Document Open Access Logo

Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models

Authors Cyril Banderier , Markus Kuba , Stephan Wagner , Michael Wallner

PDF
Thumbnail PDF

File

LIPIcs.AofA.2024.7.pdf
  • Filesize: 0.85 MB
  • 18 pages

Document Identifiers

Author Details

Cyril Banderier
  • Laboratoire d'Informatique de Paris Nord, Université Sorbonne Paris Nord, Villetaneuse, France
Markus Kuba
  • Department Applied Mathematics & Physics, University of Applied Sciences - Technikum Wien, Austria
Stephan Wagner
  • Institute of Discrete Mathematics, TU Graz, Austria
  • Department of Mathematics, Uppsala Universitet, Sweden
Michael Wallner
  • Institut für Diskrete Mathematik und Geometrie, TU Wien, Austria

Funding

  • Banderier, Cyril: supported by the French-Austrian PHC Amadeus project "Asymptotic behaviour of combinatorial structures".
  • Wagner, Stephan: supported by the Swedish research council (VR), grant 2022-04030.
  • Wallner, Michael: supported by the Austrian Science Fund (FWF): P 34142 and OeAD WTZ project FR 01/2023.

Acknowledgements

The second author warmly thanks Thomas Feierl for many discussions about watermelons and Paul Schreivogl for discussions on partition functions. We also thank Christian Krattenthaler for drawing our attention to the open problem of the phase transition of contacts in watermelons, and the two referees for their feedback. Last but not least, we are pleased to dedicate this excursion into the realm of composition schemes and limit laws to one of the greatest experts of this realm, namely Michael Drmota, at the occasion of his 60th birthday!

Cite AsGet BibTex

Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner. Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.7

Abstract

Composition schemes are ubiquitous in combinatorics, statistical mechanics and probability theory. We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of q-enumeration (this is a model where objects with a parameter of value k have a Gibbs measure/Boltzmann weight q^k). For structures enumerated by a composition scheme, we prove a phase transition for any parameter having such a Gibbs measure: for a critical value q = q_c, the limit law of the parameter is a two-parameter Mittag-Leffler distribution, while it is Gaussian in the supercritical regime (q > q_c), and it is a Boltzmann distribution in the subcritical regime (0 < q < q_c). We apply our results to fundamental statistics of lattice paths and quarter-plane walks. We also explain previously observed limit laws for pattern-restricted permutations, and a phenomenon uncovered by Krattenthaler for the wall contacts in watermelons.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Distribution functions
Keywords
  • Composition schemes
  • q-enumeration
  • generating functions
  • Gibbs distribution
  • phase transitions
  • limit laws
  • Mittag-Leffler distribution
  • chi distribution
  • Boltzmann distribution

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Florian Aigner. A new determinant for the q-enumeration of alternating sign matrices. Journal of Combinatorial Theory, Series A, 180:27 pp., 2021. URL: https://arxiv.org/abs/1810.08022.
  2. George E. Andrews. The theory of compositions. IV: Multicompositions. The Mathematics Student, 2007:25-31, 2007. URL: https://georgeandrews1.github.io/publications.html.
  3. Cyril Banderier and Michael Drmota. Formulae and asymptotics for coefficients of algebraic functions. Combinatorics, Probability and Computing, 24(1):1-53, 2015. URL: http://lipn.univ-paris13.fr/~banderier/Papers/Alg.pdf.
  4. Cyril Banderier and Philippe Flajolet. Basic analytic combinatorics of directed lattice paths. Theoretical Computer Science, 281(1-2):37-80, 2002. URL: https://doi.org/10.1016/S0304-3975(02)00007-5.
  5. Cyril Banderier, Philippe Flajolet, Gilles Schaeffer, and Michèle Soria. Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Structures & Algorithms, 19(3-4):194-246, 2001. URL: https://lipn.univ-paris13.fr/~banderier/Papers/rsa.pdf.
  6. Cyril Banderier, Markus Kuba, and Michael Wallner. Phase transitions of composition schemes: Mittag-Leffler and mixed-Poisson distributions. Annals of Applied probability, 2024. 53 pp., to appear. URL: https://doi.org/10.48550/arXiv.2103.03751.
  7. Nicholas R. Beaton, A. L. Owczarek, and Andrew Rechnitzer. Exact solution of some quarter plane walks with interacting boundaries. Electronic Journal of Combinatorics, 26(3 (P53)), 2019. URL: https://doi.org/10.37236/8024.
  8. Edward A. Bender. Central and local limit theorems applied to asymptotic enumeration. Journal of Combinatorial Theory, Series A, 15:91-111, 1973. URL: https://doi.org/10.1016/0097-3165(73)90038-1.
  9. Alexei Borodin, Vadim Gorim, and Eric M. Rains. q-distributions on boxed plane partitions. Selecta Mathematica, 16(4):731-789, 2010. URL: https://doi.org/10.1007/s00029-010-0034-y.
  10. Mireille Bousquet-Mélou. Three osculating walkers. Journal of Physics: Conference Series, 42:35-46, 2006. URL: https://doi.org/10.1088/1742-6596/42/1/005.
  11. Richard Brak, John W. Essam, and Aleksander L. Owczarek. New results for directed vesicles and chains near an attractive wall. Journal of Statistical Physics, 93:155-192, 1998. URL: https://doi.org/10.1023/B:JOSS.0000026731.35385.93.
  12. Richard Brak, John W. Essam, and Aleksander L. Owczarek. Scaling analysis for the adsorption transition in a watermelon network of n directed non-intersecting walks. Journal of Statistical Physics, 102:997-1017, 2001. URL: https://doi.org/10.1023/A:1004819507352.
  13. Aksheytha Chelikavada and Hugo Panzo. Limit theorems for fixed point biased permutations avoiding a pattern of length three. arXiv, 2023. URL: https://arxiv.org/abs/2311.04623.
  14. Mihai Ciucu and Christian Krattenthaler. Enumeration of lozenge tilings of hexagons with cut off corners. Journal of Combinatorial Theory, Series A, 100:201-231, 2002. URL: https://doi.org/10.1006/jcta.2002.3288.
  15. Michael Drmota, Omer Giménez, and Marc Noy. Degree distribution in random planar graphs. Journal of Combinatorial Theory. Series A, 118(7):2102-2130, 2011. URL: https://doi.org/10.1016/j.jcta.2011.04.010.
  16. Michael Drmota and Michèle Soria. Marking in combinatorial constructions: generating functions and limiting distributions. Theoretical Computer Science, 144(1-2):67-99, 1995. URL: https://doi.org/10.1016/0304-3975(94)00294-S.
  17. Michael Drmota and Michèle Soria. Images and preimages in random mappings. SIAM Journal on Discrete Mathematics, 10(2):246-269, 1997. URL: https://doi.org/10.1137/S0895480194268421.
  18. Philippe Duchon, Philippe Flajolet, Guy Louchard, and Gilles Schaeffer. Boltzmann samplers for the random generation of combinatorial structures. Combinatorics, Probability and Computing, 13(4-5):577-625, 2004. URL: https://doi.org/10.1017/S0963548304006315.
  19. Sergi Elizalde. Multiple pattern avoidance with respect to fixed points and excedances. Electronic Journal of Combinatorics, 11(1):40 pp., 2004. URL: https://doi.org/10.37236/1804.
  20. Thomas Feierl. The height of watermelons with wall. Journal of Physics A: Mathematical and Theoretical, 45(9), 2012. URL: https://doi.org/10.1088/1751-8113/45/9/095003.
  21. Thomas Feierl. The height and range of watermelons without wall. European Journal of Combinatorics, 34(1):138-154, 2013. URL: https://doi.org/10.1016/j.ejc.2012.07.021.
  22. Thomas Feierl. Asymptotics for the number of walks in a Weyl chamber of type B. Random Structures & Algorithms, 45(2):261-305, 2014. URL: https://doi.org/10.1002/rsa.20467.
  23. James Allen Fill, Philippe Flajolet, and Nevin Kapur. Singularity analysis, Hadamard products, and tree recurrences. Journal of Computational and Applied Mathematics, 174(2):271-313, 2005. URL: https://doi.org/10.1016/j.cam.2004.04.014.
  24. Michael E. Fisher. Walks, walls, wetting, and melting. Journal of Statistical Physics, 34:667-729, 1984. URL: https://doi.org/10.1007/BF01009436.
  25. Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. URL: http://algo.inria.fr/flajolet/Publications/book.pdf.
  26. Philippe Flajolet and Michèle Soria. General combinatorial schemas: Gaussian limit distributions and exponential tails. Discrete Mathematics, 114(1-3):159-180, 1993. URL: https://doi.org/10.1016/0012-365X(93)90364-Y.
  27. Manosij Ghosh Dastidar and Michael Wallner. Bijections and congruences involving lattice paths and integer compositions. arXiv, 2024. A short version of this work was published in the proceedings of GASCOM 2024, https://cgi.cse.unsw.edu.au/~eptcs/paper.cgi?GASCom2024.22. URL: https://arxiv.org/pdf/2402.17849.
  28. Xavier Gourdon. Largest component in random combinatorial structures. Discrete Mathematics, 180(1-3):185-209, 1998. URL: https://doi.org/10.1016/S0012-365X(97)00115-5.
  29. Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley Publishing Company, second edition, 1994. URL: https://www-cs-faculty.stanford.edu/~knuth/gkp.html.
  30. Brian Hopkins and Stéphane Ouvry. Combinatorics of multicompositions. In Combinatorial and additive number theory IV, pages 307-321. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-67996-5_16.
  31. Sam Hopkins, Alexander Lazar, and Svante Linusson. On the q-enumeration of barely set-valued tableaux and plane partitions. European Journal of Combinatorics, 2021. URL: https://doi.org/10.1016/j.ejc.2023.103760.
  32. Hsien-Kuei Hwang. On convergence rates in the central limit theorems for combinatorial structures. European Journal of Combinatorics, 19:329-343, 1998. URL: https://doi.org/10.1006/eujc.1997.0179.
  33. Christian Krattenthaler. Watermelon configurations with wall interaction: exact and asymptotic results. Journal of Physics: Conference Series, 42:179-212, 2006. URL: https://doi.org/10.1088/1742-6596/42/1/017.
  34. Christian Krattenthaler, Anthony J. Guttmann, and Xavier G. Viennot. Vicious walkers, friendly walkers and Young tableaux II: with a wall. Journal of Physics A: Mathematical and General, 33:8835-8866, 2000. URL: https://doi.org/10.1088/0305-4470/33/48/318.
  35. Sara Kropf and Stephan Wagner. On q-quasiadditive and q-quasimultiplicative functions. Electronic Journal of Combinatorics, 24(1):Paper No. 1.60, 22 pages, 2017. URL: https://doi.org/10.37236/6373.
  36. Colin L. Mallows. Non-null ranking models I. Biometrika, 44:114-130, 1957. URL: https://doi.org/10.2307/2333244.
  37. Aleksander L. Owczarek and Thomas Prellberg. Exact solution of pulled, directed vesicles with sticky walls in two dimensions. Journal of Mathematical Physics, 60:8 pp., 2019. URL: https://doi.org/10.1063/1.5083149.
  38. Valerie Roitner. Contacts and returns in 2-watermelons without wall. Bulletin of the Institute of Combinatorics and its Applications, 89, 2020. URL: http://bica.the-ica.org/Volumes/89/Reprints/BICA2020-01-Reprint.pdf.
  39. Benedikt Stufler. Gibbs partitions: a comprehensive phase diagram. Annales de l'Institut Henri Poincaré - Probabilités et Statistiques, 2023. To appear. URL: https://arxiv.org/abs/2204.06982.
  40. Rami Tabbara, Aleksander L. Owczarek, and Andrew Rechnitzer. An exact solution of two friendly interacting directed walks near a sticky wall. Journal of Physics A: Mathematical and Theoretical, 47(1), 2014. URL: https://doi.org/10.1088/1751-8113/47/1/015202.
  41. Antoine Vella. Pattern avoidance in permutations: Linear and cyclic orders. Electronic Journal of Combinatorics, 9(2):43 pp., 2003. URL: https://doi.org/10.37236/1690.
  42. Michael Wallner. A half-normal distribution scheme for generating functions. European Journal of Combinatorics, 87:103138, 2020. URL: https://doi.org/10.1016/j.ejc.2020.103138.
  43. Shuang Wu. Sachdev-Ye-Kitaev model with an extra diagonal perturbation: phase transition in the eigenvalue spectrum. Journal of Physics A: Mathematical and Theoretical, 55(41):415207, 2022. URL: https://doi.org/10.1088/1751-8121/ac93cd.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact