### Abstract

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o(1))log n/log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.

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

Pages (from-to) | 585-603 |

Number of pages | 19 |

Journal | Communications in Contemporary Mathematics |

Volume | 9 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2007 |

### Fingerprint

### Keywords

- Balls and bins
- Expanders
- Girth
- Mixing rate
- Non-backtracking random walks

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Contemporary Mathematics*,

*9*(4), 585-603. https://doi.org/10.1142/S0219199707002551

**Non-backtracking random walks mix faster.** / Alon, Noga; Benjamini, Itai; Lubetzky, Eyal; Sodin, Sasha.

Research output: Contribution to journal › Article

*Communications in Contemporary Mathematics*, vol. 9, no. 4, pp. 585-603. https://doi.org/10.1142/S0219199707002551

}

TY - JOUR

T1 - Non-backtracking random walks mix faster

AU - Alon, Noga

AU - Benjamini, Itai

AU - Lubetzky, Eyal

AU - Sodin, Sasha

PY - 2007/8

Y1 - 2007/8

N2 - We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o(1))log n/log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.

AB - We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than (1 + o(1))log n/log log n times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.

KW - Balls and bins

KW - Expanders

KW - Girth

KW - Mixing rate

KW - Non-backtracking random walks

UR - http://www.scopus.com/inward/record.url?scp=34547947639&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547947639&partnerID=8YFLogxK

U2 - 10.1142/S0219199707002551

DO - 10.1142/S0219199707002551

M3 - Article

AN - SCOPUS:34547947639

VL - 9

SP - 585

EP - 603

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 4

ER -