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Width scaling of the continuum branch threshold

Statement

Assume the Axiom of Countable Choice. Let $g_{0}$ be an even unimodal probability density on $R$ with respect to Lebesgue measure $λ$ , smooth near zero, with $g_{0} (0) > 0$ and $g_{0}^{''} (0) < 0$ , and let $σ > 0$ . For the scale family $g_{σ} (ω) = σ^{- 1} g_{0} (ω / σ)$ , in the stationary mean-field self-consistency calculation with all-to-all sinusoidal coupling $K / n$ , the threshold is $K_{c} (σ) = 2 σ / (π g_{0} (0))$ . At fixed $K > 0$ , the local supercritical side $K > K_{c} (σ)$ is equivalently $σ < π K g_{0} (0) / 2$ . The local positive-branch conclusion retains the stationary locked and drifting populations in a rotating frame and applies only sufficiently close to the threshold.

Statement references

The Axiom of Countable Choice (AC_ω)

Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ

Empirical phase measures and Fourier order statistics

A scoped continuum self-consistency input

Facts & Assumptions

[F1]

For an even unimodal density $g$ , smooth near zero with $g (0) > 0$ and $g^{''} (0) < 0$ , the stationary mean-field self-consistency threshold is $2 / (π g (0))$ , and the nondegenerate positive branch is locally on its supercritical side. This is the all-to-all sinusoidal model with coupling $K / n$ and stationary locked and drifting populations in a rotating frame (A scoped continuum self-consistency input).

[F2]

A probability density is nonnegative, measurable and integrates to one with respect to its reference measure; here that measure is Lebesgue measure $λ$ on the real line. The cited density definition also specifies evenness and unimodality (Empirical phase measures and Fourier order statistics, Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ).

[F3]

The real numbers form an ordered field, so multiplication or division of an inequality by a positive real preserves its direction (The reals form a totally ordered field).

[F4]

Under countable choice, for a $C^{1}$ diffeomorphism $T : R \to R$ , any $f \in L^{1} (λ)$ satisfies $\int_{R} f (y) d λ (y) = \int_{R} f (T (x)) ∣ T^{'} (x) ∣ d λ (x)$ , with an integrable transformed function. This is the one-dimensional case with $U = V = R$ (change of variables for integrable functions, The class L¹(μ) of integrable functions, Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ).

[F5]

Scalar multiplication and composition obey the product and chain rules for differentiation; differentiable functions are continuous (Sum, product, quotient, and chain rules from difference quotients).

[F6]

Countable choice asserts that a sequence of nonempty sets has a choice function. It is assumed here to meet the hypothesis of [F4] (The Axiom of Countable Choice (AC_ω)).

[F7]

The class $L^{1} (λ)$ consists of integrable function representatives. For a measurable real-valued $f$ , integrability means $\int ∣ f ∣ d λ < \infty$ (The class L¹(μ) of integrable functions, Integrable real and complex functions, and their integrals).

Proof

Given: Countable choice, a density $g_{0}$ , and positive $σ, K$ as in the statement, in the stationary self-consistency setting of [F1].

Goal: Verify the scaled density hypotheses, compute its threshold, and translate the supercritical inequality.

1.1F2F3F4F5F6F7given

By [F2], $g_{0}$ is nonnegative and measurable. Even unimodality gives $g_{0} (ω) \leq g_{0} (0) < \infty$ for every $ω$ , and $\int_{R} g_{0} d λ = 1$ . Thus $∣ g_{0} ∣ = g_{0}$ has finite integral, so $g_{0} \in L^{1} (λ)$ by [F7]. Apply [F4] with $U = V = R$ , $f = g_{0}$ and $T (ω) = ω / σ$ . This map is bijective with inverse $u \mapsto σ u$ ; both maps are continuously differentiable by [F5], and $∣ T^{'} ∣ = 1 / σ$ by positivity of $σ$ . With countable choice as in [F6], [F4] shows that $g_{σ}$ is integrable and $\int_{R} g_{σ} (ω) d λ (ω) = \int_{R} g_{0} (u) d λ (u) = 1$ .

2.1F1F2F3F5step 1.1given

Positive rescaling preserves evenness, nonnegativity and monotonicity on the positive half-line. Explicitly, $g_{σ} (- ω) = g_{σ} (ω)$ and $0 \leq x \leq y$ implies $g_{σ} (x) \geq g_{σ} (y)$ by [F2] and [F3]. Applying [F5] twice gives $g_{σ}^{'} (ω) = σ^{- 2} g_{0}^{'} (ω / σ)$ and $g_{σ}^{''} (ω) = σ^{- 3} g_{0}^{''} (ω / σ)$ near zero. The same rules preserve smoothness near zero. Thus $g_{σ} (0) = g_{0} (0) / σ > 0$ and $g_{σ}^{''} (0) = g_{0}^{''} (0) / σ^{3} < 0$ . Together with step 1.1, these verify the density hypotheses of [F1].

3.1F1F3step 2.1algebra∎

By [F1] and step 2.1, $K_{c} (σ) = 2 / (π g_{σ} (0)) = 2 σ / (π g_{0} (0))$ . Since $π g_{0} (0) > 0$ , [F3] gives $K > K_{c} (σ)$ exactly when $σ < π K g_{0} (0) / 2$ . The local positive-branch conclusion and the stationary-population qualifications are inherited from [F1].

Depends on

Used by · 1

Background continuum phase transition

Dependency tree · 2 levels

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