Skip to main content
SpringerLink
Log in

The Height and Range of Watermelons without Wall

(Extended Abstract)

  • Conference paper
Combinatorial Algorithms (IWOCA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5874))

Included in the following conference series:

Abstract

We determine the weak limit of the distribution of the random variables “height” and “range” on the set of p-watermelons without wall restriction as the number of steps tends to infinity. Additionally, we provide asymptotics for the moments of the random variable “height”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baik, J.: Random vicious walks and random matrices. Comm. Pure Appl. Math. 53(11), 1385–1410 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bonichon, N., Mosbah, M.: Nicolas Bonichon and Mohamed Mosbah. Watermelon uniform random generation with applications. Theoret. Comput. Sci. 307(2), 241–256 (2003); Random generation of combinatorial objects and bijective combinatorics

    Article  MATH  MathSciNet  Google Scholar 

  3. Chung, K.L.: Excursions in Brownian motion. Ark. Mat. 14(2), 155–177 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  4. Essam, J.W., Guttmann, A.J.: Vicious walkers and directed polymer networks in general dimensions. Phys. Rev. E (3) 52(6, part A), 5849–5862 (1995)

    Article  MathSciNet  Google Scholar 

  5. Feierl, T.: The height and range of watermelons without wall. Preprint, arXiv:math.CO/0806.0037

    Google Scholar 

  6. Feierl, T.: The height of watermelons with wall. Preprint, arXiv:math.CO/0802.2691

    Google Scholar 

  7. Fisher, M.E.: Walks, walls, wetting, and melting. J. Statist. Phys. 34(5-6), 667–729 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Flajolet, P., Gourdon, X., Dumas, P.: Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci. 144(1-2), 3–58 (1995); Special volume on mathematical analysis of algorithms

    Article  MATH  MathSciNet  Google Scholar 

  9. Gessel, I., Viennot, G.: Determinants, paths, and plane partitions (1989) (preprint), http://people.brandeis.edu/~gessel/homepage/papers/pp.pdf

  10. Gillet, F.: Asymptotic behaviour of watermelons. Preprint, arXiv:math.PR/0307204 (2003)

    Google Scholar 

  11. Guttmann, A.J., Owczarek, A.L., Viennot, X.G.: Vicious walkers and Young tableaux. I. Without walls. J. Phys. A 31(40), 8123–8135 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123(2), 225–280 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Katori, M., Izumi, M., Kobayashi, N.: Maximum distribution of bridges of noncolliding brownian paths. Phys. Rev. E 78, 051102 (2008) (electronic)

    Google Scholar 

  14. Katori, M., Tanemura, H.: Functional central limit theorems for vicious walkers. Stoch. Stoch. Rep. 75(6), 369–390 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kennedy, D.P.: The distribution of the maximum Brownian excursion. J. Appl. Probability 13(2), 371–376 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  16. Krattenthaler, C., Guttmann, A.J., Viennot, X.G.: Vicious walkers, friendly walkers and Young tableaux. II. With a wall. J. Phys. A 33(48), 8835–8866 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lindström, B.: On the vector representations of induced matroids. Bull. London Math. Soc. 5, 85–90 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nagao, T., Forrester, P.J.: Vicious random walkers and a discretization of Gaussian random matrix ensembles. Nuclear Phys. B 620(3), 551–565 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schehr, G., Majumdar, S.M., Comtet, A., Randon-Furling, J.: Exact distribution of the maximal height of watermelons. Preprint, arXiv:0807.0522 (2008)

    Google Scholar 

  20. Vervaat, W.: A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7, 143–149 (1979)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Nordbergstr. 15, 1090, Wien, Austria

    Thomas Feierl

Authors
  1. Thomas Feierl
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Theoretical Computer Science and Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Jiří Fiala  & Jan Kratochvíl  & 

  2. School of Electrical Engineering and Computer Science, The University of Newcastle, University Drive, NSW 2308, Callaghan, Australia

    Mirka Miller

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Feierl, T. (2009). The Height and Range of Watermelons without Wall. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10217-2_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10216-5

  • Online ISBN: 978-3-642-10217-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation