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## Gibbs distributions for random partitions generated by a fragmentation process (2006)

Citations: | 20 - 3 self |

### Citations

1197 | Enumerative Combinatorics - Stanley - 1997 |

997 |
Principles of polymer chemistry
- Flory
- 1953
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Citation Context ...mple formula: Bn,k(1!,2!,3!...) = ( ) n − 1 n! k − 1 k! (13) is known as a Lah number [5, p. 135]. The Gibbs model in this instance is a variation of Flory’s model for a linear polymerization process =-=[10]-=-. Another interpretation is 61 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 Figure 1: Cutting a rooted random segment provided by Kingman’s coalescent [1, 18]. It is easily s... |

805 |
Reversibility and stochastic networks
- Kelly
- 1979
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Citation Context ...rted in part by N.S.F. Grant DMS-0405779 11 Introduction Gibbs models for random partitions generated by random processes of coagulation and fragmentation have been widely studied ([26], [27], [28], =-=[17]-=-). They typically arise as equilibrium distributions of time-reversible processes of coagulation and fragmentation (see for instance [7], and [3] for general results about exchangeable fragmentationco... |

370 |
Origins of the coalescent:
- Kingman
- 2000
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Citation Context ...merization process [10]. Another interpretation is 61 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 Figure 1: Cutting a rooted random segment provided by Kingman’s coalescent =-=[1, 18]-=-. It is easily shown in this case that a sequence of random partitions (Πk,1 ≤ k ≤ n) such the Πk has the Gibbs(n,k) distribution is obtained as follows. Let G1 be a uniformly distributed random roote... |

282 | On the geneaology of large populations. - Kingman - 1982 |

222 | Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists
- Aldous
- 1999
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Citation Context ...merization process [10]. Another interpretation is 61 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 1 4 2 5 3 6 Figure 1: Cutting a rooted random segment provided by Kingman’s coalescent =-=[1, 18]-=-. It is easily shown in this case that a sequence of random partitions (Πk,1 ≤ k ≤ n) such the Πk has the Gibbs(n,k) distribution is obtained as follows. Let G1 be a uniformly distributed random roote... |

180 |
The sampling theory of selectively neutral alleles.
- Ewens
- 1972
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Citation Context ...ngman’s n-coalescent. Then P( ˜ Πθ = π) = θk−1 k∏ (ni − 1)! (54) [θ + 1]n−1 for each partition π of [n] into k components of sizes n1,... ,nk The distribution of ˜ Πθ defined by (54) first appears in =-=[9]-=- and is known as Ewens’ Sampling Formula with parameter θ. Note that it is a canonical Gibbs distribution on P [n] with weight sequence ((j −1)!,j ≥ 1). This distribution is a particular mixture over ... |

162 | Logarithmic Combinatorial Structures: A Probabilistic Approach, - Arratia, Barbour, et al. - 2003 |

98 | Statistical mechanics of combinatorial partitions, and their limit shapes. Functional Analysis and Its Applications, - Vershik - 1996 |

85 |
Combinatorial stochastic processes, Lecture Notes of the 2002 Ecole des Probabilites de St. Flour
- Pitman
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Citation Context ...wi/i! instead of wi, and the polynomial Zn(x1,x2,...) := n!Yn(1!x1,2!x2,...) (4) is called the canonical partition function. For textbook treatments of such models, and references to earlier work see =-=[24]-=-. Typically, the canonical Gibbs distribution (2) is derived either from thermodynamic considerations, or from a set of detailed balance equations corresponding to a reversible equilibrium between pro... |

83 | Stochastic inequalities on partially ordered spaces - Kamae, Krengel, et al. - 1977 |

60 |
Systems in stochastic equilibrium
- Whittle
- 1986
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Citation Context ... supported in part by N.S.F. Grant DMS-0405779 11 Introduction Gibbs models for random partitions generated by random processes of coagulation and fragmentation have been widely studied ([26], [27], =-=[28]-=-, [17]). They typically arise as equilibrium distributions of time-reversible processes of coagulation and fragmentation (see for instance [7], and [3] for general results about exchangeable fragmenta... |

53 | Coalescent random forests,
- Pitman
- 1999
(Show Context)
Citation Context ...f random graphs and has been studied and applied in several other contexts. The coalescent obtained by reversing the process of deleting the edges at random is the additive coalescent as discussed in =-=[23]-=-. This is what is illustrated in figure 2. 3 Fragmentation Processes Definition 5. Call a sequence of P [n]-valued random variables (Π1,Π2,... ,Πn) a fragmentation process or a refining process if wit... |

52 | Exchangeable Gibbs partitions and Stirling triangles. - Gnedin, Pitman - 2006 |

43 | Beta-coalescents and continuous stable random trees Preprint: arXiv:math.PR/0602113 v1, - Berestycki, Berestycki, et al. - 2006 |

39 | Enumerations of trees and forests related to branching processes and random walks. Microsurveys in discrete probability
- Pitman
- 1998
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Citation Context ...hen Bn,k(11−1 ,22−1,3 3−1 ,...) is the number of forests of k rooted trees labelled [n]. This time again there is a simple formula for Bn,k. As a consequence of Cayley’s enumeration of random forests =-=[22, 5]-=- Bn,k(1 1−1 ,2 2−1 ,3 3−1 ( ) n − 1 ,...) = n k − 1 n−k (14) The Gibbs model in this instance corresponds to assuming that all forests of k rooted 75 4 2 5 4 2 5 4 2 5 4 2 5 4 2 5 4 2 6 1 6 1 6 1 6 1... |

37 | Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models - Haas, Miermont, et al. - 2008 |

34 | Random graphs - Kolchin - 1999 |

33 |
Stochastic coalescence
- Marcus
- 1968
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Citation Context ...dom rooted tree). The existence of this family is implied by the work of Hendriks et al. [14], who studied a corresponding family of Markov processes governed by the Marcus-Lushnikov coalescent model =-=[1, 21, 20]-=- with collision kernel Kij = a+b(i+j), call it the affine coalescent. The discrete-time skeleton of the affine coalescent turns out to just the time reversal of the process derived here. The segment s... |

32 | Small-time behavior of betacoalescents. - Berestycki, Berestycki, et al. - 2008 |

27 | On sampling distributions for coalescent processes with simultaneous multiple collisions. - Möhle - 2006 |

26 |
On graphical partitions,
- Erdos, Richmond
- 1993
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Citation Context ...on over the set Rn of all refining sequences of partitions of [n] such that the kth term of the sequence has k components. The consequent enumeration #Rn = n!(n − 1)!/2n−1 was obtained by Erdös et al =-=[8]-=-. The fact that Πk determined by this model has the Gibbs(n,k) distribution with weight sequence wj = j! was obtained by Bayewitz et. al. [2] and Kingman [18]. Example 4. Cutting a rooted random tree.... |

25 |
Coagulation in finite systems
- Lushnikov
- 1978
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Citation Context ... be functions of t. There seems to be no standard term in the literature for a distribution on partitions of this form, which will be called here a modified Gibbs distribution. For example, Lushnikov =-=[20]-=- showed that the coalescent model with monodisperse initial condition and collision rates Kx,y = xf(y) + yf(x) leads to such modified Gibbs distributions. Hendriks et. al [14] showed that this is also... |

19 | Excheangable partitions derived from Markovian coalescents - Gnedin, Dong, et al. - 2007 |

19 | Reversible coagulation-fragmentation processes and random combinatorial structures: Asymptotics for the number of groups, Random Struct - Erlihson, Granovsky - 2004 |

19 | The morphology of partially ordered sets - Harper - 1974 |

18 | Random combinatorial structures: the convergent case - Barbour, Granovsky - 2005 |

14 | The equilibrium behavior of reversible coagulation–fragmentation processes
- DURRETT, GRANOVSKY, et al.
- 1999
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Citation Context ... and fragmentation have been widely studied ([26], [27], [28], [17]). They typically arise as equilibrium distributions of time-reversible processes of coagulation and fragmentation (see for instance =-=[7]-=-, and [3] for general results about exchangeable fragmentationcoalescence processes in equilibrium). There is a much smaller literature in which Gibbs models are derived from an irreversible Markovian... |

14 | Exact solutions for random coagulation processes, - Hendriks, Spouge, et al. - 1985 |

13 | Exchangeable fragmentation-coalescence processes and their equilibrium measures
- BERESTYCKI
- 2004
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Citation Context ...mentation have been widely studied ([26], [27], [28], [17]). They typically arise as equilibrium distributions of time-reversible processes of coagulation and fragmentation (see for instance [7], and =-=[3]-=- for general results about exchangeable fragmentationcoalescence processes in equilibrium). There is a much smaller literature in which Gibbs models are derived from an irreversible Markovian coagulat... |

9 | The extent of correlations in a stochastic coalescence process, - Bayewitz, Yerushalmi, et al. - 1974 |

9 | Lagrangian Probability Distributions - Consul, Famoye - 2006 |

8 | Probabilistic and analytical aspects of the umbral calculus”, CWI Tract 119, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, - Bucchianico - 1997 |

8 | 2006c). Conditioned galton-watson trees do not grow - Janson |

7 | The hyperbolic geometry of random transpositions. - Berestycki - 2006 |

7 | Poisson representation of a Ewens fragmentation process, - Gnedin, Pitman - 2007 |

6 |
The branching process method in the Lagrange random variate generation, cgm.cs.mcgill.ca/~luc/branchingpaper.ps Diaconis
- Devroye
- 1992
(Show Context)
Citation Context ...,...) (18) which is the Borel distribution of the total progeny of a critical Poisson-Galton-Watson process with Poisson(1) offspring distribution started with one individual. See e.g. [23, §4.1] and =-=[6]-=- for proofs and various generalizations. As a consequence of (18) and the asymptotic independence of the Xn,1,... ,Xn,k−1, the asymptotic distribution of the combined size Xn,1+...+Xn,k−1 of all but t... |

5 |
On the lengths of the pieces of a stick broken at random
- Holst
- 1980
(Show Context)
Citation Context ... ranked lengths of k subintervals of [0,1] obtained by cutting [0,1] at k − 1 points picked independently and uniformly at random from [0,1]. This asymptotic distribution has been extensively studied =-=[15]-=-. For the weight sequence wj = j j−1 of Example 4, the behavior is different again. What happens is that for each fixed k the sequence of ranked sizes 15, when normalized by n, converges in probabilit... |

5 |
Statistical processes of aggregation and polymerisation
- Whittle
- 1965
(Show Context)
Citation Context ...search supported in part by N.S.F. Grant DMS-0405779 11 Introduction Gibbs models for random partitions generated by random processes of coagulation and fragmentation have been widely studied ([26], =-=[27]-=-, [28], [17]). They typically arise as equilibrium distributions of time-reversible processes of coagulation and fragmentation (see for instance [7], and [3] for general results about exchangeable fra... |

4 |
Random set partitions
- Goh, Schmutz
- 1994
(Show Context)
Citation Context ...stributions arise naturally from an assumption of equally likely outcomes on a suitable configuration space. Related problems of enumeration and asymptotic distributions have been extensively studied =-=[16, 19, 11, 12, 25]-=-. 2.2 Some important examples We introduce here a few natural examples of Gibbs distributions and their combinatorial interpretations (for particular sequences of weights (wj)) that motivate much of t... |

4 | Use of Lagrange expansion for generating discrete generalized probability distributions - Consul, Shenton - 1972 |

4 | Convolution polynomials. Mathematica journal - Knuth - 1992 |

3 |
Random mappings. Translation series
- Kolchin
- 1986
(Show Context)
Citation Context ...stributions arise naturally from an assumption of equally likely outcomes on a suitable configuration space. Related problems of enumeration and asymptotic distributions have been extensively studied =-=[16, 19, 11, 12, 25]-=-. 2.2 Some important examples We introduce here a few natural examples of Gibbs distributions and their combinatorial interpretations (for particular sequences of weights (wj)) that motivate much of t... |

3 |
The equilibrium statistics of a clustering process in the uncondensed phase
- Whittle
- 1965
(Show Context)
Citation Context .... ∗ Research supported in part by N.S.F. Grant DMS-0405779 11 Introduction Gibbs models for random partitions generated by random processes of coagulation and fragmentation have been widely studied (=-=[26]-=-, [27], [28], [17]). They typically arise as equilibrium distributions of time-reversible processes of coagulation and fragmentation (see for instance [7], and [3] for general results about exchangeab... |

3 | Power laws for family sizes in a duplication model Ann. Probab. 33,6: 2094–2126 - Durrett, Schweinsberg - 2005 |

2 |
A series of transformations leading to convolution identities
- Gould
- 1961
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Citation Context ...n − 1 ∏n Bn,k(1,w2,w3,...) = (mc + nb) (1 ≤ k ≤ n) (22) k − 1 m=k+1 This evaluation, which can be read from [14, (19)-(21)], is shown to be a consequence of a famous convolution identity due to Gould =-=[13]-=-. 4.2 Proof of Theorem 11, first half Let (Πk,1 ≤ k ≤ n) be a P [n]-valued fragmentation process defined by the recursive Gibbs splitting rule derived from these weights, with the linear selection rul... |

2 |
On numbers related to objects of unlike partitions and occupancy problems
- Holst
- 1981
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Citation Context ..., 24]. These formulae are useful whenever the weight sequence (wj) is such that the associated polynomials admit an explicit formula as functions of n and k, or can be suitably approximated (see e.g. =-=[16]-=-). Some of these results are reviewed in [24]. The class of weight sequences (wj) for which the microcanonical Gibbs model is “solvable”, meaning there is an explicit formula for the Bn,k, is quite la... |

2 |
Probabilistic methods in combinatorial mathematics
- Rényi
- 1969
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Citation Context ...stributions arise naturally from an assumption of equally likely outcomes on a suitable configuration space. Related problems of enumeration and asymptotic distributions have been extensively studied =-=[16, 19, 11, 12, 25]-=-. 2.2 Some important examples We introduce here a few natural examples of Gibbs distributions and their combinatorial interpretations (for particular sequences of weights (wj)) that motivate much of t... |

2 | The structure of partitions of large integers - Fristedt - 1993 |

1 |
The structure of partitions of large integers
- Friedst
- 1993
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