Toshikazu Sunada (Meiji University)
sunada@math.meiji.ac.jp The $K_4$ crystal is the most symmetric realization in space of the maximal abelian covering graph of the complete graph $K_4$. It is observed that the $K_4$ crystal has the ``very strong isotropic property" and the ``energy minimizing property" among all periodic realizations which are only shared by the diamond crystal. Having these examples in mind, I will discuss how symmetry and energy of general crystals are related to asymptotic behavior of random walks.