### Abstract

We give a lower bound on the spectral gap for a class of binary collision processes. In ALEA Lat. Am. J. Probab. Math. Stat. 4, 205-222 (2008), Caputo showed that, for a class of binary collision processes given by simple averages on the complete graph, the analysis of the spectral gap of an N-component system is reduced to that of the same system for N = 3. In this paper, we give a comparison technique to reduce the analysis of the spectral gap of binary collision processes given by simple averages on d-dimensional lattice to that on the complete graph. We also give a comparison technique to reduce the analysis of the spectral gap of binary collision processes which are not given by simple averages to that given by simple averages. Combining them with Caputo's result, we give a new and elementary method to obtain spectral gap estimates. The method applies to a number of binary collision processes on the complete graph and also on d-dimensional lattice, including a class of energy exchange models which was recently introduced in arXiv:1109.2356, and zero-range processes.

Original language | English |
---|---|

Title of host publication | Springer Proceedings in Mathematics and Statistics |

Publisher | Springer New York LLC |

Pages | 543-560 |

Number of pages | 18 |

Volume | 40 |

ISBN (Print) | 9781447148623 |

DOIs | |

Publication status | Published - 2013 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Springer Proceedings in Mathematics and Statistics*(Vol. 40, pp. 543-560). Springer New York LLC. https://doi.org/10.1007/978-1-4471-4863-0_23

**On the spectral gap of the kac walk and other binary collision processes on d-dimensional lattice.** / Sasada, Makiko.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Springer Proceedings in Mathematics and Statistics.*vol. 40, Springer New York LLC, pp. 543-560. https://doi.org/10.1007/978-1-4471-4863-0_23

}

TY - GEN

T1 - On the spectral gap of the kac walk and other binary collision processes on d-dimensional lattice

AU - Sasada, Makiko

PY - 2013

Y1 - 2013

N2 - We give a lower bound on the spectral gap for a class of binary collision processes. In ALEA Lat. Am. J. Probab. Math. Stat. 4, 205-222 (2008), Caputo showed that, for a class of binary collision processes given by simple averages on the complete graph, the analysis of the spectral gap of an N-component system is reduced to that of the same system for N = 3. In this paper, we give a comparison technique to reduce the analysis of the spectral gap of binary collision processes given by simple averages on d-dimensional lattice to that on the complete graph. We also give a comparison technique to reduce the analysis of the spectral gap of binary collision processes which are not given by simple averages to that given by simple averages. Combining them with Caputo's result, we give a new and elementary method to obtain spectral gap estimates. The method applies to a number of binary collision processes on the complete graph and also on d-dimensional lattice, including a class of energy exchange models which was recently introduced in arXiv:1109.2356, and zero-range processes.

AB - We give a lower bound on the spectral gap for a class of binary collision processes. In ALEA Lat. Am. J. Probab. Math. Stat. 4, 205-222 (2008), Caputo showed that, for a class of binary collision processes given by simple averages on the complete graph, the analysis of the spectral gap of an N-component system is reduced to that of the same system for N = 3. In this paper, we give a comparison technique to reduce the analysis of the spectral gap of binary collision processes given by simple averages on d-dimensional lattice to that on the complete graph. We also give a comparison technique to reduce the analysis of the spectral gap of binary collision processes which are not given by simple averages to that given by simple averages. Combining them with Caputo's result, we give a new and elementary method to obtain spectral gap estimates. The method applies to a number of binary collision processes on the complete graph and also on d-dimensional lattice, including a class of energy exchange models which was recently introduced in arXiv:1109.2356, and zero-range processes.

UR - http://www.scopus.com/inward/record.url?scp=84883385029&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883385029&partnerID=8YFLogxK

U2 - 10.1007/978-1-4471-4863-0_23

DO - 10.1007/978-1-4471-4863-0_23

M3 - Conference contribution

AN - SCOPUS:84883385029

SN - 9781447148623

VL - 40

SP - 543

EP - 560

BT - Springer Proceedings in Mathematics and Statistics

PB - Springer New York LLC

ER -