Bounded discrete walks

Abstract : This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work.
Type de document :
Communication dans un congrès
DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.35-48, 2010, DMTCS Proceedings
Liste complète des métadonnées

Littérature citée [16 références]  Voir  Masquer  Télécharger

https://hal.inria.fr/hal-00542185
Contributeur : Coordination Episciences Iam <>
Soumis le : jeudi 20 août 2015 - 16:33:30
Dernière modification le : jeudi 11 janvier 2018 - 06:19:44
Document(s) archivé(s) le : mercredi 26 avril 2017 - 10:10:58

Fichier

dmAM0103.pdf
Fichiers éditeurs autorisés sur une archive ouverte

Licence


Distributed under a Creative Commons Paternité - Pas d'utilisation commerciale - Partage selon les Conditions Initiales 4.0 International License

Identifiants

  • HAL Id : hal-00542185, version 2

Citation

C. Banderier, P. Nicodème. Bounded discrete walks. DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.35-48, 2010, DMTCS Proceedings. 〈hal-00542185v2〉

Partager

Métriques

Consultations de la notice

221

Téléchargements de fichiers

123