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The Bernoulli-Laplace model describes a diffusion process of two types ofparticles between two urns. To analyze the finite-size dynamics of thisprocess, and for other constructive results we diagonalize the correspondingtransition matrix and calculate explicitly closed-form expressions for alleigenvalues and eigenvectors of the Markov transition matrix $T_{BL}$. This isdone by a new method based on mapping the eigenproblem for $T_{BL}$ to theassociated problem for a linear partial differential operator $L_{BL}$ actingon the vector space of homogeneous polynomials in three indeterminates. Themethod is applicable to other Two Urns models and is relatively easy to usecompared to previous methods based on orthogonal polynomials or grouprepresentations.

A Nonlinear Transform for the Diagonalization of the Bernoulli-Laplace Diffusion Model and Orthogonal Polynomials