Volker Betz, Daniel Ueltschi
We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and their cycle lengths satisfy a Poisson-Dirichlet law.
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The behavior of the Poisson-Dirichlet distribution with small mutation rate is studied through large deviations. The structure of the rate function indicates that the number of alleles is finite at the instant when mutation appears. The large deviation results are then used to study the asymptotic behavior of the homozygosity, and the Poisson-Dirichlet distribution with symmetric selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as the mutation rate approaches zero.
We study measures on random partitions, arising from condensing stochastic particle systems with stationary product distributions. We provide fairly general conditions on the stationary weights, which lead to Poisson-Dirichlet statistics of the condensed phase in the thermodynamic limit. The Poisson-Dirichlet distribution is known to be the unique reversible measure of split-merge dynamics for random partitions, which we use to characterize the limit law. We also establish concentration results for the macroscopic phase, using size-biased sampling techniques and the equivalence of ensembles to…
Large deviation principles are established for the two-parameter Poisson-Dirichlet distribution and two-parameter Dirichlet process when parameter $θ$ approaches infinity. The motivation for these results is to understand the differences in terms of large deviations between the two-parameter models and their one-parameter counterparts. New insight is obtained about the role of the second parameter $α$ through a comparison with the corresponding results for the one-parameter Poisson-Dirichlet distribution and Dirichlet process.
The two parameter Poisson-Dirichlet distribution $PD(α,θ)$ is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingman's Poisson-Dirichlet distribution. The two parameter Dirichlet process $Π_{α,θ,ν_0}$ is the law of a pure atomic random measure with masses following the two parameter Poisson-Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures $PD(α,θ)$ and $Π_{α,θ,ν_0}$. The methods used come from the theory of Diric…
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 $α=0$.
This paper has been withdrawn by the author.
Paper discusses fragmentation coagulation duality formula related to those already appearing in the literature.
Dirichlet law for factorization of integers, polynomials and permutations
本文通过多维围道积分证明整数、多项式和排列分解为 $k$ 部分时遵循狄利克雷分布，推广了 DDT 反正弦定律，并指出任意参数的狄利克雷分布可类似建模。
The two-parameter Poisson-Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and Gamma subordinators with the two-parameters, $α$ and $θ$, corresponding to the stable component and Gamma component respectively. The moderate deviation principles are established for the two-parameter Poisson-Dirichlet distribution and the corresponding homozygosity when $θ$ approaches infinity, and the large deviation principle is established for the two-parameter Poisson-Dirichlet distribution when both $α$ and $θ$ app…
This paper explores large sample properties of the two-parameter $(α,θ)$ Poisson--Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension of the Dirichlet process, we explore the consistency and weak convergence of the the two-parameter Poisson--Dirichlet posterior process. We also establish the weak convergence of properly centered two-parameter Poisson--Dirichlet processes for large $θ+nα.$ This latter result complements large $θ$ results for the Dirichlet process and Poisson--Dirichlet sequences, and …
In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology. This implies Poisson-Dirichlet statistics by general spin glass arguments. In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary …
Poisson-Dirichlet graphons and permutons
本文提出了一种基于两参数泊松-狄里赫利过程的通用图和排列极限对象，用于分析超图和超排列的渐近形状特性。
The two-parameter Poisson--Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (that is, the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Using this, we apply the theory of point processes to reveal the mathematical structure of the two-parameter Poisson--Dirichlet distribution. Also, developing the Laplace transform ap…
We present an alternative approach to the Bayesian nonparametric analysis of conditional species richness under two-parameter Poisson Dirichlet priors. We rely on a known characterization by deletion of classes property and on results for Beta-Binomial distributions. Besides leading to simplified and much more direct proofs, our proposal provides a new scale mixture representation of the conditional asymptotic law.
Attenuated Poisson Dirichlet approximations for divisibility configurations
研究调和随机样本的不同素因子归一化对数点过程，证明其向Dickman泊松云原子序列（等价于均匀衰减泊松-狄利克雷律）的定量收敛。
In Arguin & Tai (2018), the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.
The Poisson--Dirichlet distribution arises in many different areas. The parameter $θ$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $θ$ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson--Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter $…
We present new, exceptionally efficient proofs of Poisson--Dirichlet limit theorems for the scaled sizes of irreducible components of random elements in the classic combinatorial contexts of arbitrary assemblies, multisets, and selections, when the components generating functions satisfy certain standard hypotheses. The proofs exploit a new criterion for Poisson--Dirichlet limits, originally designed for rapid proofs of Billingsley's theorem on the scaled sizes of log prime factors of random integers (and some new generalizations). Unexpectedly, the technique applies in the present combinatori…
Stochastic geometry models of wireless networks based on Poisson point processes are increasingly being developed with a focus on studying various signal-to-interference-plus-noise ratio (SINR) values. We show that the SINR values experienced by a typical user with respect to different base stations of a Poissonian cellular network are related to a specific instance of the so-called two-parameter Poisson-Dirichlet process. This process has many interesting properties as well as applications in various fields. We give examples of several results proved for this process that are of immediate or …
Asymptotic behavior of the Dirichlet energy on Poisson point clouds
该研究针对平面泊松云上的二次对相互作用，通过尺度变换与离散到连续极限分析，证明其几乎必然可由确定性常数乘Dirichlet能量近似，并提出适配的粗粒度函数收敛概念以规避异常区域影响。
Poisson approximations of the number of fixed points in random multiset permutations
该研究针对随机多重集合排列的不动点数量问题，采用斯坦因方法开展分析，拓展了普通排列不动点数量的泊松近似相关结论。
We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v_1,v_2,...), where each v_j is the sum of j independent Exponential(1) random variables and the different v_i need not be independent. In particular, our analysis applies to the Poisson-Dirichlet branching random walk and to the Poisson-weighted infinite tree. As a corollary, we also determine the expected height of a random recursive tree to within O(1).
We show that a slight modification of a theorem of Ruzmaikina and Aizenman on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $PD(a,0)$. Precisely, let $s$ be a proper random mass-partition i.e. a random sequence $(s_i, i\in\N)$ such that $s_1\geq s_2\geq ...$ and $\sum_i s_i=1$ a.s. Consider ${W_i}_{i\in\N}$, an iid sequence of positive random variables with a density and such that $E[W^λ]$ is finite for all $λ\in\R$. It is shown that if the law of $s$ is invariant under a random multiplicative shift $s_i W_i$ of the atoms followed by…
Poisson-Dirichlet scaling limits of Kemp's supertrees
本文确定了Kemp的$d$维二叉树及其他超树模型的缩放与局部极限，该极限通过基于两参数Poisson-Dirichlet过程缩放并粘合无穷多个独立稳定树于根处来构造，并探讨了其普适性与相图。
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…
The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincare inequality but not the log-Sobolev one.
Poisson-Dirichlet approximation for the stationary distribution of the inclusion process
本文通过改进Stein方法，利用传统导数简化了对包含过程平稳分布的泊松-狄利克雷分布近似误差界分析。
We show that the chaos representation of some Compound Poisson Type processes displays an underlying intrinsic combinatorial structure, partly independent of the chosen process. From the computational viewpoint, we solve the arising combinatorial complexity by means of the moments/cumulants duality for the laws of the corresponding processes, themselves measures on distributional spaces, and provide a combinatorial interpretation of the associated 'extended' Fock spaces. From the theoretical viewpoint, in the case of the Gamma measure, we trace back such complexity to its 'simplicial part', i.…
Poisson approximation for large permutation groups
本文研究大置换群$G_{k,n}$随机元素的不动点、循环、逆序数等统计量的极限分布，证明其收敛于具有依赖结构的复合泊松分布。
The large deviation principle is established for the Poisson--Dirichlet distribution when the parameter $θ$ approaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson--Dirichlet distribution with selection. A phase transition occurs depending on the growth rate of the selection intensity. If the selection intensity grows sublinearly in $θ$, then the large deviation rate function is the same as the neutral model; if the selection intensity grows at a linear or greater rate in $θ$, then the large deviation rate function includes an additiona…
We consider the sequential sampling of species, where observed samples are classified into the species they belong to. We are particularly interested in studying some quantities describing the sampling process when there is a new species discovery. We assume that the observations and species are organized as a two-parameter Poisson-Dirichlet Process, which is commonly used as a Bayesian prior in the context of entropy estimation, and we use the computation of the mean posterior entropy given a sample developed in [4]. Our main result shows the existence of a monotone functional, constructed fr…
Several results of large deviations are obtained for distributions that are associated with the Poisson--Dirichlet distribution and the Ewens sampling formula when the parameter $θ$ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of $θ$ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of $θ$ going to infinity in a Poisson--Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. …
We consider a Markov chain 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 (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). We prove a conjecture of Vershik stating that the Poisson-Dirichlet law with parameter theta=1 is the unique invariant distribution for this Markov chain. Our proof uses a combination of probabilistic, combinatoric, and representation-theoretic arguments.
We provide a general theorem bounding the error in the approximation of a random measure of interest--for example, the empirical population measure of types in a Wright-Fisher model--and a Dirichlet process, which is a measure having Poisson-Dirichlet distributed atoms with i.i.d. labels from a diffuse distribution. The implicit metric of the approximation theorem captures the sizes and locations of the masses, and so also yields bounds on the approximation between the masses of the measure of interest and the Poisson-Dirichlet distribution. We apply the result to bound the error in the approx…
Thermal Concentration and Poisson-Dirichlet Edge Statistics for Random-Lattice Gibbs Ensembles
研究高维Haar随机幺模格上的吉布斯测度，证明在边缘窗口内吉布斯质量的泊松点过程极限，并建立原始方向吉布斯系综的固定因子能见度曲线。
The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson--Dirichlet process $\operatorname {PD}(α,θ)$. We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson--Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in…
We construct a pair of related diffusions on a space of interval partitions of the unit interval $[0,1]$ that are stationary with the Poisson-Dirichlet laws with parameters (1/2,0) and (1/2,1/2) respectively. These are two particular cases of a general construction of such processes obtained by decorating the jumps of a spectrally positive Lévy process with independent squared Bessel excursions. The processes of ranked interval lengths of our partitions are members of a two parameter family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). The latter diffusions are continu…
We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters $(α,0)$ and $(α,α)$. The construction has two steps. The first is a general construction of interval partition processes obtained previously, by decorating the jumps of a Lévy process with independent excursions. Here, we focus on the second step, which requires explicit transition kernels and what we call pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper, we…
Many low temperature disordered systems are expected to exhibit Poisson-Dirichlet (PD) statistics. In this paper, we focus on the case when the underlying disorder is a logarithmically correlated Gaussian process $ϕ_N$ on the box $[-N,N]^d\subset\mathbb{Z}^d$. Canonical examples include branching random walk, $*$-scale invariant fields, with the central example being the two dimensional Gaussian free field (GFF), a universal scaling limit of a wide range of statistical mechanics models. The corresponding Gibbs measure obtained by exponentiating $β$ (inverse temperature) times $ϕ_N$ is a discre…
Dual process in the two-parameter Poisson-Dirichlet diffusion
本文为双参数 Poisson-Dirichlet 扩散识别了对偶过程并求得转移概率，利用带突变的 Kingman 凝结和 Polya 罐方案扩展，给出了扩散转移密度的全新概率证明。
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 wit…
Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a ``typical'' compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces associated with random cubic graphs; by a theorem of Belyi, these are ``dense'' in the space of compact Riemann surfaces. The question as to how these surfaces are distributed in the Teichmüller spaces depends on the study of oriented cycles in random cubic graphs with random orientation; Brooks and Makover conjectur…
We give a pathwise construction of a two-parameter family of purely-atomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet$(α,θ)$ distributions, for $α\in (0,1)$ and $θ\ge 0$. This resolves a conjecture of Feng and Sun (2010). We build on our previous work on $(α,0)$- and $(α,α)$-interval partition evolutions. Indeed, we first extract a self-similar superprocess from the levels of stable processes whose jumps are decorated with squared Bessel excursions and distinct allelic types. We complete our construction by time-change and normalisation …
The recently introduced two-parameter infinitely-many neutral alleles model extends the celebrated one-parameter version, related to Kingman's distribution, to diffusive two-parameter Poisson-Dirichlet frequencies. Here we investigate the dynamics driving the species heterogeneity underlying the two-parameter model. First we show that a suitable normalization of the number of species is driven by a critical continuous-state branching process with immigration. Secondly, we provide a finite-dimensional construction of the two-parameter model, obtained by means of a sequence of Feller diffusions …
Time series data may exhibit clustering over time and, in a multiple time series context, the clustering behavior may differ across the series. This paper is motivated by the Bayesian non--parametric modeling of the dependence between the clustering structures and the distributions of different time series. We follow a Dirichlet process mixture approach and introduce a new class of multivariate dependent Dirichlet processes (DDP). The proposed DDP are represented in terms of vector of stick-breaking processes with dependent weights. The weights are beta random vectors that determine different …
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…
The two-parameter Poisson--Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingman's one-parameter Poisson--Dirichlet distribution and to certain Fleming--Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by formulating a $K$-allele Wright--Fisher model for a population of size $N$, involving a uniform mutation pattern and a specific sta…
Characteristic polynomial of generalized Ewens random permutations
研究广义Ewens分布下随机排列矩阵特征多项式的收敛性，揭示其与泊松级数及高斯解析混沌的关系。
We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson-Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin system, the probability that distant sites form an even partition is given by the Poisson-Dirichlet counterpart.
Directed Spatial Permutations on Asymmetric Tori
本文研究二维非对称环面上的随机空间置换，利用空间偏置、动态重抽样与 Schramm 方法证明大循环渐近服从参数为一的 Poisson–Dirichlet 分布。
Dynamical models for the two-parameter Poisson-Dirichlet distribution and the Pitman-Yor process
本文引入并研究了一族与Pitman-Yor过程和双参数Poisson-Dirichlet分布相关的扩散过程，通过Dirichlet形式构造，并揭示了多样性指数在种群演化中的关键作用。
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…
When $S=(S_t)_{t\ge 0}$ is an $α$-stable subordinator, the sequence of ordered jumps of $S$, up till time $1$, omitting the $r$ largest of them, and taken as proportions of their sum $^{(r)}S_t$, defines a 2-parameter distribution on the infinite dimensional simplex, $\nabla_{\infty}$, which we call the $\mathrm{PD}_α^{(r)}$ distribution. When $r=0$ it reduces to the $\mathrm{PD}_α$ distribution introduced by Kingman in 1975. We observe a serendipitous connection between $\mathrm{PD}_α^{(r)}$ and the negative binomial point process of Gregoire (1984), which we exploit to analyse in detail a si…
The recently introduced two-parameter Poisson-Dirichlet diffusion extends the infinitely-many-neutral-alleles model, related to Kingman's distribution and to Fleming-Viot processes. The role of the additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by providing a finite-population construction, with finitely-many species, of the two-parameter infinite-dimensional diffusion. The costruction is obtained in terms of Wright-Fisher chai…
Modulated Poisson-Dirichlet diffusions arising from inclusion processes with a slow phase
本文研究带慢速相的平均场包含过程，证明其依分布收敛到一个推广Poisson-Dirichlet扩散的双分量无穷维随机扩散，并建立两相间的质量交换与极限动力学的Feller适定性。
Let $p_1 \ge p_2 \ge \dots$ be the prime factors of a random integer chosen uniformly from $1$ to $n$, and let $$ \frac{\log p_1}{\log n}, \frac{\log p_2}{\log n}, \dots $$ be the sequence of scaled log factors. Billingsley's Theorem (1972), in its modern formulation, asserts that the limiting process, as $n \to \infty$, is the Poisson-Dirichlet process with parameter $θ=1$. In this paper we give a new proof, inspired by the 1993 proof by Donnelly and Grimmett, and extend the result to factorizations of elements of normed arithmetic semigroups satisfying certain growth conditions, for which th…
We propose a new method to apply the Lipschitz functional calculus of local Dirichlet forms to Poisson random measures.
Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavaré (1992) extended Feller's coupling, to associate cycles of random permutations governed by the Ewens $(θ)$ distribution with spacings derived from independent Bernoulli trials with success probability $θ/(n-1+θ)$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacing…
We prove some general results about the asymptotics of the distribution of the number of cycles of given length of a random permutation whose distribution is invariant under conjugation. These results were first established to be applied in a forthcoming paper (Cycles of free words in several random permutations with restricted cycles lengths), where we prove results about cycles of random permutations which can be written as free words in several independent random permutations. However, we also apply them here to prove asymptotic results about random permutations with restricted cycle length…
Poisson contingency tables, Gibbs permutations, and entropic optimal transport
该研究通过将泊松列联表与Mallows型吉布斯排列耦合，在有限维及渐近尺度上分析其性质，证明了Pal和Mukherjee的相关猜想，涉及熵最优传输等理论。
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