagulated by $PD(β,θ/α)$ for $0<α<1$, $0 \leqβ<1$ and $-β<θ/α$. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of $PD(α,θ)$ coagulated with some law $Q$. The analysis exploits distributional relationships between compositions of species sampling random probability measures and coagulation operators and recent work on Cauchy-Stieltjes transforms of random probability measures by Vershik, Yor and Tsilevich (2004) and James (2002). We use this to obtain explicit descriptions in the case where {\footnotesize $Q$} corresponds to a large class of power tempered Poisson Kingman models analyzed in James~(2002). That is, explicit results are obtained for models outside of the $PD(β,θ/α)$ family.
The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes.
First we establish explosion criteria for jump processes with an arbitrary locally compact separable metric state space. Then these results are applied to two stochastic coagulation-fragmentation models--the direct simulation model and the mass flow model. In the pure coagulation case, there is almost sure explosion in the mass flow model for arbitrary homogeneous coagulation kernels with exponent bigger than 1. In the case of pure multiple fragmentation with a continuous size space, explosion occurs in both models provided the total fragmentation rate grows sufficiently fast at zero. However,…
We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes, simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters $θ_{\star}\leq θ^{\star}\in…
A hierarchical system of equations is introduced to describe dynamics of `sizes' of infinite clusters which coagulate and fragmentate with homogeneous rates of certain form. We prove that this system of equations is solved weakly by correlation measures for stochastic dynamics of interval partitions evolving according to some split-merge transformations. Regarding those processes, a sufficient condition for a distribution to be reversible is given. Also, an asymptotic result for properly rescaled processes is shown to obtain a solution to a nonlinear equation called the coagulation-fragmentati…
We establish the central limit theorem for the number of groups at the equilibrium of a coagulation-fragmentation process given by a parameter function with polynomial rate of growth. The result obtained is compared with the one for random combinatorial structures obeying the logarithmic condition.
We consider Markov chains on the space of (countable) partitions of the interval $[0,1]$, obtained first by size biased sampling twice (allowing repetitions) and then merging the parts with probability $β_m$ (if the sampled parts are distinct) or splitting the part with probability $β_s$ according to a law $σ$ (if the same part was sampled twice). We characterize invariant probability measures for such chains. In particular, if $σ$ is the uniform measure then the Poisson-Dirichlet law is an invariant probability measure, and it is unique within a suitably defined class of ``analytic'' invarian…
Gibbs partitions of the integers generated by stable subordinators of index $α\in(0,1)$ form remarkable classes of random partitions where in principle much is known about their properties, including practically effortless obtainment of otherwise complex asymptotic results potentially relevant to applications in general combinatorial stochastic processes, random tree/graph growth models and Bayesian statistics. This class includes the well-known models based on the two-parameter Poisson-Dirichlet distribution which forms the bulk of explicit applications. This work continues efforts to provide…
We consider a coagulation multiple-fragmentation equation, which describes the concentration $c\_t(x)$ of particles of mass $x \in (0,\infty)$ at the instant $t \geq 0$ in a model where fragmentation and coalescence phenomena occur. We study the existence and uniqueness of measured-valued solutions to this equation for homogeneous-like kernels of homogeneity parameter $λ\in (0,1]$ and bounded fragmentation kernels, although a possibly infinite total fragmentation rate, in particular an infinite number of fragments, is considered. This work relies on the use of a Wasserstein-type distance, whic…
We establish a characterization of coagulation-fragmentation processes, such that the induced birth and death processes depicting the total number of groups at time $t\ge 0$ are time homogeneous. Based on this, we provide a characterization of mean-field Gibbs coagulation-fragmentation models, which extends the one derived by Hendriks et al. As a by- product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to mean-field models.
Consider two urns, $A$ and $B$, where initially $A$ contains a large number $n$ of balls and $B$ is empty. At each step, with equal probability, either we pick a ball at random in $A$ and place it in $B$, or vice-versa (provided of course that $A$, or $B$, is not empty). The number of balls in $B$ after $n$ steps is of order $\sqrt n$, and this number remains essentially the same after $\sqrt n$ further steps. Observe that each ball in the urn $B$ after $n$ steps has a probability bounded away from $0$ and $1$ to be placed back in the urn $A$ after $\sqrt n$ further steps. So, even though the …
We consider the exchangeable fragmentation-coagulation (EFC) processes, where the coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and the fragmentations dislocate at finite rate an individual block into sub-blocks of infinite size. Sufficient conditions are found for the block-counting process to explode (i.e. to reach $\infty$) or not and for infinity to be an exit boundary or an entrance boundary. In a case of regularly varying fragmentation and coagulation mechanisms, we find regimes where the boundary $\infty$ can be either an exit, an entrance or a regular boundary…
The isotropic 4-wave kinetic equation is considered in its weak formulation using model (simplified) homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting where the kernels have a rate of growth at most linear. We also consider finite stochastic particle systems undergoing instantaneous coagulation- fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).
Coagulation-fragmentation processes describe the stochastic association and dissociation of particles in clusters. Cluster dynamics with cluster-cluster interactions for a finite number of particles has recently attracted attention especially in stochastic analysis and statistical physics of cellular biology, as novel experimental data is now available, but their interpretation remains challenging.} We derive here probability distribution functions for clusters that can either aggregate upon binding to form clusters of arbitrary sizes or a single cluster can dissociate into two sub-clusters. U…
We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [dgg] for the above class of processes. The method used goes back to A.Khintchine.
In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson--Dirichlet distributions $\operatorname {PD}(α,θ)$ that take values in this space. We introduce families of random fragmentation and coagulation operators $\mathrm {Frag}_α$ and $\mathrm {Coag}_{α,θ}$, respectively, with the following property: if the input to $\mathrm {Frag}_α$ has $\operatorname {PD}(α,θ)$ distribution, then the output has $\opera…
We establish necessary and sufficient conditions for convergence (in the sense of finite dimensional distributions) of multiplicative measures on the set of partitions. We show that this convergence is equivalent to asymptotic independence of finite sizes of components. The multiplicative measures depict component spectra of random structures, the equilibrium of classic models of statistical mechanics and stochastic processes of coagulation-fragmentation. We then apply Schur's tauberian lemma and some results from additive number theory and enumerative combinatorics, in order to verify the con…
We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, $a_k\sim Ck^{p-1}, k\to\infty, p>0$,where $C$ is a positive constant. The measures considered are associated with reversible coagulation-fragmentation processes and certain combinatorial structures, known as assemblies. We prove the functional central limit theorem for the fluctuations of a scaled random partition from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbe…
We introduce a reversible Markovian coagulation-fragmentation process on the set of partitions of $\{1,\ldots,L\}$ into disjoint intervals. Each interval can either split or merge with one of its two neighbors. The invariant measure can be seen as the Gibbs measure for a homogeneous pinning model \cite{cf:GBbook}. Depending on a parameter $λ$, the typical configuration can be either dominated by a single big interval (delocalized phase), or be composed of many intervals of order $1$ (localized phase), or the interval length can have a power law distribution (critical regime). In the three case…
We develop a unified approach to the problem of clustering in the three different fields of applications, as indicated in the title the paper. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fawler method in statistical physics. Our main result is the derivation of asymptotic formulae for probabilities of clusters (= groups) of certain sizes as the number of particles goes to infinity. Based on these formulae we prove the zero-one law for the distribution of the largest cluster and establish the threshold function in the phase transition from 0 to 1 in th…
We show that for $0<α<1$ and $θ>-α$, the Poisson-Dirichlet distribution with parameter $(α, θ)$ is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split and merge transformations and the parameter $α=0$.
We investigate the connection between two classical models of phase transition phenomena, the (discrete size) stochastic Becker-Döring, a continous time Markov chain model, and the (continuous size) deterministic Lifshitz-Slyozov model, a nonlinear transport partial differential equation. For general coefficients and initial data, we introduce a scaling parameter and prove that the empirical measure associated to the stochastic Becker-Döring system converges in law to the weak solution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previous studies, we use a weak to…
These notes give a mathematical introduction to two seemingly unrelated topics: (i) quantum spin systems and their cycle and loop representations, due to Tóth and Aizenman-Nachtergaele; (ii) coagulation-fragmentation stochastic processes. These topics are nonetheless related, as we argue that the lengths of cycles and loops satisfy an effective coagulation-fragmentation process. This suggests that their joint distribution is Poisson-Dirichlet. These ideas are far from being proved, but they are backed by several rigorous results, notably of Dyson-Lieb-Simon and Schramm.
Some models of clustering processes are formulated and analytically solved employing generating functions methods. Those models include events which result from combined action of the coagulation and fragmentation processes. Fragmentation processes of two kinds, so-called similar and arbitrary, ones, are brought forward, and the explicit forms of their solutions are produced. This implies some possibility of existence of different aggregation mechanisms for clusters creation differing in their inner structure. All the models are based on "the three-level bunch" scheme of interaction between th…
We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford's trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to es…
Exchangeable sequences of random probability measures (partitions of mass) and their corresponding exchangeable bridges play an important role in a variety of areas in probability, statistics and related areas, including Bayesian statistics, physics, finance and machine learning. An area of theoretical as well as practical interest, is the study of coagulation and fragmentation operators on partitions of mass. In this regard, an interesting but formidable question is the identification of operators and distributional families on mass partitions that exhibit interesting duality relations. In th…
We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous' beta-splitting model, which has an extended parameter range $β>-2$ with respect to the ${\rm beta}(β+1,β+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson--Dirichlet models for exchangeable random partitions of $\mathbb {N}$, with an extended parameter range $0\leα\le1$, $θ\ge-2α$ and $α<0$, $θ=-mα$, $m\in \mathbb {N}$.
NICTA（澳大利亚国家信息与通信技术研究院）；Australian National University（澳大利亚国立大学）
The two parameter Poisson-Dirichlet Process (PDP), a generalisation of the Dirichlet Process, is increasingly being used for probabilistic modelling in discrete areas such as language technology, bioinformatics, and image analysis. There is a rich literature about the PDP and its derivative distributions such as the Chinese Restaurant Process (CRP). This article reviews some of the basic theory and then the major results needed for Bayesian modelling of discrete problems including details of priors, posteriors and computation. The PDP allows one to build distributions over countable partitions…
Fractional Poisson processes, a rapidly growing area of non-Markovian stochastic processes, are useful in statistics to describe data from counting processes when waiting times are not exponentially distributed. We show that the fractional Kolmogorov-Feller equations for the probabilities at time t can be representated by an infinite linear system of ordinary differential equations of first order in a transformed time variable. These new equations resemble a linear version of the discrete coagulation-fragmentation equations, well-known from the non-equilibrium theory of gelation, cluster-dynam…
