Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-07T10:59:14.216Z Has data issue: false hasContentIssue false
  • English
  • Français

Random Sampling of Plane Partitions

Published online by Cambridge University Press:  02 November 2009

OLIVIER BODINI ,
ÉRIC FUSY  and
CARINE PIVOTEAU
OLIVIER BODINI
Affiliation:
LIP6 – Équipe Spiral, 104 avenue du Président Kennedy, 75016 Paris, France (e-mail: bodini@calfor.lip6.frcarine.pivoteau@calfor.lip6.fr)
ÉRIC FUSY
Affiliation:
LIX, École Polytechnique, 91128 Palaiseau Cedex, France (e-mail: fusy@lix.polytechnique.fr)
CARINE PIVOTEAU
Affiliation:
LIP6 – Équipe Spiral, 104 avenue du Président Kennedy, 75016 Paris, France (e-mail: bodini@calfor.lip6.frcarine.pivoteau@calfor.lip6.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

This article presents uniform random generators of plane partitions according to size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is O(n(ln n)3) in approximate-size sampling and O(n4/3) in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for (a × b)-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 2 , March 2010 , pp. 201 - 226
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bender, E. A. and Knuth, D. E. (1972) Enumeration of plane partitions. J. Combin. Theory Ser. A 13 40–54.CrossRefGoogle Scholar
[2]Bressoud, D. M. (1999) Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, Cambridge University Press, New York.CrossRefGoogle Scholar
[3]Cerf, R. and Kenyon, R. (2001) The low-temperature expansion of the Wulff crystal in the 3D Ising model. Comm. Math. Phys. 222 147–179.CrossRefGoogle Scholar
[4]Cohn, H., Larsen, M. and Propp, J. (1998) The shape of a typical boxed plane partition. New York J. Math. 4 137–166.Google Scholar
[5]Duchon, P., Flajolet, P., Louchard, G. and Schaeffer, G. (2004) Boltzmann samplers for the random generation of combinatorial structures. Combin. Probab. Comput. 13 577–625.CrossRefGoogle Scholar
[6]Flajolet, P., Fusy, É. and Pivoteau, C. (2007) Boltzmann sampling of unlabelled structures. In Proc. 4th Workshop on Analytic Algorithms and Combinatorics: ANALCO'07 (New Orleans), SIAM, pp. 201–211.Google Scholar
[7]Flajolet, P., Gourdon, X. and Dumas, P. (1995) Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 3–58.CrossRefGoogle Scholar
[8]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.CrossRefGoogle Scholar
[9]Flajolet, P., Zimmerman, P. and Van Cutsem, B. (1994) A calculus for the random generation of labelled combinatorial structures. Theoret. Comput. Sci. 132 1–35.CrossRefGoogle Scholar
[10]Fusy, É. (2007) Uniform random sampling of planar graphs in linear time. Random Struct. Alg., to appear. arXiv:0705.1287Google Scholar
[11]Gansner, E. R. (1981) Matrix correspondences of plane partitions. Pac. J. Math. 92 295–315.CrossRefGoogle Scholar
[12]Knuth, D. E. (1970) Permutations, matrices, and generalized Young tableaux. Pacific J. Math. 34 709–727.CrossRefGoogle Scholar
[13]Krattenthaler, C. (1999) Another involution principle-free bijective proof of Stanley's hook-content formula. J. Combin. Theory Ser. A 88 66–92.CrossRefGoogle Scholar
[14]Ma, J. and Janse van Rensburg, E. J. (2005) Rectangular vesicles in three dimensions. J. Phys. A 38 4115–4147.CrossRefGoogle Scholar
[15]MacMahon, P. A. (1912) Memoir on the theory of the partitions of numbers VI: Partitions in two-dimensional space, to which is added an adumbration of the theory of partitions in three-dimensional space. Phil. Trans. Roy. Soc. London Ser. A 211 345–373.Google Scholar
[16]Maeda, T. and Nakatsu, T. (2007) Amoebas and instantons. Int. J. Mod. Phys. A 22 937–984.CrossRefGoogle Scholar
[17]Meinardus, G. (1954) Asymptotische Aussagen über Partitionen. Math. Z. 59 388–398.CrossRefGoogle Scholar
[18]Mutafchiev, L. (2006) The size of the largest part of random plane partitions of large integers. Integers 6 A13.Google Scholar
[19]Mutafchiev, L. and Kamenov, E. (2006) Asymptotic formula for the number of plane partitions of positive integers. CR Acad. Bulgare Sci. 59 361–366.Google Scholar
[20]Nijenhuis, A. and Wilf, H. S. (1978) Combinatorial Algorithms, 2nd edn, Academic Press.Google Scholar
[21]Novelli, J.-C., Pak, I. and Stoyanovskii, A. V. (1997) A direct bijective proof of the hook-length formula. Discrete Math. Theor. Comput. Sci. 1 53–67.CrossRefGoogle Scholar
[22]Okounkov, A. and Reshetikhin, N. (2003) Correlation function of Schur process with application to local geometry of a random 3-dimensional young diagram. J. Amer. Math. Soc. 16 581–603.CrossRefGoogle Scholar
[23]Okounkov, A. and Reshetikhin, N. (2007) Random skew plane partitions and the Pearcey process. Comm. Math. Phys. 269.Google Scholar
[24]Pak, I. (2001/02) Hook length formula and geometric combinatorics. Séminaire Lotharingien de Combinatoire 46 #B46f.Google Scholar
[25]Pivoteau, C., Salvy, B. and Soria, M. (2008) Boltzmann oracle for combinatorial systems. In Algorithms, Trees, Combinatorics and Probabilities: Proc. 5th Colloquium on Mathematics and Computer Science, DMTCS Proceedings, pp. 475–488.Google Scholar
[26]Propp, J. (1997) Generating random elements of finite distributive lattices. Electron. J. Combin. 4 #15.CrossRefGoogle Scholar
[27]Vershik, A. M. (1996) Statistical mechanics of combinatorial partitions, and their limit configurations. Funct. Anal. Appl. 30 90–105.CrossRefGoogle Scholar
[28]Wilson, D. B. (1997) Determinant algorithms for random planar structures. In SODA ’97: Proc. 8th Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, USA, pp. 258–267.Google Scholar
[29]Wright, E. M. (1931) Asymptotic partition formulae I: Plane partitions. Quart. J. Math. Oxford Ser. 2 177–189.CrossRefGoogle Scholar
[30]Young, A. (1901) On quantitative substitutional analysis. Proc. Lond. Math. Soc. 33 97–146.Google Scholar
[31]Zeilberger, D. (1996) Proof of the alternating sign matrix conjecture. Electron. J. Combin. 3 #13.Google Scholar