Transition under q noise in the Sznajd model on square lattice

In order to describe more realistically the formation
of a consensus in human opinion dynamics, in this work we study the Sznajd
model with probabilistic noise in two dimensions on square lattice.
The time evolution of this system is performed via Monte Carlo simulations.
This social behavior model with
noise presents a well defined second-order phase transition.
For small enough noise $q<0.33$ most agents end up sharing the same
opinion. We performed Monte Carlo simulation on
the square lattice
with various systems of size $L=30$, $100$, $300$, $500$,
$1000$, and $3000$. We take $1.5\times 10^5$ Monte Carlo steps per spin (MCS)
to make the system reach the steady state, and then the time averages are
estimated over the next $3\times 10^5$ MCS. One MCS is
accomplished after
all the $N$ spins are investigated whether they flip or not.
The results are averaged over $N_{run}$
$(20\le N_{run}\le 100)$ independent
simulation runs for each lattice and for
given set of parameters $(q,N)$. In our study of
the Sznajd model on square lattices with
probabilistic noise $q$ we found a phase transition
when the initial concentrations $p$ of spins up are different from
$p=0.5$ Our results show the introduction of a
probabilistic noise ($0<q<1$) to produce a phase transition
independent of the initial concentrations $p$ of spins up
and down. Therefore, we found out a strong second-order
phase transition when the size
lattice increases, at critical noise $q_{c}= 0.329(4)$.
This means that strong noise, $q>q_c$, destroys the consensus found in
deterministic Sznajd opinion dynamics, while for small noise nearly all
agents share the same opinion.