Bounded Discrete Walks

Cyril Banderier (1) , Pierre Nicodeme (2, 3)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 AMIB - Algorithms and Models for Integrative Biology

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 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.

Mots clés: Random walks; analytic combinatorics; Brownian bridge; Rayleigh distribution; genes ranking statistics

Domaines: Mathématique discrète [cs.DM]

Cyril Banderier, Pierre Nicodeme. Bounded Discrete Walks. AofA2010 - Conference on Analysis of Algorithms 2010, Jun 2010, Vienna, Austria. pp.35-48. ⟨hal-00542185v1⟩