LemmaStatement: AI-adaptedProof: AI-generated

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Almost antisymmetry and summed kernel bounds

PDF p. 25: Proof of Lemma 4.3; PDF p. 26: Proof of Lemma 4.3

Statement

For $a, b \in (- π, π]$ , let $K (a, b) = \sin (∣ a ∣ - \min {∣ b ∣, π / 2})$ . Then $K (a, b) + K (b, a) \leq 1_{{∣ a ∣ \geq π / 2}} + 1_{{∣ b ∣ \geq π / 2}}$ , where each indicator is $1$ when its condition holds and $0$ otherwise.

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 $θ$ be a stable phase configuration, meaning a local minimum on $R^{V} / (2 π Z)^{V}$ of $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x, y} (1 - \cos (θ_{x} - θ_{y}))$ . Choose principal representatives $θ_{x} \in (- π, π]$ and suppose $∣ V ∣^{- 1} \sum_{x} e^{i θ_{x}} \in [0, 1]$ . For $0 < γ < β \leq π / 2$ , write $C_{ψ} = {x \in V : ∣ θ_{x} ∣ \geq ψ}$ and $e (X, Y) = \sum_{x \in X} \sum_{y \in Y} A_{x, y}$ , with $X^{c} = V ∖ X$ . Then

$\sin (β - γ) e (C_{β}, C_{γ}^{c}) \leq \sum_{x, y \in C_{β}} A_{x, y} K (θ_{x}, θ_{y}) \leq e (C_{π / 2}, C_{β}) \leq e (C_{π / 2}, V) .$

If additionally $G$ is an $(n, d^{*}, α, c^{-}, c^{+})$ -expander, then $e (C_{π / 2}, V) \leq (1 + c^{+} + α) d^{*} ∣ C_{π / 2} ∣$ .

Statement references

Kernel for stability amplification

Phase tail sets for amplification

Oriented edge count between sets

Two-sided spectral expander — Definition 1.9

Kernel stability condition — Lemma 2.8

The Axiom of Countable Choice (AC_ω)

Finite phase tori, local lifts and the flat metric

Homogeneous Kuramoto energy

Phasors, trigonometric identities and periodic derivatives

Facts & Assumptions

[F1]

The kernel has the formula in the Statement and, when $∣ b ∣ \geq π / 2$ , $K (a, b) = - \cos a$ (Kernel for stability amplification).

[F2]

Sine is odd and bounded above by $1$ ; cosine is even and nonnegative on $[- π / 2, π / 2]$ (Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine). The quarter-turn identity gives $\sin (t - π / 2) = - \cos t$ (Quarter-turn values and shifts by pi/2 and pi).

[F3]

The phase tails and ordered edge counts are as defined in the Statement; $A$ is symmetric and nonnegative, and $C_{π / 2} \subseteq C_{β} \subseteq C_{γ}$ (Phase tail sets for amplification, Oriented edge count between sets).

[F4]

Under the graph, stability, representative, order-parameter and countable-choice hypotheses in the Statement, $\sum_{x \in V} A_{x, y} K (θ_{x}, θ_{y}) \geq 0$ for every $y$ (Kernel stability condition — Lemma 2.8, The Axiom of Countable Choice (AC_ω)).

[F5]

For $y \in C_{β}$ , the kernel is at most $0$ on $C_{γ} ∖ C_{β}$ and at most $\sin (γ - β)$ on $C_{γ}^{c}$ (Kernel bounds on three phase regions).

[F6]

For the stated expander and any $X \subseteq V$ , the maximum-degree consequence of the incident-edge bound gives $e (X, V) \leq (1 + c^{+} + α) d^{*} ∣ X ∣$ (Two-sided spectral expander — Definition 1.9, Total incident edge bounds — Lemma 3.6).

Proof

Given: The angles $a, b$ , and, for the summed bounds, the graph, stable configuration and angles $γ, β$ in the Statement.

Goal: Prove the pointwise inequality, the chain of summed bounds and the expander consequence.

1.1F1F2algebra

Put $u = ∣ a ∣$ and $v = ∣ b ∣$ . If both are below $π / 2$ , [F1] and oddness give $K (a, b) + K (b, a) = \sin (u - v) + \sin (v - u) = 0$ . If $u \geq π / 2 > v$ , then $K (a, b) \leq 1$ and $K (b, a) = \sin (v - π / 2) = - \cos v \leq 0$ . The reversed case is symmetric. If both are at least $π / 2$ , each kernel value is at most $1$ . These cases include equality at $π / 2$ and give the stated indicator bound.

2.1F3step 1.1algebra

Let $S = \sum_{x, y \in C_{β}} A_{x, y} K (θ_{x}, θ_{y})$ . Interchanging the finite summation indices and using symmetry of $A$ yields twice $S$ as the sum of $A_{x, y} (K (θ_{x}, θ_{y}) + K (θ_{y}, θ_{x}))$ . Apply step 1.1 termwise. The two indicator sums are equal by symmetry, and the first counts precisely $e (C_{π / 2}, C_{β})$ because $C_{π / 2} \subseteq C_{β}$ . Thus $S \leq e (C_{π / 2}, C_{β}) \leq e (C_{π / 2}, V)$ , the last inequality using nonnegative entries.

3.1F2F3F4F5F6step 2.1algebra∎

Sum [F4] over $y \in C_{β}$ and split the $x$ -sum into $C_{β}$ , $C_{γ} ∖ C_{β}$ , and $C_{γ}^{c}$ . By [F5] and nonnegativity of $A$ , the total is at most $S + \sin (γ - β) e (C_{γ}^{c}, C_{β})$ . Symmetry of $A$ and oddness of sine therefore give $S \geq \sin (β - γ) e (C_{β}, C_{γ}^{c})$ . Apply [F6] with $X = C_{π / 2}$ for the final expander bound. Together with step 2.1 this proves the chain.

Source notes

The pointwise inequality and its use in the proof of Lemma 4.3 are attributed to the cited article. The line on PDF page 25 omits $d^{*}$ from the incident-edge bound; the following page displays it. The expander bound stated here follows from the separately declared incident-edge supplier.

Depends on

Used by · 1

Large-set amplification — Lemma 4.3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Expander graphs are globally synchronizing · PDF pp. 25–26, proof of Lemma 4.3