Analytical expression for the exit probability of the q-voter model in one dimension

We present in this paper an approximation that is able to give an analytical expression for the exit probability of the q-voter model in one dimension. This expression gives a better fit for the more recent data about simulations in large networks [A. M. Timpanaro and C. P. C. do Prado, Phys. Rev. E 89, 052808 (2014)] and as such departs from the expression ρ(q)/ρ(q)+(1-ρ)(q) found in papers that investigated small networks only [R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008); P. Przybyła et al., Phys. Rev. E 84, 031117 (2011); F. Slanina et al., Europhys. Lett. 82, 18006 (2008)]. The approximation consists in assuming a large separation on the time scales at which active groups of agents convince inactive ones and the time taken in the competition between active groups. Some interesting findings are that for q=2 we still have ρ(2)/ρ(2)+(1-ρ)(2) as the exit probability and for q>2 we can obtain a lower-order approximation of the form ρ(s)/ρ(s)+(1-ρ)(s) with s varying from q for low values of q to q-1/2 for large values of q. As such, this work can also be seen as a deduction for why the exit probability ρ(q)/ρ(q)+(1-ρ)(q) gives a good fit, without relying on mean-field arguments or on the assumption that only the first step is nondeterministic, as q and q-1/2 will give very similar results when q→∞.