Hostname: page-component-f554764f5-68cz6 Total loading time: 0 Render date: 2025-04-09T18:21:12.390Z Has data issue: false hasContentIssue false
  • English
  • Français

Area Limit Laws for Symmetry Classes of Staircase Polygons

Published online by Cambridge University Press:  27 January 2010

U. SCHWERDTFEGER ,
C. RICHARD  and
B. THATTE
U. SCHWERDTFEGER
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany (e-mail: [email protected])
C. RICHARD
Affiliation:
Department für Mathematik, Universität Erlangen–Nürnberg, Bismarckstraße 1 1/2, 91054 Erlangen, Germany (e-mail: [email protected])
B. THATTE
Affiliation:
Mathematics and Computer Science Building, University of Canterbury, Private Bag 40800, Christchurch, New Zealand (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive area limit laws for the various symmetry classes of staircase polygons on the square lattice, in a uniform ensemble where, for fixed perimeter, each polygon occurs with the same probability. This complements a previous study by Leroux and Rassart, where explicit expressions for the area and perimeter generating functions of these classes have been derived.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 3 , May 2010 , pp. 441 - 461
Copyright
Copyright © Cambridge University Press 2010

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.)

Article purchase

Temporarily unavailable

References

[1]Aldous, D. (1991) The continuum random tree I. Ann. Probab. 19 1–28.CrossRefGoogle Scholar
[2]Aldous, D. (1991) The continuum random tree II: An overview. In Stochastic Analysis, Vol. 167 of London Math. Soc. Lecture Notes, Cambridge University Press, Cambridge, pp. 23–70.CrossRefGoogle Scholar
[3]Aldous, D. (1993) The continuum random tree III. Ann. Probab. 21 248–289.CrossRefGoogle Scholar
[4]Belkin, B. (1972) An invariance principle for conditioned recurrent random walk attracted to a stable law. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 21 45–64.CrossRefGoogle Scholar
[5]Billingsley, P. (1995) Probability and Measure, 3rd edn, Wiley, New York.Google Scholar
[6]Bousquet-Mélou, M. (1996) A method for the enumeration of various classes of column-convex polygons. Discrete Math. 154 1–25.CrossRefGoogle Scholar
[7]Bousquet-Mélou, M. and Viennot, X. G. (1992) Empilements de segments et q-énumération de polyominoes convexes dirigés. J. Combin. Theory Ser. A 60 196–224.CrossRefGoogle Scholar
[8]Chung, K. L. (1974) A Course in Probability Theory, 2nd edn, Academic Press, New York.Google Scholar
[9]Cifarelli, D. M. and Regazzini, E. (1973) Sugli stimatori non distorti a varianza minima di una classe di funzioni di probabilità troncate. Giorn. Econom. Ann. Econom. 32 492–501.Google Scholar
[10]Constantine, G. M. and Savits, T. H. (1996) A multivariate Faà di Bruno formula with applications. Trans. Amer. Math. Soc. 348 503–520.CrossRefGoogle Scholar
[11]Delest, M.-P., Gouyou-Beauchamps, D. and Vauquelin, B. (1987) Enumeration of parallelogram polyominoes with given bond and site perimeter. Graphs Combin. 3 325–339.CrossRefGoogle Scholar
[12]Delest, M.-P. and Viennot, X. G. (1984) Algebraic languages and polyominoes enumeration. Theor. Comput. Sci. 34 169–206.CrossRefGoogle Scholar
[13]Drmota, M. (2004) Stochastic analysis of tree-like data structures. Proc. Royal Soc. London Ser. A 460 271–307.CrossRefGoogle Scholar
[14]Duchon, P. (1999) Q-grammars and wall polyominoes. Ann. Combin. 3 311–321.CrossRefGoogle Scholar
[15]Feller, W. (1970) An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn, Wiley, New York.Google Scholar
[16]Fill, J. A., Flajolet, P. and Kapur, N. (2005) Singularity analysis, Hadamard products, and tree recurrences. J. Comput. Appl. Math. 174 271–313.CrossRefGoogle Scholar
[17]Flajolet, P. and Louchard, G. (2001) Analytic variations on the Airy distribution. Algorithmica 31 361–377.CrossRefGoogle Scholar
[18]Flajolet, P. and Odlyzko, A. M. (1990) Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216–240.CrossRefGoogle Scholar
[19]Flajolet, P., Poblete, P. and Viola, A. (1998) On the analysis of linear probing hashing. Algorithmica 22 37–71.CrossRefGoogle Scholar
[20]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[21]Gessel, I. (1980) A noncommutative generalization and q-analog of the Lagrange inversion formula. Trans. Amer. Math. Soc. 257 455–482.Google Scholar
[22]Gittenberger, B. (1999) On the contour of random trees. SIAM J. Discrete Math. 12 434–458.CrossRefGoogle Scholar
[23]Gouyou-Beauchamps, D. and Leroux, P. (2005) Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice. Theor. Comput. Sci. 346 307–334.CrossRefGoogle Scholar
[24]Gutjahr, W. and Pflug, G. C. (1992) The asymptotic contour process of a binary tree is a Brownian excursion. Stochastic Process. Appl. 41 69–89.CrossRefGoogle Scholar
[25]Guttmann, A. J., ed. (2009) Polygons, Polyominoes and Polycubes, Vol. 775 of Lecture Notes in Physics, Springer, Berlin/Heidelberg. pp. 247–299.CrossRefGoogle Scholar
[26]Iglehart, D. L. (1974) Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2 608–619.CrossRefGoogle Scholar
[27]Janson, S. (2007) Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Prob. Surv. 4 80–145.CrossRefGoogle Scholar
[28]Kearney, M. J., Majumdar, S. N. and Martin, R. J. (2007) The first-passage area for drifted Brownian motion and the moments of the Airy distribution. J. Phys. A: Math. Theor. 40 F863–F869.CrossRefGoogle Scholar
[29]Leroux, P. and Rassart, É. (2001) Enumeration of symmetry classes of parallelogram polyominoes. Ann. Sci. Math. Québec 25 71–90.Google Scholar
[30]Leroux, P., Rassart, É. and Robitaille, A. (1998) Enumeration of symmetry classes of convex polyominoes in the square lattice. Adv. Appl. Math. 21 343–380.CrossRefGoogle Scholar
[31]Lin, K. Y. (2007) Rigorous derivation of the radius of gyration generating function for staircase polygons. J. Phys. A: Math. Theor. 40 1419–1426.CrossRefGoogle Scholar
[32]Louchard, G. (1984) The Brownian excursion area: A numerical analysis. Comput. Math. Appl. 10 413–417.CrossRefGoogle Scholar
[33]Louchard, G. (1986) Erratum: ‘The Brownian excursion area: A numerical analysis’. Comput. Math. Appl. 12 375.Google Scholar
[34]Madras, N. and Slade, G. (1993) The Self-Avoiding Walk, Birkhäuser, Boston.Google Scholar
[35]Meir, A. and Moon, J. W. (1978) On the altitude of nodes in random trees. Canad. J. Math. 30 997–1015.CrossRefGoogle Scholar
[36]Nguyễn Thế, M. (2003) Area of Brownian Motion with Generatingfunctionology. In Discrete Random Walks: DRW'03 (Banderier, C. and Krattenthaler, C., eds), Discr. Math. and Theoret. Comput. Sci. Proceedings AC, pp. 229–242.CrossRefGoogle Scholar
[37]Perman, M. and Wellner, J. A. (1996) On the distribution of Brownian areas. Ann. Appl. Probab. 6 1091–1111.CrossRefGoogle Scholar
[38]Prellberg, T. and Brak, R. (1995) Critical exponents from non-linear functional equations for partially directed cluster models. J. Statist. Phys. 78 701–730.CrossRefGoogle Scholar
[39]Revuz, D. and Yor, M. (1998) Continuous Martingales and Brownian Motion, Springer, New York.Google Scholar
[40]Richard, C. (2002) Scaling behaviour of two-dimensional polygon models. J. Statist. Phys. 108 459–493.CrossRefGoogle Scholar
[41]Richard, C. (2006) Staircase polygons: Moments of diagonal lengths and column heights. J. Phys.: Conf. Ser. 42 239–257.Google Scholar
[42]Richard, C. (2009) On q-functional equations and excursion moments. Discrete Math. 309 207–230.CrossRefGoogle Scholar
[43]Richard, C. (2009) Limit distributions and scaling functions. In Polygons, Polyominoes and Polycubes (Guttmann, A. J., ed.), Vol. 775 of Lecture Notes in Physics, Springer, Berlin/Heidelberg, pp. 247–299.CrossRefGoogle Scholar
[44]Richard, C., Guttmann, A. J. and Jensen, I. (2001) Scaling function and universal amplitude combinations for self-avoiding polygons. J. Phys. A: Math. Gen. 34 L495–L501.CrossRefGoogle Scholar
[45]Shepp, L. A. (1982) On the integral of the absolute value of the pinned Wiener process, Ann. Probab. 10 234–239.CrossRefGoogle Scholar
[46]Shepp, L. A. (1991) Acknowledgment of priority: ‘On the integral of the absolute value of the pinned Wiener process’. Ann. Probab. 19 1397.CrossRefGoogle Scholar
[47]Takács, L. (1991) A Bernoulli excursion and its various applications. Adv. Appl. Probab. 23 557–585.CrossRefGoogle Scholar
[48]Takács, L. (1995) Limit distributions for the Bernoulli meander. J. Appl. Probab. 32 375–395.CrossRefGoogle Scholar
[49]Vöge, M. and Guttmann, A. J. (2002) On the number of benzenoid hydrocarbons. J. Chem. Inf. Comput. Sci. 42 456–466.CrossRefGoogle ScholarPubMed