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

## TL;DR

This paper introduces a novel method for diagonalizing the transition matrix of the Bernoulli-Laplace diffusion model, enabling explicit calculation of eigenvalues and eigenvectors through a mapping to a differential operator, simplifying analysis of such Markov processes.

## Contribution

A new approach mapping the eigenproblem to a differential operator, making the diagonalization of Bernoulli-Laplace and similar models more straightforward and broadly applicable.

## Key findings

- Explicit eigenvalues and eigenvectors derived for the Bernoulli-Laplace model.
- Method applicable to other two-urn diffusion models.
- Simplifies analysis compared to previous orthogonal polynomial methods.

## Abstract

The Bernoulli-Laplace model describes a diffusion process of two types of particles between two urns. To analyze the finite-size dynamics of this process, and for other constructive results we diagonalize the corresponding transition matrix and calculate explicitly closed-form expressions for all eigenvalues and eigenvectors of the Markov transition matrix $T_{BL}$. This is done by a new method based on mapping the eigenproblem for $T_{BL}$ to the associated problem for a linear partial differential operator $L_{BL}$ acting on the vector space of homogeneous polynomials in three indeterminates. The method is applicable to other Two Urns models and is relatively easy to use compared to previous methods based on orthogonal polynomials or group representations.