Spectral geometry of crystal lattices
Motoko Kotani, Toshikazu Sunada · Contemporary mathematics - American Mathematical Society · 2003
The aim of this expository article is to exhibit several interesting interactions among geometry, graph theory and probability through a brief survey of a series of our recent work on geometry of crystal lattices ([37], [39], [40], [44], [45]). Emphasis is put on the underlying ideas and concepts such as Albanese tori and Albanese maps associated with crystal lattices which originate in classical algebraic geometry and Riemannian geometry. Several geometric properties of crystal lattices are derived from asymptotic properties of random walks. The spectral theory of transition operators and their magnetic version on crystal lattices is also explained with suggestions for further studies ([6], [17], [28], [29]).