  The one-parameter two-dimensional cellular automaton with the Margolus
neighbourhood is analyzed based on the considering the projection of the
stochastic movements of a single particle. Introducing the auxiliary random
variable associated with the direction of the movement, we reduce the problem
under consideration to the study of a two-dimensional Markov chain. The master
equation for the probability distribution is derived and solved exactly using
the probability generating function method. The probability distribution is
expressed analytically in terms of Jacobi polynomials. The moments of the
obtained solution allowed us to derive the exact analytical formula for the
parametric dependence of the diffusion coefficient in the two-dimensional
cellular automaton with the Margolus neighbourhood. Our analytic results agree
with earlier empirical results of other authors and refine them. The results
are of interest for the modelling two-dimensional diffusion using cellular
automata especially for the multicomponent problem.
