Skip to Main content Skip to Navigation
Conference papers

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.
Complete list of metadata

Cited literature [16 references]  Display  Hide  Download

https://hal.inria.fr/hal-00542185
Contributor : Coordination Episciences Iam <>
Submitted on : Thursday, August 20, 2015 - 4:33:30 PM
Last modification on : Thursday, March 5, 2020 - 6:31:07 PM
Long-term archiving on: : Wednesday, April 26, 2017 - 10:10:58 AM

File

dmAM0103.pdf
Publisher files allowed on an open archive

Licence


Distributed under a Creative Commons Attribution - NonCommercial - ShareAlike 4.0 International License

Identifiers

  • HAL Id : hal-00542185, version 2

Citation

C. Banderier, P. Nicodème. Bounded discrete walks. pp.35-48. ⟨hal-00542185v2⟩

Share

Metrics

Record views

1532

Files downloads

1329