Electronic Journal of Probability

Random partitions in statistical mechanics

Nicholas Ercolani, Sabine Jansen, and Daniel Ueltschi

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We consider a family of distributions on spatial random partitions that provide a coupling between different models of interest: the ideal Bose gas; the zero-range process; particle clustering; and spatial permutations. These distributions are invariant for a "chain of Chinese restaurants" stochastic process. We obtain results for the distribution of the size of the largest component.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 82, 37 pp.

Accepted: 9 September 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B05: Classical equilibrium statistical mechanics (general)

Spatial random partitions Bose-Einstein condensation (inhomogeneous) zero-range process chain of Chinese restaurants sums of independent random variables heavy-tailed variables infinitely divisible laws

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Ercolani, Nicholas; Jansen, Sabine; Ueltschi, Daniel. Random partitions in statistical mechanics. Electron. J. Probab. 19 (2014), paper no. 82, 37 pp. doi:10.1214/EJP.v19-3244. https://projecteuclid.org/euclid.ejp/1465065724

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  • Aigner, Martin. A course in enumeration. Graduate Texts in Mathematics, 238. Springer, Berlin, 2007. x+561 pp. ISBN: 978-3-540-39032-9.
  • Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3–48.
  • Armendáriz, Inés; Loulakis, Michail. Thermodynamic limit for the invariant measures in supercritical zero range processes. Probab. Theory Related Fields 145 (2009), no. 1-2, 175–188.
  • Armendáriz, Inés; Grosskinsky, Stefan; Loulakis, Michail. Zero-range condensation at criticality. Stochastic Process. Appl. 123 (2013), no. 9, 3466–3496.
  • Armendáriz, Inés; Loulakis, Michail. Thermodynamic limit for the invariant measures in supercritical zero range processes. Probab. Theory Related Fields 145 (2009), no. 1-2, 175–188.
  • Arratia, Richard; Barbour, A. D.; Tavaré, Simon. Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2003. xii+363 pp. ISBN: 3-03719-000-0.
  • T. Bak sajeva, E. Manstavi cius, On statistics of permutations chosen from the Ewens distribution, arXiv:1303.4540 [math.CO] (2013)
  • Ball, J. M.; Carr, J.; Penrose, O. The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions. Comm. Math. Phys. 104 (1986), no. 4, 657–692.
  • R. Becker, W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen, Ann. Phys. (Leipzig) 24, 719–752 (1935)
  • Benfatto, Giuseppe; Cassandro, Marzio; Merola, I.; Presutti, E. Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas. J. Math. Phys. 46 (2005), no. 3, 033303, 38 pp.
  • van den Berg, M.; Lewis, J. T.; Pulé, J. V. The large deviation principle and some models of an interacting boson gas. Comm. Math. Phys. 118 (1988), no. 1, 61–85.
  • Bertoin, Jean. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2.
  • Betz, Volker; Ueltschi, Daniel. Spatial random permutations and infinite cycles. Comm. Math. Phys. 285 (2009), no. 2, 469–501.
  • V. Betz, D. Ueltschi, Critical temperature of dilute Bose gases, Phys. Rev. A 81, 023611 (2010)
  • Betz, Volker; Ueltschi, Daniel. Spatial random permutations and Poisson-Dirichlet law of cycle lengths. Electron. J. Probab. 16 (2011), no. 41, 1173–1192.
  • Betz, Volker; Ueltschi, Daniel; Velenik, Yvan. Random permutations with cycle weights. Ann. Appl. Probab. 21 (2011), no. 1, 312–331.
  • L. Bogachev, D. Zeindler, Asymptotic statistics of cycles in surrogate-spatial permutations, to appear in Commun. Math. Phys. (2014); doi:10.1007/s00220-014-2110-1
  • Buffet, E.; Pulé, J. V. Fluctuation properties of the imperfect Bose gas. J. Math. Phys. 24 (1983), no. 6, 1608–1616.
  • Burke, C. J.; Rosenblatt, M. A Markovian function of a Markov chain. Ann. Math. Statist. 29 1958 1112–1122.
  • Chatterjee, Sourav; Diaconis, Persi. Fluctuations of the Bose-Einstein condensate. J. Phys. A 47 (2014), no. 8, 085201, 23 pp.
  • M. Chen, Dirichlet forms and symmetrizable jump processes, Chinese Quart. J. Math. 6, 83–104 (1991)
  • Chleboun, Paul; Grosskinsky, Stefan. Condensation in stochastic particle systems with stationary product measures. J. Stat. Phys. 154 (2014), no. 1-2, 432–465.
  • A. Cipriani, D. Zeindler, The limit shape of random permutations with polynomially growing cycle weights, arXiv.org/1312.3517 [math.PR]
  • Denisov, D.; Dieker, A. B.; Shneer, V. Large deviations for random walks under subexponentiality: the big-jump domain. Ann. Probab. 36 (2008), no. 5, 1946–1991.
  • S. Dereich, P. Mörters, Cycle length distributions in random permutations with diverging cycle weights, to appear in Random Struct. Algor. (2014)
  • Doney, R. A. A large deviation local limit theorem. Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 3, 575–577.
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8.
  • Durrett, Rick. Probability models for DNA sequence evolution. Probability and its Applications (New York). Springer-Verlag, New York, 2002. viii+240 pp. ISBN: 0-387-95435-X.
  • Durrett, Richard; Granovsky, Boris L.; Gueron, Shay. The equilibrium behavior of reversible coagulation-fragmentation processes. J. Theoret. Probab. 12 (1999), no. 2, 447–474.
  • Embrechts, Paul; Hawkes, John. A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory. J. Austral. Math. Soc. Ser. A 32 (1982), no. 3, 412–422.
  • Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas. Modelling extremal events. For insurance and finance. Applications of Mathematics (New York), 33. Springer-Verlag, Berlin, 1997. xvi+645 pp. ISBN: 3-540-60931-8.
  • N. Ercolani, S. Jansen, D. Ueltschi, Lindelöf integrals for combinatorics and probability theory, in preparation
  • Ercolani, Nicholas M.; Ueltschi, Daniel. Cycle structure of random permutations with cycle weights. Random Structures Algorithms 44 (2014), no. 1, 109–133.
  • Erlihson, Michael M.; Granovsky, Boris L. Reversible coagulation-fragmentation processes and random combinatorial structures: asymptotics for the number of groups. Random Structures Algorithms 25 (2004), no. 2, 227–245.
  • Erlihson, Michael M.; Granovsky, Boris L. Limit shapes of Gibbs distributions on the set of integer partitions: the expansive case. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 5, 915–945.
  • M. R. Evans, Phase transitions in one-dimensional non-equilibrium systems, Braz. J. Phys. 30, 42–57 (2000)
  • Flajolet, Philippe; Sedgewick, Robert. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. ISBN: 978-0-521-89806-5.
  • Ford, Walter B. Studies on divergent series and summability & The asymptotic developments of functions defined by Maclaurin series. Chelsea Publishing Co., New York 1960 x+342 pp.
  • Gnedenko, B. V.; Kolmogorov, A. N. Limit distributions for sums of independent random variables. Translated from the Russian, annotated, and revised by K. L. Chung. With appendices by J. L. Doob and P. L. Hsu. Revised edition Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills., Ont. 1968 ix+293 pp.
  • Godrèche, C.; Luck, J. M. Condensation in the inhomogeneous zero-range process: an interplay between interaction and diffusion disorder. J. Stat. Mech. Theory Exp. 2012, no. 12, P12013, 45 pp.
  • Grosskinsky, Stefan; Schätz, Gunter M.; Spohn, Herbert. Condensation in the zero range process: stationary and dynamical properties. J. Statist. Phys. 113 (2003), no. 3-4, 389–410.
  • Ibragimov, I. A.; Linnik, Yu. V. Independent and stationary sequences of random variables. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov. Translation from the Russian edited by J. F. C. Kingman. Wolters-Noordhoff Publishing, Groningen, 1971. 443 pp.
  • Jansen, Sabine; König, Wolfgang. Ideal mixture approximation of cluster size distributions at low density. J. Stat. Phys. 147 (2012), no. 5, 963–980.
  • S. Jansen, W. König, B. Metzger, Large deviations for cluster size distributions in a continuous classical many-body system, preprint, arXiv:1107.3670 (2011)
  • Maples, Kenneth; Nikeghbali, Ashkan; Zeindler, Dirk. On the number of cycles in a random permutation. Electron. Commun. Probab. 17 (2012), no. 20, 13 pp.
  • Nagaev, A. V. Local limit theorems with regard to large deviations when Cramér's condition is not satisfied. (Russian) Litovsk. Mat. Sb. 8 1968 553–579.
  • Nikeghbali, Ashkan; Zeindler, Dirk. The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 4, 961–981.
  • Nikeghbali, Ashkan; Zeindler, Dirk. The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 4, 961–981.
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • Pitman, Jim; Yor, Marc. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 (1997), no. 2, 855–900.
  • Sator, N. Clusters in simple fluids. Phys. Rep. 376 (2003), no. 1, 1–39.
  • Sütó, András. Percolation transition in the Bose gas. II. J. Phys. A 35 (2002), no. 33, 6995–7002.
  • Vershik, A. M. Statistical mechanics of combinatorial partitions, and their limit configurations. (Russian) Funktsional. Anal. i Prilozhen. 30 (1996), no. 2, 19–39, 96; translation in Funct. Anal. Appl. 30 (1996), no. 2, 90–105
  • B. Waclaw, L. Bogacz, Z. Burda, W. Janke, Condensation in zero-range processes on inhomogeneous networks, Phys. Rev. E 76, 046114 (2007)
  • Zhao, James Y. Universality of asymptotically Ewens measures on partitions. Electron. Commun. Probab. 17 (2012), no. 16, 11 pp.