Eigenfunctions and the Integrated Density of States on Archimedean Tilings

We study existence and absence of $\ell^2$-eigenfunctions of the combinatorial Laplacian on the 11 Archimedean tilings of the Euclidean plane by regular convex polygons. We show that exactly two of these tilings (namely the $(3.6)^2$ "Kagome" tiling and the $(3.12^2)$ tiling) have $\ell^2$-eigenfunctions. These eigenfunctions are infinitely degenerate and are constituted of explicitly described eigenfunctions which are supported on a finite number of vertices of the underlying graph (namely on the hexagons and $12$-gons in the tilings, respectively). Furthermore, we provide an explicit expression for the Integrated Density of States (IDS) of the Laplacian on Archimedean tilings in terms of eigenvalues of Floquet matrices and deduce integral formulas for the IDS of the Laplacian on the $(4^4)$, $(3^6)$, $(6^3)$, $(3.6)^2$, and $(3.12^2)$ tilings. Our method of proof can be applied to other $\mathbb{Z}^d$-periodic graphs as well.