Analysis on Graphs and Fractals

The K_4 crystal --a new crystal structure similar to the diamond lattice

T Sunada, Meiji University, Tokyo, Japan

Abstract

I will discuss how symmetry and energy of general crystals are related to asymptotic behavior of random walks on crystal lattices. As an application, we construct the $K_4$ crystal, 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.