### Abstract

It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let G be a random graph on n vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on G, and show that, with high probability, it exhibits cutoff at time h^{−1} log n, where h is the asymptotic entropy for simple random walk on a Galton–Watson tree that approximates G locally. (Previously this was only known for typical starting points.) Furthermore, we show this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton–Watson tree.

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

Pages (from-to) | 1116-1130 |

Number of pages | 15 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 55 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2019 |

### Fingerprint

### Keywords

- Mixing times of Markov chains
- Nonbacktracking vs. Simple random walk
- Random graphs
- Random walks

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*55*(2), 1116-1130. https://doi.org/10.1214/18-AIHP911

**Comparing mixing times on sparse random graphs.** / Ben-Hamou, Anna; Lubetzky, Eyal; Peres, Yuval.

Research output: Contribution to journal › Article

*Annales de l'institut Henri Poincare (B) Probability and Statistics*, vol. 55, no. 2, pp. 1116-1130. https://doi.org/10.1214/18-AIHP911

}

TY - JOUR

T1 - Comparing mixing times on sparse random graphs

AU - Ben-Hamou, Anna

AU - Lubetzky, Eyal

AU - Peres, Yuval

PY - 2019/5/1

Y1 - 2019/5/1

N2 - It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let G be a random graph on n vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on G, and show that, with high probability, it exhibits cutoff at time h−1 log n, where h is the asymptotic entropy for simple random walk on a Galton–Watson tree that approximates G locally. (Previously this was only known for typical starting points.) Furthermore, we show this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton–Watson tree.

AB - It is natural to expect that nonbacktracking random walk will mix faster than simple random walks, but so far this has only been proved in regular graphs. To analyze typical irregular graphs, let G be a random graph on n vertices with minimum degree 3 and a degree distribution that has exponential tails. We determine the precise worst-case mixing time for simple random walk on G, and show that, with high probability, it exhibits cutoff at time h−1 log n, where h is the asymptotic entropy for simple random walk on a Galton–Watson tree that approximates G locally. (Previously this was only known for typical starting points.) Furthermore, we show this asymptotic mixing time is strictly larger than the mixing time of nonbacktracking walk, via a delicate comparison of entropies on the Galton–Watson tree.

KW - Mixing times of Markov chains

KW - Nonbacktracking vs. Simple random walk

KW - Random graphs

KW - Random walks

UR - http://www.scopus.com/inward/record.url?scp=85066120499&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066120499&partnerID=8YFLogxK

U2 - 10.1214/18-AIHP911

DO - 10.1214/18-AIHP911

M3 - Article

VL - 55

SP - 1116

EP - 1130

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 2

ER -