### Abstract

The Propp Machine is a deterministic process that simulates a random walk. Instead of distributing chips randomly, each position makes the chips move according to the walk's possible steps in a fixed order. A random walk is called Proppian if at each time at each position the number of chips differs from the expected value by at most a constant, independent of time or the initial configuration of chips. The simple walk where the possible steps are 1 or -1 each with probability p=12 is Proppian, with constant approximately 2.29. The equivalent simple walks on
^{Zd} are also Proppian. Here, we show the same result for a larger class of walks on Z, allowing an arbitrary number of possible steps with some constraint on their probabilities.

Original language | English (US) |
---|---|

Pages (from-to) | 349-361 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 5 |

DOIs | |

State | Published - Mar 6 2011 |

### Fingerprint

### Keywords

- Derandomization
- Propp machine
- Random walk

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*311*(5), 349-361. https://doi.org/10.1016/j.disc.2010.11.001

**Proppian random walks in ℤ.** / Freire, Juliana; Spencer, Joel.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 311, no. 5, pp. 349-361. https://doi.org/10.1016/j.disc.2010.11.001

}

TY - JOUR

T1 - Proppian random walks in ℤ

AU - Freire, Juliana

AU - Spencer, Joel

PY - 2011/3/6

Y1 - 2011/3/6

N2 - The Propp Machine is a deterministic process that simulates a random walk. Instead of distributing chips randomly, each position makes the chips move according to the walk's possible steps in a fixed order. A random walk is called Proppian if at each time at each position the number of chips differs from the expected value by at most a constant, independent of time or the initial configuration of chips. The simple walk where the possible steps are 1 or -1 each with probability p=12 is Proppian, with constant approximately 2.29. The equivalent simple walks on Zd are also Proppian. Here, we show the same result for a larger class of walks on Z, allowing an arbitrary number of possible steps with some constraint on their probabilities.

AB - The Propp Machine is a deterministic process that simulates a random walk. Instead of distributing chips randomly, each position makes the chips move according to the walk's possible steps in a fixed order. A random walk is called Proppian if at each time at each position the number of chips differs from the expected value by at most a constant, independent of time or the initial configuration of chips. The simple walk where the possible steps are 1 or -1 each with probability p=12 is Proppian, with constant approximately 2.29. The equivalent simple walks on Zd are also Proppian. Here, we show the same result for a larger class of walks on Z, allowing an arbitrary number of possible steps with some constraint on their probabilities.

KW - Derandomization

KW - Propp machine

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=79953784786&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953784786&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2010.11.001

DO - 10.1016/j.disc.2010.11.001

M3 - Article

VL - 311

SP - 349

EP - 361

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 5

ER -