Noisy voter model as a generalized Ehrenfest urn model and q -Gaussian stationary laws

Abstract

We study a generalized Ehrenfest urn model that interpolates between the Ehrenfest and voter dynamics through a mixing parameter 𝛼. This model can be interpreted in two different ways: adding noise to the voter model, where 𝛼 represents the intensity of the noise; or adding interaction to the Ehrenfest model, where (1−𝛼) represents the level of the interaction. We focus on a thermodynamic limit where the system size 𝑁→∞and 𝛼→0 with 𝑁⁢𝛼 held constant, and show that the stationary distribution converges to a 𝑞-Gaussian law. In this regime, the entropic index 𝑞 is determined explicitly by the constant 𝑁⁢𝛼. The definition of 𝑞-Gaussians with compact support is extended to include boundary-singular but integrable densities, thereby allowing two equivalent representations: a compact-support and a real-line 𝑞-Gaussian, establishing a duality between them. Moreover, after a suitable change of variable, we prove that the extended version of the 𝑞-Gaussian is the symmetric beta distribution. The analysis also reveals an order-disorder phase transition structure, being the Cauchy´s distribution, (supported on the entire real line); and the arcsine distribution, (their equivalent with compact support), the limit distributions when the susceptibility becomes maximal. These results provide a direct microscopic link between interacting urn models and generalized entropies.