Mean-field analysis of the majority-vote model broken-ergodicity steady state

Abstract

We study analytically a variant of the one-dimensional majority-vote model in which the individual retains its opinion in case there is a tie among the neighbors' opinions. The individuals are fixed in the sites of a ring of size $L$ and can interact with their nearest neighbors only. The interesting feature of this model is that it exhibits an infinity of spatially heterogeneous absorbing configurations for $L\to \infty$ whose statistical properties we probe analytically using a mean-field framework based on the decomposition of the $L$-site joint probability distribution into the $n$-contiguous-site joint distributions, the so-called $n$-site approximation. To describe the broken-ergodicity steady state of the model we solve analytically the mean-field dynamic equations for arbitrary time $t$ in the cases n=3 and 4. The asymptotic limit $t\to \infty$ reveals the mapping between the statistical properties of the random initial configurations and those of the final absorbing configurations. For the pair approximation ($n=2$) we derive that mapping using a trick that avoids solving the full dynamics. Most remarkably, we find that the predictions of the 4-site approximation reduce to those of the 3-site in the case of expectations involving three contiguous sites. In addition, those expectations fit the Monte Carlo data perfectly and so we conjecture that they are in fact the exact expectations for the one-dimensional majority-vote model.

References in corpus (9)

• Statistical physics of social dynamics
• Dynamics of Majority Rule
• Nonequilibrium phase transition in a model for social influence
• Heterogeneous believes, segregation and extremism in the making of public opinions
• Dynamics of Non-Conservative Voters
• Ordering phase transition in the one-dimensional Axelrod model
• Complementarity and diversity in a soluble model ecosystem
• Culture-area relation in Axelrod's model for culture dissemination
• Statistics of opinion domains of the majority-vote model on a square lattice

Cited by in corpus (2)

• Exact solution of the isotropic majority-vote model on complete graphs
• Revisiting the effect of external fields in Axelrod's model of social dynamics