Exact solution of the Nonconsensus Opinion Model on the line

TL;DR

This paper provides an exact solution to the Nonconsensus Opinion Model on a line, revealing rapid convergence to steady state, long-range correlations, and differences in phase transition behavior compared to regular percolation.

Contribution

It offers the first exact analytical solution of the NCO model on a line, connecting it with invasion percolation with trapping and analyzing phase transitions on Bethe lattices.

Findings

System evolves exponentially fast to steady state.

Average cluster size scales as the square of initial size.

NCO model on Bethe lattices exhibits a different phase transition diagram.

Abstract

The nonconcensus opinion model (NCO) introduced recently by Shao et al., [Phys. Rev. Lett.103, 018701 (2009)] is solved exactly on the line. Although, as expected, the model exhibits no phase transition in one dimension, its study is interesting because of the connection with invasion percolation with trapping. The system evolves exponentially fast to the steady-state, rapidly developing long-range correlations: The average cluster size in the steady state scales as the square of the initial cluster size, of the (uncorrelated) initial state. We also discuss briefly the NCO model on Bethe lattices, arguing that its phase transition diagram is different than that of regular percolation.