Walk of a point defect in a model with macroscopic ground state degeneracy

Walks of the point defect in the two- and three-dimensional lattice models of colored hard spheres mimicking the q-state antiferromagnetic Potts model are considered. The walks are generated by moves of neighboring vertices which conserve the ground state energy. Various ''traps'' confining the walks are observed, while for the models with q greater than or equal to 4 and q greater than or equal to 5 on the square and cubic lattices respectively nonvanishing probabilities that all lattice vertices are attainable in a walk are found.