Noisy voter model: Explicit expressions for finite system size

Abstract

Urn models are classic stochastic models that have been used to describe a diverse kind of complex systems. Voter and Ehrenfest's models are very well-known urn models. An opinion model that combines these two models is presented in this work and it is used to study a noisy voter model. In particular, at each temporal step, an Ehrenfest's model step is done with probability α or a voter step is done with probability 1-α. The parameter α plays the role of noise. By performing a spectral analysis, it is possible to obtain explicit expressions for the order parameter, susceptibility, and Binder's fourth-order cumulant. Recursive expressions in terms of the dual Hahn polynomials are given for first passage and return distributions to consensus and the equal coexistence of opinions. In the cases where they follow power-law distributions, their exponents are computed. This model has a pseudocritical noise value that depends on the system size; a discussion about thermodynamic limits is given.