Positive first order parameter — Corollary 2.2

PDF p. 13: Corollary 2.2

Assume countable choice. Let $G=(V,E)$ be a finite connected undirected graph with $n=\mathrm{\mid }V\mathrm{\mid }>0$, symmetric binary adjacency matrix $A$, and possible self-loops. Let $d^{\ast }\ge 0$ and $0\le \alpha <1\mathrm{/}4$, and suppose $G$ is an $(n,d^{\ast },\alpha )$-expander: $\mathrm{\parallel }A-(d^{\ast }\mathrm{/}n)J\mathrm{\parallel }\le \alpha d^{\ast }$, where $J$ is the all-ones matrix and the norm is the Euclidean operator norm. Let $\theta$ be a stable state, meaning a local minimum on the phase torus of the energy ${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y\in V}{A}_{xy}(1-\cos ({\theta }_x-{\theta }_y))$. For $k\ge 1$ put ${\rho }_k(\theta )=n^{-1}{\sum }_{x\in V}{e}^{ik{\theta }_x}$, where phases are angles modulo $2\pi$. Use the common-rotation convention under which ${\rho }_1$ is real and nonnegative and phases have representatives in $(-\pi ,\pi ]$. Then ${\rho }_1(\theta )>0$.

Adjacency, degrees, graph Laplacian, and spectral notation

Adjacency spectral expander — Definition 1.3

Equilibria, stable states, synchronized states, and spurious minima

Daido order parameters

Bounds, saturation, and rotation of order parameters

First and second order parameter inequality — Lemma 2.1

The Axiom of Countable Choice (AC_ω)

Corollary 2.2 of the cited article states this positivity conclusion under the article's preceding connectedness and phase-normalization conventions. The citation is attribution; the argument above uses [F1].