Poisson Dirichlet$(α,θ)$-Bridge Equations and Coagulation-Fragmentation Duality

可逆凝聚-碎裂过程和随机组合结构：群数的渐近性

Reversible coagulation-fragmentation processes and random combinatorial structures:asymptotics for the number of groups

本文建立了由多项式增长率的参数函数给出的凝聚-破碎过程平衡点群数的中心极限定理。所得结果与服从对数条件的随机组合结构的结果进行了比较。

某些凝聚-碎裂过程的渐近性与不变Poisson-Dirichlet测度

Asymptotics of certain coagulation-fragmentation processes and invariant Poisson-Dirichlet measures

我们考虑马尔可夫链的间隔$[0,1]$的（可数）分区的空间，首先通过大小偏置采样两次（允许重复），然后合并的部分概率$β_m$（如果采样的部分是不同的）或分裂的部分概率$β_s$根据法律$σ$（如果同一部分被采样两次）。我们刻画了这种链的不变概率测度。特别地，如果$σ$是一致测度，则泊松-狄利克雷定律是不变的概率测度，并且它在适当定义的“解析”不变测度类中是唯一的。我们还推导出这些链的瞬时性和递归性准则。

Inverse clustering of Gibbs Partitions via independent fragmentation and dual dependent coagulation operators

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…

Well-posedness for a coagulation multiple-fragmentation equation

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…

混凝破碎过程的时间动力学

On time dynamics of coagulation-fragmentation processes

我们建立了一个凝固-碎裂过程的表征，使得描述在时间$t\ge 0$的组总数的诱导出生和死亡过程是时间均匀的。在此基础上，我们给出了平均场Gibbs凝聚-碎裂模型的一个特征，它推广了Hendriks等人的结果. Berestycki和皮特曼的回答，限制平均场模型。

A two-time-scale phenomenon in a fragmentation-coagulation process

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 …

On the explosion of the number of fragments in the simple exchangeable fragmentation-coalescence processes

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…

Isotropic Wave Turbulence with simplified kernels: existence, uniqueness and mean-field limit for a class of instantaneous coagulation-fragmentation processes

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).

Stochastic coagulation-fragmentation processes with a finite number of particles and applications

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…

可逆凝聚-破碎过程分拆函数的渐近公式

Asymptotic formula for a partition function of reversible coagulation -fragmentation processes

我们构建了一个概率模型，似乎无关的考虑凝聚和破碎的随机过程。通过证明该模型的局部极限定理，我们建立了过程的一类参数函数的平衡测度的分拆函数的渐近公式。这个公式证明了在[dgg]中对上述过程的猜想。使用的方法可以追溯到A. Khintchine。

凝聚-破碎对偶性、Poisson-Dirichlet分布和随机递归树

Coagulation--fragmentation duality, Poisson--Dirichlet distributions and random recursive trees

本文给出了分裂算子与凝聚算子之间对偶性的一个新例子。考虑质量分区的空间（即，递减序列的非负实数的总和是1）和两个参数家庭的Poisson-狄利克雷分布$\operatorname {PD}(α,θ)$，采取的价值在这个空间。我们引入家庭的随机碎片和凝聚操作$\mathrm {Frag}_α$和$\mathrm {Coag}_{α,θ}$，分别具有以下性质：如果输入$\mathrm {Frag}_α$有$\operatorname {PD}(α,θ)$分布，那么输出有$\operatorname {PD}(α,θ+1)$分布，而相反的是真实的$\mathrm {Coag}_{α,θ}$。这一结果可以证明使用从属表示，它提供了一个同伴的一组关系，Pitman之间的$\operatorname {PD}(α,θ)$和$\operatorname {PD}(αβ,θ)$。重复应用$\mathrm {Frag}_α$算子产生了一系列碎片链。我们表明，这些马尔可夫链可以自然地编码的某些随机递归树，并使用这种表示给出了一个替代的和更具体的证明凝聚-分裂的对偶性。

Asymptotics of counts of small components in random structures and models of coagulation-fragmentation

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…

凝聚-碎裂过程和随机组合结构中乘法测度的极限形状

Limit shapes of multiplicative measures associated with coagulation-fragmentation processes and random combinatorial structures

我们发现极限形状的一个家庭的乘法措施的集合上的分区，引起的指数母函数的扩展参数，$a_k\sim Ck^{p-1}, k\to\infty, p>0$，其中$C$是一个正的常数。考虑的措施与可逆的凝聚-碎裂过程和某些组合结构，称为组件。从尺度随机分拆的极限形状出发，证明了其涨落的泛函中心极限定理。我们证明，当组件的大小超过门限值，组件的数量的独立性转换成它们的条件独立性。除其他事项外，本文还讨论了，在一般情况下，极限形状，阈值和凝胶化之间的相互作用。

A one-dimensional coagulation-fragmentation process with a dynamical phase transition

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…

无限可交换分区的凝结-碎裂双重性：来自耦合混合泊松物种采样模型的双重性

Coagulation-Fragmentation Duality of Infinitely Exchangeable Partitions from Coupled Mixed Poisson Species Sampling Models

本文提出了一种基于耦合混合泊松物种采样模型的新框架，通过四组件系统实现了无限可交换分区的凝结-碎裂双重性，解决了传统分区分析的难题。

可逆凝并-破碎模型中的凝聚与涨落研究

On the Condensation and fluctuations in reversible coagulation-fragmentation models

该研究针对超临界密度下可逆凝并-破碎平均场模型的不变测度，证明次指数尾条件下存在单个巨型凝聚粒子，其余粒子渐近独立同分布，并给出凝聚质量涨落条件与粒子质量经验测度的大偏差原理。

凝聚-破碎过程、随机组合结构和可加数系统中的聚类：渐近公式和极限定律

Clustering in coagulation -fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws

我们开发了一个统一的方法，在三个不同的应用领域的聚类问题，如标题中所示的文件。该方法是基于欣钦的概率方法，该方法源于统计物理学中的达尔文-福勒方法。我们的主要结果是推导出的渐近公式的概率集群（=组）的某些大小的粒子数趋于无穷大。在此基础上，我们证明了最大团簇分布的零一律，并建立了在该定律中从0到1相变的门限函数。

通过碎裂与凝并实现具有分层粒度的数据的网络与交互模型

Network and interaction models for data with hierarchical granularity via fragmentation and coagulation

提出一类具有分层粒度的贝叶斯非参数网络与交互模型，利用整数分拆的碎裂-凝并对偶性连接 Hollywood 与广义伽马 graphex 模型，并给出贝叶斯推断。

双参数Poisson-Dirichlet测度与可逆可交换破碎-合并过程

Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes

我们证明了对于$0<α<1$和$θ>-α$，参数为$(α, θ)$的Poisson-Dirichlet分布是一个相当自然的破碎-合并过程的唯一可逆分布.这完成了文献中某些分裂和合并变换以及参数$α=0$的早期结果。

Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model

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…

Quantum Heisenberg models and their probabilistic representations

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.

Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions

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…

Coag-Frag duality for a class of stable Poisson-Kingman mixtures

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…

A Bayesian View of the Poisson-Dirichlet Process

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 and their representation by infinite systems of ordinary differential equations

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…

通过准共轭性在双参数泊松-狄利克雷隐马尔可夫模型中实现精确推断

Exact inference via quasi-conjugacy in two-parameter Poisson-Dirichlet hidden Markov models

本文提出了一种非参数模型，用于从离散时间数据中推断时间演变的未观察概率分布，数据由无标签的划分组成。潜在过程是双参数泊松-狄利克雷扩散过程，观测通过可交换抽样产生。应用包括社会和遗传数据，其中仅观察到聚类汇总信息。为了解决不可行的似然，我们开发了一个可计算的推断框架，避免了标签枚举和直接模拟潜在状态。我们利用扩散过程与在划分上的纯死亡过程之间的对偶性，以及编码新数据影响的凝集算子，从而得到前向和后向推断的闭式递归更新。我们计算了任意时间点潜在状态的精确后验分布和未来或插值划分的预测分布。这使我们能够进行在线和离线推断和预测，并完全量化不确定性，绕过MCMC和序列蒙特卡罗方法。与粒子滤波相比，我们的方法在准确性、方差和计算效率方面都有显著优势。我们通过合成实验和社会网络应用展示了该方法，恢复了时间变化异质性的可解释模式。