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

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 Nα 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 Nα. 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.