  Cellular automata are often used to model systems in physics, social
sciences, biology that are inherently asynchronous. Over the past 20 years,
studies have demonstrated that the behavior of cellular automata drastically
changed under asynchronous updates. Still, the few mathematical analyses of
asynchronism focus on one-dimensional probabilistic cellular automata, either
on single examples or on specific classes. As for other classic dynamical
systems in physics, extending known methods from one- to two-dimensional
systems is a long lasting challenging problem.
  In this paper, we address the problem of analysing an apparently simple 2D
asynchronous cellular automaton: 2D Minority where each cell, when fired,
updates to the minority state of its neighborhood. Our experiments reveal that
in spite of its simplicity, the minority rule exhibits a quite complex response
to asynchronism. By focusing on the fully asynchronous regime, we are however
able to describe completely the asymptotic behavior of this dynamics as long as
the initial configuration satisfies some natural constraints. Besides these
technical results, we have strong reasons to believe that our techniques
relying on defining an energy function from the transition table of the
automaton may be extended to the wider class of threshold automata.
