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.
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References
Cancrini, N., Caputo, P., Martinelli, F.: Relaxation time for L-reversal chains and other chromosome shuffles. Ann. Appl. Probab. 16, 1506–1527 (2006)
Caputo, P.: Spectral gap inequalities in product spaces with conservation laws. In: Funaki, T., Osada, H. (eds.) Stochastic Analysis on Large Interacting Systems, pp. 53–88. Math. Soc. Japan, Tokyo (2004)
Caputo, P.: On the spectral gap of the Kac walk and other binary collision processes. ALEA Lat. Am. J. Probab. Math. Stat. 4, 205–222 (2008)
Carlen, E.A., Carvalho, M.C., Loss, M.: Determination of the spectral gap in Kac’s master equation and related stochastic evolutions. Acta Math. 191, 1–54 (2003)
Giroux, G., Ferland, R.: Global spectral gap for Dirichlet-Kac random motions. J. Stat. Phys. 132, 561–567 (2008)
Grigo, A., Khanin, K., Szasz, D.: Mixing rates of particle systems with energy exchange. arXiv:1109.2356
Kac, M.: Foundation of kinetic theory. In: Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability (1954–1955), vol. III, pp. 171–197. University of California Press, Berkeley (1956)
Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)
Landim, C., Sethuraman, S., Varadhan, S.: Spectral gap for zero-range dynamics. Ann. Probab. 24, 1871–1902 (1995)
Lu, S.L., Yau, H.T.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys. 156, 399–433 (1993)
Morris, B.: Spectral gap for the zero range process with constant rate. Ann. Probab. 34, 1645–1664 (2006)
Nagahata, Y., Sasada, M.: Spectral gap for multi-species exclusion processes. J. Stat. Phys. 143, 381–398 (2011)
Sasada, M.: Spectral gap for particle systems with degenerate energy exchange rates (in preparation)
Sasada, M., Tsuboi, T.: On a remarkable sequence of transition probability matrices (in preparation)
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Sasada, M. (2013). On the Spectral Gap of the Kac Walk and Other Binary Collision Processes on d-Dimensional Lattice. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_23
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DOI: https://doi.org/10.1007/978-1-4471-4863-0_23
Publisher Name: Springer, London
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