Comparison Theory for Markov Chains on Different State Spaces and Application to Random Walk on Derangements

We consider two Markov chains on state spaces . In this paper, we prove bounds on the eigenvalues of the chain on the smaller state space based on the eigenvalues of the chain on the larger state space. This generalizes work of Diaconis, Saloff-Coste, and others on comparison of chains in the case . The main tool is the extension of functions from the smaller space to the larger, which allows comparison of the entire spectrum of the two chains. The theory is used to give quick analyses of several chains without symmetry. We apply this theory to analyze the mixing properties of a 'random transposition' walk on derangements.