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Uniform bound on average H mass — Corollary 2.6

PDF p. 16: Corollary 2.6

Statement

Assume countable choice. Let $G = (V, E)$ be a finite connected undirected graph, possibly with loops, with $n = ∣ V ∣ > 0$ and symmetric binary adjacency matrix $A$ . Connected means that every two vertices can be joined by a finite edge path. Let $d^{*} \geq 0$ and $α \leq 1 / 5$ , and suppose $G$ is an $(n, d^{*}, α)$ -expander, meaning that $∥ A - (d^{*} / n) J ∥ \leq α d^{*}$ for the all-ones matrix $J$ and the Euclidean operator norm. Let $θ$ be a stable state: a local minimum on the phase torus of $E_{G} (θ) = \frac{1}{2} \sum_{x, y \in V} A_{x y} (1 - \cos (θ_{x} - θ_{y}))$ . Choose phase representatives in $(- π, π]$ after a common rotation for which $ρ_{1} = n^{- 1} \sum_{x \in V} e^{i θ_{x}}$ is real and nonnegative. Define $H (t) = \sin^{2} (∣ t ∣)$ for $∣ t ∣ \leq π / 2$ and $H (t) = 1$ for $π / 2 \leq ∣ t ∣ \leq π$ . Then

$\frac{1}{n} \sum_{x \in V} H (θ_{x}) \leq \frac{5 α^{2}}{2} .$

Statement references

Adjacency, degrees, graph Laplacian, and spectral notation

Adjacency spectral expander — Definition 1.3

Equilibria, stable states, synchronized states, and spurious minima

Finite phase tori, local lifts and the flat metric

Daido order parameters

Truncated sine-square function

The Axiom of Countable Choice (AC_ω)

Facts & Assumptions

[F1]

Under the statement’s connectedness, stability and normalization hypotheses, Lemma 2.3 gives $n α^{2} \geq ρ_{1}^{2} \sum_{x} H (θ_{x})$ (Order parameter control of total H mass — Lemma 2.3).

[F2]

When additionally $0 \leq α \leq 1 / 5$ , Lemma 2.5 gives $ρ_{1}^{2} \geq 1 - 3 α$ (Tractable order parameter inequalities — Lemma 2.5). Both suppliers carry countable choice (The Axiom of Countable Choice (AC_ω)).

[F3]

The adjacency entries record edges; the expander condition is the stated operator-norm bound (Adjacency, degrees, graph Laplacian, and spectral notation, Adjacency spectral expander — Definition 1.3). For real $t$ , the phasor $e^{i t}$ has unit modulus and is $2 π$ -periodic (Phasors, trigonometric identities and periodic derivatives). The definition of $H$ gives $H (0) = 0$ (Truncated sine-square function).

[F4]

Phase classes modulo $2 π$ are identified bijectively with unit-circle points by $[t] \mapsto (\cos t, \sin t)$ (Finite phase tori, local lifts and the flat metric). Since $e^{i t} = \cos t + i \sin t$ and $e^{i 0} = 1$ , equality $e^{i t} = 1$ therefore implies $[t] = [0]$ (Phasors, trigonometric identities and periodic derivatives).

Proof

Given: The connected expander, normalized stable state and parameters in the Statement, under countable choice.

Goal: Bound the average of $H$ by $5 α^{2} / 2$ .

1.1F1F2givenalgebra

If $d^{*} > 0$ , the nonnegative operator norm and the expander inequality imply $α \geq 0$ . By [F2], $ρ_{1}^{2} \geq 1 - 3 α \geq 2 / 5$ . Combining this positive lower bound with [F1] and dividing by $n ρ_{1}^{2}$ gives $n^{- 1} \sum_{x} H (θ_{x}) \leq α^{2} / ρ_{1}^{2} \leq 5 α^{2} / 2$ .

2.1F3F4givenstep 1.1algebra∎

If $d^{*} = 0$ , the expander bound gives $∥ A ∥ = 0$ , so $A$ is the zero matrix. Connectedness and nonemptiness then force $n = 1$ . The normalized first order parameter is the single unit phasor $e^{i θ_{1}}$ , which is real and nonnegative, hence equals $1$ . By [F4], $[θ_{1}] = [0]$ , so $θ_{1} = 2 π k$ for an integer $k$ . The representative condition $- π < θ_{1} \leq π$ gives $- 1 / 2 < k \leq 1 / 2$ , forcing $k = 0$ . Hence $θ_{1} = 0$ and $H (θ_{1}) = 0$ . Thus the claimed bound also holds in this case, regardless of the sign of $α$ . Together with step 1.1 this proves the claim.

Source notes

The original Corollary 2.6 is attributed to the article, PDF p. 16. Section 2.1, PDF p. 11, supplies its standing connectedness and normalization conventions. The argument above uses the completed mathematical suppliers; the article citation is attribution rather than a premise.

Depends on

Used by · 1

Contradiction threshold for amplification

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 p. 11, Section 2.1 standing conventions; p. 16, Corollary 2.6