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.
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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
T3 - Springer Proceedings in Mathematics and Statistics
SP - 543
EP - 560
BT - Symmetries, Integrable Systems and Representations
PB - Springer New York LLC
ER -