Kernel bounds on three phase regions

PDF p. 24: Proof of Lemma 4.2; equations (4.3), (4.4)

Assume countable choice. Let $G=(V,E)$ be a finite undirected graph with nonempty vertex set, possible self-loops and symmetric binary adjacency matrix $A$. Let $\theta$ be a stable phase configuration, meaning a local minimum on $T_V={\mathbb{R}}^V\mathrm{/}(2\pi \mathbb{Z}{)}^V$ of ${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y\in V}{A}_{x,y}(1-\cos ({\theta }_x-{\theta }_y))$. Choose each ${\theta }_x\in (-\pi ,\pi ]$ and suppose ${\rho }_1=\mathrm{\mid }V{\mathrm{\mid }}^{-1}{\sum }_{x\in V}{e}^{i{\theta }_x}\in [0,1]$.

For $0<\gamma <\beta \le \pi \mathrm{/}2$, let ${\mathcal{C}}_{\psi }=\{x\in V:\mathrm{\mid }{\theta }_x\mathrm{\mid }\ge \psi \}$, and put $K(\alpha ,\eta )=\sin (\mathrm{\mid }\alpha \mathrm{\mid }-\min \{\mathrm{\mid }\eta \mathrm{\mid },\pi \mathrm{/}2\})$. For every $y\in {\mathcal{C}}_{\beta }$, the following bounds hold: $K({\theta }_x,{\theta }_y)\le 1$ if $x\in {\mathcal{C}}_{\beta }$; $K({\theta }_x,{\theta }_y)\le 0$ if $x\in {\mathcal{C}}_{\gamma }\setminus {\mathcal{C}}_{\beta }$; and $K({\theta }_x,{\theta }_y)\le \sin (\gamma -\beta )$ if $x\in {\mathcal{C}}_{\gamma }^c$.

With $e(X,Y)={\sum }_{x\in X}{\sum }_{y\in Y}{A}_{x,y}$ and complements relative to $V$, consequently $e({\mathcal{C}}_{\beta },{\mathcal{C}}_{\beta })+\sin (\gamma -\beta )e({\mathcal{C}}_{\beta },{\mathcal{C}}_{\gamma }^c)\ge 0$.

The three-region bounds and their summed use are attributed to the proof of Lemma 4.2 in Expander graphs are globally synchronizing, PDF page 24. The proof above uses the stated mathematical suppliers.