Tractable order parameter inequalities — Lemma 2.5

PDF p. 15: Lemma 2.5 and proof; PDF p. 16: Lemma 2.5 and proof

Assume countable choice. Let $G=(V,E)$ be a finite connected undirected graph, possibly with loops, with $n=\mathrm{\mid }V\mathrm{\mid }>0$ and symmetric binary adjacency matrix $A$. Connected means that every two vertices are joined by a finite edge path. Let $d^{\ast }\ge 0$ and $0\le \alpha \le 1\mathrm{/}5$, 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 ${\rho }_k(\theta )=n^{-1}{\sum }_{x\in V}{e}^{ik{\theta }_x}$, using real representatives of the phases, and choose a common rotation so that ${\rho }_1$ is real and nonnegative. Then ${\rho }_1^2\ge 1-3\alpha$ and $\mathrm{\mid }{\rho }_2\mathrm{\mid }\ge 1-5{\alpha }^2$.

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

The Axiom of Countable Choice (AC_ω)

The statement is attributed to Expander graphs are globally synchronizing, Lemma 2.5. The cubic supplier treats the endpoint $\alpha =0$ separately; the argument above uses its proved estimate rather than importing the article's claim as a premise.