We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non- random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit self-organised criticality in the cluster size distribution, with universal exponent 3 2.
The critical behaviour of the stationary cluster size distribution, found in Theorem , has also been seen in a different form of fragmentation-coalescence process studied by Bressaud and Fournier @cite_2 . This was a mean-field approximation to a one-dimensional forest-fire model on @math . Edges (each of weight one) are glued together if the site between them is occupied, which happens at rate 1, and clusters of @math edges have all sites removed at rate @math . A mean-field approximation to this model, ignoring correlations, follows Then, they found that the stationary cluster size distribution converges weakly to @math as @math ( fragmentation' rate converges to zero) where which matches the distribution in Theorem in the case where @math .
@cite_2
2097808961 2065794571 2022131627 1986567569
Clustering at multiple critical scales may be common for plants since many different factors and processes may cause clustering. This is especially true for tropical rain forests for which theories explaining species coexistence and community structure rest heavily on spatial patterns. We used point pattern analysis to analyze the spatial structure of Shorea congestiflora, a dominant species in a 25-ha forest dynamics plot in a rain forest at Sinharaja World Heritage Site (Sri Lanka), which apparently shows clustering at several scales. We developed cluster processes incorporating two critical scales of clustering for exploring the spatial structure of S. congestiflora and interpret it in relation to factors such as competition, dispersal limitation, recruitment limitation, and Janzen-Connell effects. All size classes showed consistent large-scale clustering with a cluster radius of ;25 m. Inside the larger clusters, small-scale clusters with a radius of 8 m were evident for recruits and saplings, weak for intermediates, and disappeared for adults. The pattern of all trees could be divided into two independent patterns: a random pattern (nearest neighbor distance . 8m ) comprising ;12 of the trees and a nested double-cluster pattern. This finding suggests two independent recruitment and or seed dispersal mechanisms. Saplings were several times as abundant as recruits and may accumulate several recruit generations. Recruits were only weakly associated with adults and occupied about half of the large-scale clusters, but saplings almost all. This is consistent with recruitment limitation. For ;70 (95 ) of all juveniles the nearest adult was less than 26 m away (53 m), suggesting a dispersal limitation that may also be related to the critical large-scale clustering. Our example illustrates the manner in which the use of a specific and complex null hypothesis of spatial structure in point pattern analysis can help us better understand the biology of a species and generate specific hypotheses to be further investigated in the field. k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's coalescent, which has numerous applications in population genetics, is the δ0-coalescent for δ0 a unit mass at 0. The coalescent recently derived by Bolthausen and Sznit- man from Ruelle's probability cascades, in the context of the Sherrington- Kirkpatrick spin glass model in mathematical physics, is the U-coalescent for U uniform on � 0� 1� .F or� = U, and whenever an infinite number of masses are present, each collision in a � -coalescent involves an infinite number of masses almost surely, and the proportion of masses involved exists as a limit almost surely and is distributed proportionally to � . The two-parameter Poisson-Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable ran- dom partitions ofgoverned by a generalization of the Ewens sampling formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the U-coalescent and its time reversal. 1. Introduction. Markovian coalescent models for the evolution of a sys- tem of masses by a random process of binary collisions were introduced by Marcus (29) and Lushnikov (28). See (3) for a recent survey of the scientific lit- erature of these models and their relation to Smoluchowski's mean-field theory of coagulation phenomena. Evans and Pitman (15) gave a general framework for the rigorous construction of partition-valued and discrete measure-valued coalescent Markov processes allowing infinitely many massses and treated the binary coalescent model where each pair of masses x and y is subject to a coa- lescent collision at rate κ� xyfor a suitable rate kernel κ. This paper studies a family of partition-valued Markov processes, with state space the compact set of all partitions of � �= � 1� 2 ���� � , such that the restriction of the partition to each finite subset ofis a Markov chain with transition rates of a simple form determined by the moments of a finite measureon the unit interval. The case � = δ 0 , a unit mass at 0, is Kingman's coalescent in which every Regard an element of the set of ranked discrete distributions Δ := (x1, x2,…):x1≥x2≥…≥ 0, ∑ i x i = 1 as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the Δ-valued Markov process in which pairs of clusters of masses x i , x j merge into a cluster of mass x i + x j at rate x i + x j . Aldous and Pitman (1998) showed that a version of this process starting from time −∞ with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991, 1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees. Regard an element of the set Δ := (x 1 , x 2 , . . .): x 1 ≥ x 2 ≥ ⋯ ≥ 0, ∑ i x i = 1 as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman is the Δ-valued Markov process in which pairs of clusters of masses x i , x j merge into a cluster of mass x i + x j at rate x i + x j . They showed that a version (X ∞ (t), -∞ < t < ∞) of this process arises as a n → ∞ weak limit of the process started at time -1 2 log n with n clusters of mass 1 n. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous by Poisson splitting along the skeleton of the tree. We describe the distribution of X∞(t) on Δ at a fixed time t. We show that the size of the cluster containing a given atom, as a process in t, has a simple representation in terms of the stable subordinator of index 1 2. As t → -∞, we establish a Gaussian limit for (centered and normalized) cluster sizes and study the size of the largest cluster.