Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Gregory A. Freiman and Boris L. Granovsky
Journal: Trans. Amer. Math. Soc. 357 (2005), 2483-2507
MSC (2000): Primary 60K35, 05A15; Secondary 05A16, 05C80, 11M45
DOI: https://doi.org/10.1090/S0002-9947-04-03617-7
Published electronically: November 23, 2004
MathSciNet review: 2140447
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fowler method in statistical physics. Our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters (= components), as the total size of a structure (= number of particles) goes to infinity. We discover that $n^{\frac{1}{l+1}}$ is the threshold for the limiting distribution of the largest cluster. As a by-product of our study, we prove the independence of the numbers of groups of fixed sizes, as $n\to \infty.$ This is in accordance with the general principle of asymptotic independence of sites in mean-field models. The latter principle is commonly accepted in statistical physics, but not rigorously proved.


References [Enhancements On Off] (What's this?)

  • 1. M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers. Commun. Math. Phys. 65 (1979), 203-230. MR 80d:80008
  • 2. D. J. Aldous, Deterministic and stochastic models for coalescence (aggregation, coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), 3-48. MR 2001c:60153
  • 3. D. J. Aldous, Emergence of the giant component in special Marcus-Lushnikov processes. Random Structures Algorithms, 12 (1998), 179-196. MR 99g:60128
  • 4. Richard Arratia, A. D. Barbour, and Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003. MR 2032426
  • 5. R. Arratia, S. Tavaré, Independent process approximations for random combinatorial structures. Adv. Math. 104 (1994), 90-154. MR 95c:60010
  • 6. A. D. Barbour, B. Granovsky, Random combinatorial structures: the convergent case. Preprint, math.Pr/0305031, math@arXiv.org. (2003).
  • 7. J. Bell, Sufficient conditions for zero-one laws. Trans. Amer. Math. Soc. 354 (2002), 613-630. MR 2002j:60057
  • 8. J. Bell, S. Burris, Asymptotics for logical limit laws: When the growth of the components is in RT class. Trans. Amer. Math. Soc. 355 (2003), 3777-3794. MR 2004h:03072
  • 9. J. Bertoin, Homogeneous fragmentation processes. Probab. Theory Relat. Fields 121 (2001), 301-318. MR 2002j:60127
  • 10. N. H. Bingham, C. M. Goldie, J. L. Teugels. Regular variation. Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, 1987. MR 88i:26004
  • 11. S. Burris, Number theoretic density and logical limit laws. Mathematical surveys and monographs 86, American Mathematical Society, Providence, RI, 2001. MR 2002c:03060
  • 12. J. Deshouillers, G. Freiman, W. Moran, On series of discrete random variables,1: real trinomial ditributions with fixed probabilities. Asterisque 258 (1999), 411-423. MR 2000j:60028
  • 13. R. Durrett, B. Granovsky, S. Gueron, The equilibrium behaviour of reversible coagulation-fragmentation processes. J. of Theoretical Probability 12 (1999), 447-474. MR 2000g:82013
  • 14. A. Eibeck, W. Wagner, Stochastic particle approximations for Smoluchowski's coagulation equation. Annals of Applied Probab. 11 (2001), 1137-1165. MR 2002k:60208
  • 15. S. Evans, J. Pitman, Construction of Markovian coalescents. Ann. Inst. Henri Poincaré 34 (1998), 339-383. MR 99k:60184
  • 16. W. J. Ewens, Remarks on the law of succession. Athens conference on applied probability and time series analysis, v. 1, 229-244, 1995, Lecture Notes in Statistics, 114, Springer, NY, 1996. MR 98g:92001
  • 17. H. Eyring, D. Henderson, B. J. Stover, E. M. Eyring, Statistical mechanics and dynamics. NY, 1964.
  • 18. W. Feller, An introduction to probability theory and its applications, v. II, Wiley, NY, 1966. MR 35:1048
  • 19. G. Freiman, Waring's problem with an increasing number of terms. Elabuz. Goz. Ped. Inst. Ucen. Zap. 3, 105-119, 1958 (in Russian). MR 41:166
  • 20. G. Freiman, B. Granovsky, Asymptotic formula for a partition function of reversible coagulation-fragmentation processes. J. Isr. Math., 130 (2002), 259-279. MR 2003i:60036
  • 21. G. Freiman, J. Pitman, Partitions into distinct large parts. J. Austral. Math. Soc. (Series A) 57 (1994), 386-416. MR 95h:11110
  • 22. G. Freiman, A. Vershik, Yu. Yakubovitz, A local limit theorem for random strict partitions. Th. Probab. Appl. 44 (2000), 453-468. MR 2002c:11134
  • 23. B. V. Gnedenko, A. N. Kolmogorov, Limit distributions for sums of independent random variables. Addison-Wesley, 1954. MR 16:52d
  • 24. S. Gueron, The steady-state distributions of coagulation-fragmentation processes. J. Math. Biol. 1 (1998), 1-27. MR 99g:82055
  • 25. U. Hirth, A Poisson approximation for the Dirichlet law, the Ewens sampling formula and the Griffith-Engen-McCloskey law by the Stein-Chen coupling method. Bernoulli 3 1997, 225-232. MR 98m:60026
  • 26. I. A. Ibragimov, Yu. V. Linnik, Independent and stationary sequences of random variables. Walters-Noordhoff, Groningen, 1971. MR 48:1287
  • 27. I. Jeon, Existence of gelling solutions for coagulation-fragmentation equations. Commun. Math. Phys., 194 (1998), 541-567. MR 99g:82056
  • 28. I. Jeon, P. March, B. Pittel, Size of the largest cluster under zero-range invariant measures. Ann. Probab. 28 (2000), 1162-1194. MR 2002j:60183
  • 29. A. I. Khinchin, Mathematical foundations of quantum statistics. Graylock Press, Albany, N.Y., 1960. MR 22:2081
  • 30. F. Kelly, Reversibility and stochastic networks. Wiley, 1979. MR 81j:60105
  • 31. V. Kolchin, Random graphs. Encyclopedia of Mathematics and its Applications, 53, Cambridge Univ.Press, 1999. MR 2001h:60015
  • 32. P. Laurencot, D. Wrzosek, The discrete coagulation equations with collisional breakage. J. Stat. Phys. 104 (2001), 193-220. MR 2002k:82056
  • 33. L. Mutafchiev, Local limit theorems for sums of power series distributed random variables and for the number of components in labelled relational structures. Random structures and Algorithms, 3 (1992), 404-426. MR 93k:60060
  • 34. J. Norris, Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent. Ann. Appl. Probab. 9 (1999), 78-109. MR 2000d:82034
  • 35. J. Norris, Cluster coagulation. Comm. Math. Phys., 209 (2000), 407-435. MR 2002c:82079
  • 36. A. G. Postnikov, Introduction to analytic number theory. Translations of Mathematical Monographs, 68, AMS, 1987. MR 89a:11001
  • 37. E. Seneta, Functions of regular variation. Lecture Notes in Mathematics, 506, Springer, NY, 1976. MR 56:12189
  • 38. M. V. Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 92 (1917), 129-168.
  • 39. R. P. Stanley, Enumerative combinatorics, Volume 2. Cambridge University Press, 1999. MR 2000k:05026
  • 40. N. Wax, ed., Selected papers on noise and stochastic processes. Dover, 1954. MR 15:970a
  • 41. P. Whittle, Statistical processes of aggregation and polymerization. Proc. Camb. Phil. Soc., 61 (1965), 475-495. MR 31:3151
  • 42. P. Whittle, Systems in stochastic equilibrium. Wiley, 1986. MR 88c:60162

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60K35, 05A15, 05A16, 05C80, 11M45

Retrieve articles in all journals with MSC (2000): 60K35, 05A15, 05A16, 05C80, 11M45


Additional Information

Gregory A. Freiman
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel
Email: grisha@math.tau.ac.il

Boris L. Granovsky
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel
Email: mar18aa@techunix.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-04-03617-7
Keywords: Coagulation-fragmentation process, random combinatorial structures, local limit theorem, distributions on the set of partitions, additive number systems
Received by editor(s): July 18, 2002
Received by editor(s) in revised form: January 7, 2004
Published electronically: November 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society