Discrete Mathematics & Theoretical Computer Science |

- 1 Institut für Geometrie
- 2 Department of Computer Science [Purdue]

We study a random walk with positive drift in the first quadrant of the plane. For a given connected region $\mathcal{C}$ of the first quadrant, we analyze the number of paths contained in $\mathcal{C}$ and the first exit time from $\mathcal{C}$. In our case, region $\mathcal{C}$ is bounded by two crossing lines. It is noted that such a walk is equivalent to a path in a tree from the root to a leaf not exceeding a given height. If this tree is the parsing tree of the Tunstall or Khodak variable-to-fixed code, then the exit time of the underlying random walk corresponds to the phrase length not exceeding a given length. We derive precise asymptotics of the number of paths and the asymptotic distribution of the exit time. Even for such a simple walk, the analysis turns out to be quite sophisticated and it involves Mellin transforms, Tauberian theorems, and infinite number of saddle points.

Source: HAL:hal-01184773v1

Volume: DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07)

Section: Proceedings

Published on: January 1, 2007

Imported on: May 10, 2017

Keywords: Random walk in the plane,Tunstall's code,number of paths,exit time,Khodak code,Mellin transform,Tauberian theorems.,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]

Funding:

- Source : OpenAIRE Graph
*Combinatorial &Probabilistic Methods for Biol Sequences*; Funder: National Institutes of Health; Code: 5R01GM068959-04

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