Mass-angle lower bound — Lemma 4.1

PDF p. 23: Lemma 4.1

Assume countable choice. Let $G=(V,E)$ be a finite connected undirected graph with nonempty vertex set, possibly with self-loops, and let $\theta$ be a nonsynchronized stable state whose first order parameter is real and nonnegative. A stable state is a local minimum on the phase torus of the graph energy $\frac{1}{2}{\sum }_{x,y\in V}{A}_{x,y}(1-\cos ({\theta }_x-{\theta }_y))$; nonsynchronized means that the phases are not all equal modulo $2\pi$. The first order parameter is $\mathrm{\mid }V{\mathrm{\mid }}^{-1}{\sum }_{x\in V}{e}^{i{\theta }_x}$. Use the representatives ${\theta }_x\in (-\pi ,\pi ]$.

Let ${\beta }_0,\dots ,{\beta }_M$ and $c_0,\dots ,c_M$ be an amplification sequence: $M\ge 0$ is an integer, $\pi \mathrm{/}2={\beta }_0>\cdots >{\beta }_M\ge 0$, $c_0<\cdots <c_M$, and $\mathrm{\mid }{\mathcal{C}}_{{\beta }_s}\mathrm{\mid }\ge c_s$ for $0\le s\le M$, where ${\mathcal{C}}_{\psi }=\{x\in V:\mathrm{\mid }{\theta }_x\mathrm{\mid }\ge \psi \}$ and $c_0=\mathrm{\mid }{\mathcal{C}}_{\pi \mathrm{/}2}\mathrm{\mid }$. For $t\in (-\pi ,\pi ]$, let $H(t)={\sin }^2\mathrm{\mid }t\mathrm{\mid }$ if $\mathrm{\mid }t\mathrm{\mid }\le \pi \mathrm{/}2$, and $H(t)=1$ if $\pi \mathrm{/}2\le \mathrm{\mid }t\mathrm{\mid }\le \pi$. Then

${\sum }_{x\in V}H({\theta }_x)\ge c_M{\sin }^2({\beta }_M)\mathrm{.}$

The statement is attributed to Lemma 4.1 of the article cited in the frontmatter. The proof above uses monotonicity of $H$ directly: the article's intermediate comparison of ${\sin }^2({\theta }_x)$ with ${\sin }^2({\beta }_M)$ need not hold for vertices whose absolute angle exceeds $\pi \mathrm{/}2$.