Spectral gap for Kac's model of Boltzmann equation

Abstract

We consider a random walk on $S^{n-1}$ , the standard sphere of dimension $n -1$, generated by random rotations on randomly selected coordinate planes $i,j$ with $1 \le i < j \le n$. This dynamic was used by Marc Kac as a model for the spatially homogeneous Boltzmann equation. We prove that the spectral gap on $S^{n-1}$ is $n^{-1}$ up to a constant independent of $n$.