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

We give a lower bound on the spectral gap for a class of binary collision processes. In 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 by Grigo et al., and zero-range processes.