Nonequilibrium phase transition and cultural drift in the continuous-trait Axelrod model

The standard Axelrod model of cultural dissemination, based on discrete cultural traits, exhibits a nonequilibrium phase transition but is inherently limited by its inability to continuously probe the critical behavior. We address this limitation by investigating a generalized Axelrod model that employs continuous cultural traits confined to the interval [0,1]. This model utilizes a similarity threshold d that serves as a continuous control parameter representing cultural tolerance. This framework allows for a robust quantitative analysis of the model's critical properties and its dynamics under cultural drift (copying noise). For this quantitative characterization, we primarily focus on the case of F=2 cultural features. For the perfect copying scenario, we precisely locate the critical threshold d_{c}, which separates the disordered (fragmented) and ordered (polarized) phases. Through finite-size scaling, we find that the mean domain density vanishes continuously at d_{c} with the exponent β=1/3. Simultaneously, the largest domain fraction displays a surprising discontinuous jump at d_{c}. We find that the finite-size effects in the critical region are governed by the exponent ν=2 for both the continuous and discontinuous transitions. Under imperfect copying, persistent noise introduces a powerful selective pressure on the trait space, leading to the emergence of two symmetry-related attractors at the trait values d and 1-d. However, these noise-induced attractors prove fragile in the thermodynamic limit, becoming unstable at large lattice sizes, which directly accounts for the observed failure of the dynamics to freeze under sustained cultural drift. This suggests that in large, continuously evolving societies, true cultural convergence is highly unlikely, leading instead to sustained fragmentation and nonstationary dynamics where cultural domains never fully stabilize.