  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.
