The role of density in phase transitions and new perspectives on theta functions

Let $z\in \mathbb{H}:=\{z\in\mathbb{C}:\Im(z)>0\}$ and $Λ:=\frac{1}{\sqrt{\Im(z)}}\big({\mathbb Z}\oplus z{\mathbb Z}\big)$, and set $ θ(α;z):=θ(α;Λ)=\sum_{(m,n)\in\mathbb{Z}^2}e^{-πα\frac{|m+nz|^2}{\Im(z)}}. $ We study the minimization of the functional $θ(α;z)-bθ(2α;z)$ and establish a complete phase diagram for all $α>0$ and $b\in(0,\sqrt{2})$. For $0