Refined order estimates for regular graphs — Remark 2.7

PDF p. 16: Remark 2.7; Section 5 discussion; PDF p. 29: Remark 2.7; Section 5 discussion

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 self-loops allowed and counted once in each row sum. Suppose every row sum is $d\ge 0$, let $\alpha \ge 0$, and assume $\mathrm{\parallel }\mathrm{\Delta }\mathrm{\parallel }\le \alpha d$, where $\mathrm{\Delta }=A-(d\mathrm{/}n)J$ and $J$ is the all-ones matrix; the norm is the Euclidean operator norm. Let $\theta$ be a stable state, meaning a local minimum on the phase torus of ${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y}{A}_{xy}(1-\cos ({\theta }_x-{\theta }_y))$. Choose representatives in $(-\pi ,\pi ]$ after common rotation so that ${\rho }_1\ge 0$ is real, where ${\rho }_k=n^{-1}{\sum }_x{e}^{ik{\theta }_x}$. Put $X={\rho }_1^2$, $r=\mathrm{\mid }{\rho }_2\mathrm{\mid }$, and $M=n^{-1}{\sum }_{x}H({\theta }_x)$, where $H(t)={\sin }^2\mathrm{\mid }t\mathrm{\mid }$ for $\mathrm{\mid }t\mathrm{\mid }\le \pi \mathrm{/}2$ and $H(t)=1$ for $\pi \mathrm{/}2\le \mathrm{\mid }t\mathrm{\mid }\le \pi$.

For $0\le \alpha \le 0.24585$, one has ${\rho }_1^2\ge 1-3\alpha$, $\mathrm{\mid }{\rho }_2\mathrm{\mid }\ge 1-5{\alpha }^2$, and $M\le 5{\alpha }^2\mathrm{/}4$. For $0\le \alpha \le 0.1$, the stronger first bound ${\rho }_1^2\ge 1-2\alpha$ holds. The regular version of the article's bootstrap uses, for $x>0$, $f_{\mathrm{r}\mathrm{e}\mathrm{g}}(x,\alpha )=2x+3\alpha -1-(1-2{\alpha }^2\mathrm{/}x{)}^2$; its associated polynomial is $Q_{\alpha }(x)=x^2{f}_{\mathrm{r}\mathrm{e}\mathrm{g}}(x,\alpha )=2x^3+(3\alpha -2)x^2+4{\alpha }^{2}x-4{\alpha }^4$. The article prints the last sign as positive; the corrected algebra does not change the quantitative conclusions above.

As a separately attributed assertion, the final sentence of Remark 2.7 gives $n^{-1}{\sum }_{x\in V}H({\theta }_x)\le 5{\alpha }^2\mathrm{/}4$ for a $d^{\ast }$-regular $(n,d^{\ast },\alpha )$-expander and a stable state $\theta$, under the article's standing connectedness and phase-normalization conventions. That sentence states no explicit restriction on $\alpha$. Its scope beyond $0\le \alpha \le 0.24585$ is retained here as a source assertion; the local proof establishes the range-qualified estimates above.

Remark 2.7 on original PDF p. 16 asserts all the estimates retained here. Its printed cubic is $2x^3+(3\alpha -2)x^2+4{\alpha }^{2}x+4{\alpha }^4$. The last sign is a typo: multiplying $2x+3\alpha -1-(1-2{\alpha }^2\mathrm{/}x{)}^2$ by $x^2$ gives the corrected $Q_{\alpha }(x)$ in the Statement. This algebraic defect does not refute any order-parameter conclusion. The present proof keeps the source's regular Hessian inequality and pointwise-to-norm method, strengthens the norm step by removing the constant phase-vector component, and supplies a finite rational bootstrap in place of the source's unprovided endpoint root calculation. The source's final sentence attributes $n^{-1}{\sum }_{x\in V}H({\theta }_x)\le 5{\alpha }^2\mathrm{/}4$ to the proof of Corollary 2.6 for a $d^{\ast }$-regular $(n,d^{\ast },\alpha )$-expander, without an explicit $\alpha$ restriction in that sentence. The preceding sentences discuss $\alpha \le 0.1$ and $\alpha \le 0.24585$, while Corollary 2.6 itself assumes $\alpha \le 1\mathrm{/}5$. The unrestricted reading of the final sentence is therefore preserved as attributed coverage, with its intended inherited range left unspecified. The local mass estimate is established for $0\le \alpha \le 0.24585$, including the later application $\alpha =0.0816$. The broader reading is neither established nor refuted here, and no failure of an order-parameter or synchrony conclusion is inferred.