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

Erlihson, Michael M.; Granovsky, Boris L.

Erlihson, Michael M. ; Granovsky, Boris L.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 915-945 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

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.

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.

Publié le : 2008-01-01

MR 2453776 | Zbl 1181.60146

DOI : https://doi.org/10.1214/07-AIHP129
Classification: 60J27, 60K35, 82C22, 82C26

@article{AIHPB_2008__44_5_915_0,
     author = {Erlihson, Michael M. and Granovsky, Boris L.},
     title = {Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {44},
     year = {2008},
     pages = {915-945},
     doi = {10.1214/07-AIHP129},
     mrnumber = {2453776},
     zbl = {1181.60146},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2008__44_5_915_0}
}

Erlihson, Michael M.; Granovsky, Boris L. Limit shapes of Gibbs distributions on the set of integer partitions : the expansive case. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 915-945. doi : 10.1214/07-AIHP129. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_5_915_0/

Bibliographie

[1] G. Andrews. The Theory of Partitions, Vol. 2. Addison-Wesley, 1976. | MR 557013 | Zbl 0655.10001

[2] R. Arratia and S. Tavare. Independent process appoximation for random combinatorial structures. Adv. Math. 104 (1994) 90-154. | MR 1272071 | Zbl 0802.60008

[3] A. Barbour, R. Arratia and S. Tavar'E. Logarithmic Combinatorial Structures: A Probabilistic Approach. European Mathematical Society Publishing House, Zurich, 2004. | MR 2032426 | Zbl 1040.60001

[4] A. Barbour and B. Granovsky. Random combinatorial structures: The convergent case. J. Combin. Theory Ser. A 109 (2005) 203-220. | MR 2121024 | Zbl 1065.60143

[5] S. Burris. Number Theoretic Density and Logical Limit Laws. Amer. Math. Soc., Providence, RI, 2001. | MR 1800435 | Zbl 0995.11001

[6] J. Bell. Sufficient conditions for zero-one laws. Trans. Amer. Math. Soc. 354 (2002) 613-630. | MR 1862560 | Zbl 0981.60030

[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. | MR 1990173 | Zbl 1021.03022

[8] N. Berestycki and J. Pitman. Gibbs distributions for random partitions generated by a fragmentation process. J. Stat. Phys. 127 (2007) 381-418. | MR 2314353 | Zbl 1126.82013

[9] L. Bogachev and Z. Su. Central limit theorem for random partitions under the Plancherel measure, preprint. Available at math.PR/0607635, 2006. | Zbl 1157.60012

[10] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1995. | MR 1609153 | Zbl 0709.60002

[11] R. Durrett, B. Granovsky and S. Gueron. The equilibrium behaviour of reversible coagulation-fragmentation processes. J. Theoret. Probab. 12 (1999) 447-474. | MR 1684753 | Zbl 0930.60094

[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. | MR 2076340 | Zbl 1060.60020

[13] G. Freiman and B. Granovsky. Asymptotic formula for a partition function of reversible coagulation-fragmentation processes. J. Israel Math. 130 (2002) 259-279. | MR 1919380 | Zbl 1003.60009

[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. | MR 2140447 | Zbl 1062.60097

[15] G. Freiman and J. Pitman. Partitions into distinct large parts. J. Aust. Math. Soc. (Ser. A) 57 (1994) 386-416. | MR 1297011 | Zbl 0824.11064

[16] B. Fristedt. The structure of random partitions of large integers. Tran. Amer. Math. Soc. 337 (1993) 703-735. | MR 1094553 | Zbl 0795.05009

[17] A. Gnedin and J. Pitman. Exchangeable Gibbs partitions and Stirling triangles. J. Math. Sci. 138 (2006) 5674-5685. | MR 2160320 | Zbl 1293.60010 | Zbl pre02214054

[18] B. Granovsky. Asymptotics of counts of small components in random structures and models of coagulation-fragmentation. Available at math.Pr/0511381, 2006. | Zbl 1303.60013

[19] B. Granovsky and D. Stark. Asymptotic enumeration and logical limit laws for expansive multisets. J. London Math. Soc. (2) 73 (2006) 252-272. | MR 2197382 | Zbl 1086.60006

[20] B. Granovsky and N. Madras. The noisy voter model. Stochastic Process. Appl. 55 (1995) 23-43. | MR 1312146 | Zbl 0813.60096

[21] D. Griffeath. Additive and Cancellative Interacting Particle Systems. Springer, New York, 1979. | MR 538077 | Zbl 0412.60095

[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. | MR 1876169 | Zbl 0996.60001

[24] N. Kazimirov. On some conditions for absence of a giant component in the generalized allocation scheme. Discrete Math. Appl. 12 (2002) 291-302. | MR 1937012 | Zbl 1046.60018

[25] F. Kelly. Reversibility and Stochastic Networks. Wiley, New York, 1979. | MR 554920 | Zbl 0422.60001

[26] S. Kerov. Coherent random allocations, and the Ewens-Pitman formula, J. Math. Sci. 138 (2006) 5699-5710. | MR 2160323 | Zbl 1077.60007

[27] A. Khinchin. Mathematical Foundations of Quantum Statistics. Graylock Press, Albany, NY, 1960. | MR 111217 | Zbl 0089.45004

[28] V. Kolchin. Random Graphs. Cambridge Univ. Press, 1999. | MR 1728076 | Zbl 0918.05001

[29] B. Logan and L. Shepp. A variational problem for random Young tableaux. Adv. Math. 26 (1977) 206-222. | MR 1417317 | Zbl 0363.62068

[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. | MR 1179830 | Zbl 0769.60021

[31] O. Milenkovic and K. Compton. Probabilistic transforms for combinatorial urn models. Combin. Probab. Comput. 13 (2004) 645-675. | MR 2095977 | Zbl 1079.60017

[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. | MR 2059364 | Zbl 1017.05103

[33] J. Pitman. Combinatorial Stochastic Processes. Springer, Berlin, 2006. | MR 2245368 | Zbl 1103.60004

[34] B. Pittel. On a likely shape of the random Ferrers diagram. Adv. in Appl. Math. 18 (1997) 432-488. | MR 1445358 | Zbl 0894.11039

[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. | MR 1928098 | Zbl 1016.60030

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

[37] A. Shiryaev. Probability. Springer, New York, 1984. | MR 737192 | Zbl 0536.60001

[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. | MR 1757963 | Zbl 1052.82003

[39] S. Shlosman. Wulf construction in statistical mechanics and combinatorics. Russian Math. Surveys. 56 (2001) 709-738. | MR 1861442 | Zbl 1035.82013

[40] D. Stark. Logical limit laws for logarithmic structures. Math. Proc. Cambridge Philos. Soc. 140 (2005) 537-544. | MR 2225646 | Zbl 1096.03035

[41] H. Temperley. Statistical mechanics and the partition of numbers. The form of the crystal surfaces. Proc. Cambridge Philos. Soc. 48 (1952) 683-697. | MR 53036 | Zbl 0048.19802

[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. | MR 480398 | Zbl 0406.05008

[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. | MR 1480142 | Zbl 0927.60089

[44] A. Vershik. Statistical mechanics of combinatorial partitions and their limit configurations. Funct. Anal. Appl. 30 (1996) 90-105. | MR 1402079 | Zbl 0868.05004

[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. | MR 1805818 | Zbl 0969.60034

[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. | MR 1877604 | Zbl 0996.05006

[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. | MR 2197546 | Zbl 1113.82010

[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. | MR 2083230 | Zbl 1081.60009

[49] P. Whittle. Systems in Stochastic Equilibrium. Wiley, New York, 1986. | MR 850012 | Zbl 0665.60107

[50] Y. Yakubovich. Asymptotics of random partitions of a set. J. Math. Sci. 87 (1997) 4124-4137. | MR 1374322 | Zbl 0909.60017