Bounds on Mixing Times in the Bernoulli-Laplace Diffusion Model-Elementary proofs for variation distance and first passage times

The Bernoulli-Laplace model describes a diffusion process of two types of particles. To analyze the finite-size dynamics of this process, we apply a generating function method for diagonalizing the corresponding transition matrix. The method is a generalization of the generating function approach of Mark Kac [1] to diagonalize the Ehrenfest model. We then apply this solution to provide elementary proofs of the classical bounds of the mixing times in terms of total variational distance of the m-step probability distribution to the stationary distribution. We also provide large N and finite-size solutions to the first passage times of achieving stationarity. Unlike the classical proofs which are based on group representations, we construct new proofs purely from the properties of the eigenvalues and eigenvectors of the transition matrix.