Irreducibility of the Dispersion Polynomial for Periodic Graphs

Given a ZZ^d-periodic graph G, a periodic potential together with a weighted graph Laplacian defines a discrete periodic operator, which acts on functions on the vertices of G. Floquet theory allows us to study the spectrum of this operator through a finite matrix with Laurent polynomial entries. The zero set of the corresponding characteristic polynomial is called the Bloch variety. We will focus on the reducibility of this variety, which provides insights into various spectral properties, such as quantum ergodicity. In particular, we will study how the reducibility of the Bloch variety is affected as one varies the period of the potential.