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Mosc. Math. J., 2001, Volume 1, Number 3, Pages 457–468 (Mi mmj31)  

This article is cited in 18 scientific papers (total in 19 papers)

The limit shape and fluctuations of random partitions of naturals with fixed number of summands

A. M. Vershik, Yu. V. Yakubovich

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: We consider the uniform distribution on the set of partitions of integer $n$ with $c\sqrt n$ numbers of summands, $c>0$ is a positive constant. We calculate the limit shape of such partitions, assuming $c$ is constant and $n$ tends to infinity. If $c\to\infty$ then the limit shape tends to known limit shape for unrestricted number of summands (see references). If the growth is slower than $\sqrt n$ then the limit shape is universal ($e^{-t}$). We prove the invariance principle (central limit theorem for fluctuations around the limit shape) and find precise expression for correlation functions. These results can be interpreted in terms of statistical physics of ideal gas, from this point of view the limit shape is a limit distribution of the energy of two dimensional ideal gas with respect to the energy of particles. The proof of the limit theorem uses partially inversed Fourier transformation of the characteristic function and refines the methods of the previous papers of authors (see references).

Key words and phrases: Young diagram, partition of integer, limit shape, fluctuations.


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MSC: 05A17, 11P82, 82B05
Received: June 20, 2001

Citation: A. M. Vershik, Yu. V. Yakubovich, “The limit shape and fluctuations of random partitions of naturals with fixed number of summands”, Mosc. Math. J., 1:3 (2001), 457–468

Citation in format AMSBIB
\by A.~M.~Vershik, Yu.~V.~Yakubovich
\paper The limit shape and fluctuations of random partitions of naturals with fixed number of summands
\jour Mosc. Math.~J.
\yr 2001
\vol 1
\issue 3
\pages 457--468

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    This publication is cited in the following articles:
    1. A. M. Vershik, Yu. V. Yakubovich, “Asymptotics of the Uniform Measures on Simplices and Random Compositions and Partitions”, Funct. Anal. Appl., 37:4 (2003), 273–280  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. Vershik A., “Two lectures on the asymptotic representation theory and statistics of Young diagrams”, Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001), Lecture Notes in Math., 1815, Springer, Berlin, 2003, 161–182  crossref  mathscinet  zmath  isi
    3. Yu. V. Yakubovich, “On the coincidence of limit shapes for integer partitions and compositions, and a slicing of Young diagrams”, J. Math. Sci. (N. Y.), 131:2 (2005), 5569–5577  mathnet  crossref  mathscinet  zmath  elib  elib
    4. Picard J., Combinatorial stochastic processes, Lectures from the 32nd Summer School on Probability Theory (Saint-Flour, July 7–24, 2002), Lecture Notes in Math., 1875, Springer-Verlag, Berlin, 2006, x+256 pp.  mathscinet  isi
    5. Comtet A., Majumdar S.N., Ouvry S., Sabhapandit S., “Integer partitions and exclusion statistics: limit shapes and the largest parts of young diagrams”, J. Stat. Mech. Theory Exp., 2007, P10001, 13 pp.  crossref  mathscinet  zmath  isi  elib
    6. Erlihson M.M., Granovsky B.L., “Limit shapes of Gibbs distributions on the set of integer partitions: the expansive case”, Ann. Inst. Henri Poincaré Probab. Stat., 44:5 (2008), 915–945  crossref  mathscinet  zmath  adsnasa  isi
    7. Alain Comtet, Satya N. Majumdar, Sanjib Sabhapandit, “A note on limit shapes of minimal difference partitions”, Zhurn. matem. fiz., anal., geom., 4:1 (2008), 24–32  mathnet  mathscinet  zmath
    8. F. Petrov, “Limits shapes of Young diagrams. Two elementary approaches”, J. Math. Sci. (N. Y.), 166:1 (2010), 63–74  mathnet  crossref
    9. Betz V., Ueltschi D., Velenik Y., “Random Permutations with Cycle Weights”, Ann Appl Probab, 21:1 (2011), 312–331  crossref  mathscinet  zmath  isi
    10. Yakubovich Yu., “Ergodicity of Multiplicative Statistics”, J. Comb. Theory Ser. A, 119:6 (2012), 1250–1279  crossref  mathscinet  zmath  isi  elib
    11. Eriksson K., Sjostrand J., “Limiting Shapes of Birth-and-Death Processes on Young Diagrams”, Adv. Appl. Math., 48:4 (2012), 575–602  crossref  mathscinet  zmath  isi  elib
    12. Dan Beltoft, Cédric Boutillier, Nathanaël Enriquez, “Random Young Diagrams in a Rectangular Box”, Mosc. Math. J., 12:4 (2012), 719–745  mathnet  mathscinet
    13. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. Funaki T., “Equivalence of Ensembles Under Inhomogeneous Conditioning and its Applications to Random Young Diagrams”, J. Stat. Phys., 154:1-2 (2014), 588–609  crossref  mathscinet  zmath  isi  elib
    15. Bogachev L.V., “Unified Derivation of the Limit Shape For Multiplicative Ensembles of Random Integer Partitions With Equiweighted Parts”, Random Struct. Algorithms, 47:2 (2015), 227–266  crossref  mathscinet  zmath  isi  elib
    16. P. S. Bocharov, A. P. Goryashko, “O sposobakh analiza igr razbienii”, UBS, 61 (2016), 6–40  mathnet  elib
    17. P. S. Bocharov, A. P. Goryashko, “O suboptimalnykh resheniyakh antagonisticheskikh igr razbienii”, UBS, 70 (2017), 6–24  mathnet  elib
    18. Bureaux J., Enriquez N., “Asymptotics of Convex Lattice Polygonal Lines With a Constrained Number of Vertices”, Isr. J. Math., 222:2 (2017), 515–549  crossref  zmath  isi  scopus
    19. Goryashko A., 2017 Seminar on Systems Analysis, Itm Web of Conferences, 10, eds. Nikulchev E., Bubnov G., E D P Sciences, 2017  crossref  isi
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