Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case

Erlihson, Michael M. ; Granovsky, Boris L.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 915-945.

Abstract (VO)

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, $a_k\sim C{k}^{p-1},k\to \infty ,p>0$, where $C$ is a positive constant. The measures considered are associated with the generalized Maxwell-Boltzmann models in statistical mechanics, reversible coagulation-fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation.

Résumé

Nous trouvons des formes limites pour une famille de mesures multiplicatives sur l’ensemble des partitions, induites par des fonctions génératrices exponentielles avec des paramètres d’expansion $a_k\sim C{k}^{p-1},k\to \infty ,p>0$, où $C$ est une constante positive. Les mesures considérées sont associées aux modèles Maxwell-Boltzmann généralisés de la mécanique statistique, des processus de coagulation-fragmentation réversibles et des structures combinatoires connues sous le nom d’assemblées. Nous prouvons un théorème de limite centrale pour les fluctuations d’une partition qui est mise à l'échelle convenablement et choisie aléatoirement selon la mesure ci-dessus. Nous démontrons que, quand la taille des composantes dépasse la valeur seuil, l’indépendance des nombres de composants se transforme en leur indépendance conditionnelle. Entre autres, cet article traite, dans un cadre général, des relations entre la forme limite, le seuil et la congélation.

Détail

DOI : 10.1214/07-AIHP129

Classification : 60J27, 60K35, 82C22, 82C26
Keywords: Gibbs distributions on the set of integer partitions, limit shapes, random combinatorial structures and coagulation-fragmentation processes, local and integral central limit theorems

Bibliographie

[1] G. Andrews. The Theory of Partitions, Vol. 2. Addison-Wesley, 1976. | Zbl | MR

[2] R. Arratia and S. Tavare. Independent process appoximation for random combinatorial structures. Adv. Math. 104 (1994) 90-154. | Zbl | MR

[3] A. Barbour, R. Arratia and S. Tavar'E. Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society Publishing House, Zurich, 2004. | Zbl | MR

[4] A. Barbour and B. Granovsky. Random combinatorial structures: The convergent case. J. Combin. Theory Ser. A 109 (2005) 203-220. | Zbl | MR

[5] S. Burris. Number Theoretic Density and Logical Limit Laws. Amer. Math. Soc., Providence, RI, 2001. | Zbl | MR

[6] J. Bell. Sufficient conditions for zero-one laws. Trans. Amer. Math. Soc. 354 (2002) 613-630. | Zbl | MR

[7] J. Bell and 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. | Zbl | MR

[8] N. Berestycki and J. Pitman. Gibbs distributions for random partitions generated by a fragmentation process. J. Stat. Phys. 127 (2007) 381-418. | Zbl | MR

[9] L. Bogachev and Z. Su. Central limit theorem for random partitions under the Plancherel measure, preprint. Available at math.PR/0607635, 2006. | Zbl

[10] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1995. | Zbl | MR

[11] R. Durrett, B. Granovsky and S. Gueron. The equilibrium behaviour of reversible coagulation-fragmentation processes. J. Theoret. Probab. 12 (1999) 447-474. | Zbl | MR

[12] M. Erlihson and B. Granovsky. Reversible coagulation-fragmentation processes and random combinatorial structures: Asymptotics for the number of groups. Random Structures Algorithms 25 (2004) 227-245. | Zbl | MR

[13] G. Freiman and B. Granovsky. Asymptotic formula for a partition function of reversible coagulation-fragmentation processes. J. Israel Math. 130 (2002) 259-279. | Zbl | MR

[14] G. Freiman and B. Granovsky. Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws. Trans. Amer. Math. Soc. 357 (2005) 2483-2507. | Zbl | MR

[15] G. Freiman and J. Pitman. Partitions into distinct large parts. J. Aust. Math. Soc. (Ser. A) 57 (1994) 386-416. | Zbl | MR

[16] B. Fristedt. The structure of random partitions of large integers. Tran. Amer. Math. Soc. 337 (1993) 703-735. | Zbl | MR

[17] A. Gnedin and J. Pitman. Exchangeable Gibbs partitions and Stirling triangles. J. Math. Sci. 138 (2006) 5674-5685. | MR | Zbl

[18] B. Granovsky. Asymptotics of counts of small components in random structures and models of coagulation-fragmentation. Available at math.Pr/0511381, 2006. | Zbl

[19] B. Granovsky and D. Stark. Asymptotic enumeration and logical limit laws for expansive multisets. J. London Math. Soc. (2) 73 (2006) 252-272. | Zbl | MR

[20] B. Granovsky and N. Madras. The noisy voter model. Stochastic Process. Appl. 55 (1995) 23-43. | Zbl | MR

[21] D. Griffeath. Additive and Cancellative Interacting Particle Systems. Springer, New York, 1979. | Zbl | MR

[22] E. Hendriks, J. Spouge, M. Eibl and M. Schreckenberg. Exact solutions for random coagulation processes. Z. Phys. B - Cond. Matter. 58 (1985) 219-227.

[23] O. Kallenberg. Foundations of Modern Probability. Springer, New York, 2001. | Zbl | MR

[24] N. Kazimirov. On some conditions for absence of a giant component in the generalized allocation scheme. Discrete Math. Appl. 12 (2002) 291-302. | Zbl | MR

[25] F. Kelly. Reversibility and Stochastic Networks. Wiley, New York, 1979. | Zbl | MR

[26] S. Kerov. Coherent random allocations, and the Ewens-Pitman formula, J. Math. Sci. 138 (2006) 5699-5710. | Zbl | MR

[27] A. Khinchin. Mathematical Foundations of Quantum Statistics. Graylock Press, Albany, NY, 1960. | Zbl | MR

[28] V. Kolchin. Random Graphs. Cambridge Univ. Press, 1999. | Zbl | MR

[29] B. Logan and L. Shepp. A variational problem for random Young tableaux. Adv. Math. 26 (1977) 206-222. | Zbl | MR

[30] 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 Algorithms 3 (1992) 404-426. | Zbl | MR

[31] O. Milenkovic and K. Compton. Probabilistic transforms for combinatorial urn models. Combin. Probab. Comput. 13 (2004) 645-675. | Zbl | MR

[32] A. Okounkov. Symmetric functions and random partitions. Symmetric Functions 2001: Surveys of Developments and Perspectives 223-252. NATO Sci. Ser. II Math. Phys. Chem. 74. Kluwer Acad. Publ., Dordrecht, 2002. | Zbl | MR

[33] J. Pitman. Combinatorial Stochastic Processes. Springer, Berlin, 2006. | Zbl | MR

[34] B. Pittel. On a likely shape of the random Ferrers diagram. Adv. in Appl. Math. 18 (1997) 432-488. | Zbl | MR

[35] B. Pittel. On the distribution of the number of Young tableaux for a uniformly random diagram. Adv. in Appl. Math. 29 (2002) 184-214. | Zbl | MR

[36] D. Romik. Identities arising from limit shapes of costrained randiom partitions, preprint, 2003.

[37] A. Shiryaev. Probability. Springer, New York, 1984. | Zbl | MR

[38] S. Shlosman. Geometric variational problems of statistical mechanics and of combinatorics, probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 (2000) 1364-1370. | Zbl | MR

[39] S. Shlosman. Wulf construction in statistical mechanics and combinatorics. Russian Math. Surveys. 56 (2001) 709-738. | Zbl | MR

[40] D. Stark. Logical limit laws for logarithmic structures. Math. Proc. Cambridge Philos. Soc. 140 (2005) 537-544. | Zbl | MR

[41] H. Temperley. Statistical mechanics and the partition of numbers. The form of the crystal surfaces. Proc. Cambridge Philos. Soc. 48 (1952) 683-697. | Zbl | MR

[42] A. Vershik and S. Kerov. Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables. Dokl. Akad. Nauk SSSR 233 (1977) 1024-1027. | Zbl | MR

[43] A. Vershik. Limit distribution of the energy of a quantum ideal gas from the viewpoint of the theory of partitions of natural numbers. Russian Math. Surveys 52 (1997) 139-146. | Zbl | MR

[44] A. Vershik. Statistical mechanics of combinatorial partitions and their limit configurations. Funct. Anal. Appl. 30 (1996) 90-105. | Zbl | MR

[45] A. Vershik, G. Freiman and Yu. Yakubovich. A local limit theorem for random partitions of natural numbers. Theory Probab. Appl. 44 (2000) 453-468. | Zbl | MR

[46] A. Vershik and Y. Yakubovich. The limit shape and fluctuations of random partitions of naturals with fixed number of summands. Mosc. Math. J. 1 (2001) 457-468. | Zbl | MR

[47] A. Vershik and Y. Yakubovich. Fluctuations of the maximal particle energy of the quantum ideal gas and random partitions. Comm. Math. Phys. 261 (2006) 759-769. | Zbl | MR

[48] A. Vershik and Y. Yakubovich. Asymptotics of the uniform measure on the simplex, random compositions and partitions. Funct. Anal. Appl. 37 (2003) 39-48. | Zbl | MR

[49] P. Whittle. Systems in Stochastic Equilibrium. Wiley, New York, 1986. | Zbl | MR

[50] Y. Yakubovich. Asymptotics of random partitions of a set. J. Math. Sci. 87 (1997) 4124-4137. | Zbl | MR