Bounds for Kac's master equation (Q1971576)

The authors consider a random walk on the orthogonal group SO\((n)\) which was firstly introduced by M. Kac to study Boltzmann's derivation of a basic equation of kinetic theory. The random walk describes a system of velocities for which the total energy is conserved -- hence the restriction to the sphere. It is a Markov chain on the \(n\)-sphere based on random rotations in randomly chosen coordinate planes. This random walk has also been studied by Hastings as a simple way of generating an approximately random equation. The present authors show that the walk has a spectral gap bounded below by \(c/n^3\). They use this result and curvature information to bound the rate of convergence to stationarity.