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On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution

Authors Andrei Asinowski , Cyril Banderier

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Author Details

Andrei Asinowski
  • University of Klagenfurt, Austria
Cyril Banderier
  • Université Paris 13 (LIPN, UMR CNRS 7030), France

Funding

  • Asinowski, Andrei: supported by the Austrian Science Fund (FWF), in the framework of the project https://www.math.aau.at/P28466/ (P 28466-N35).

Acknowledgements

We thank the referees for their feedback.

Cite AsGet BibTex

Andrei Asinowski and Cyril Banderier. On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.AofA.2020.1

Abstract

In this article, we analyse the joint distribution of some given set of patterns in fundamental combinatorial structures such as words and random walks (directed lattice paths on ℤ²). Our method relies on a vectorial generalization of the classical kernel method, and on a matricial generalization of the autocorrelation polynomial (thus extending the univariate case of Guibas and Odlyzko). This gives access to the multivariate generating functions, for walks, meanders (walks constrained to be above the x-axis), and excursions (meanders constrained to end on the x-axis). We then demonstrate the power of our methods by obtaining closed-form expressions for an infinite family of models, in terms of simple combinatorial quantities. Finally, we prove that the joint distribution of the patterns in walks/bridges/excursions/meanders satisfies a multivariate Gaussian limit law.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Distribution functions
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Grammars and context-free languages
Keywords
  • Lattice path
  • Dyck path
  • Motzkin path
  • generating function
  • algebraic function
  • kernel method
  • context-free grammar
  • multivariate Gaussian distribution

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