LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Uniform constant-two adjacency control above a large logarithmic density

Statement

For every fixed $e > 0$ there is $g_{e} > 1$ such that $\sup_{p \in [g_{e} \log n / n, 1]} P (∥ A - p J ∥ > (2 + e) \sqrt{n p}) ⟶ 0.$ Here $A$ is the zero-diagonal adjacency matrix of $G (n, p)$ ; for small $n$ an empty parameter interval imposes no requirement. All matrix norms are Euclidean operator norms.

Facts & Assumptions

[F1]

Normalize by Uniform centered normalization across the full Bernoulli density range. On the uniform degree good event, Uniform squared-row control including dense graphs gives $r \leq \sqrt{1 + δ}$ .

[F2]

The uniform tail at fixed $t > 0$ is A parameter-uniform nonbacktracking spectral tail. The probability of a union of the two bad events is at most the sum of their probabilities by The finite union bound.

[F3]

The graph law is Erdős–Rényi random graph and high probability; adjacency and the matrices $J, I$ are defined in Adjacency, degrees, graph Laplacian, and spectral notation. Uniform limits, including the empty-interval convention, are Asymptotic quantifiers and uniform parameter ranges.

[F4]

The operator norm is the maximum over unit vectors in The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖. The vector norm is homogeneous and satisfies the triangle inequality by The inner-product norm is definite, homogeneous, and satisfies the triangle inequality.

[F5]

The integral logarithm $L$ agrees with $\log$ on $(0, \infty)$ by The integral logarithm L is the published natural logarithm, is strictly increasing by The integral logarithm is continuous and strictly increasing on (0,∞), and is unbounded above by The integral logarithm is unbounded above and below. The natural numbers are cofinal in the reals by Every complete ordered field is Archimedean.

External input

BBK Theorem 2.2 gives $∥ H ∥ \leq r f (ρ (B) / r) + 7 b$ for a finite Hermitian $H$ , where $r = \max_{i} (\sum_{j} ∣ H_{i j} ∣^{2})^{1 / 2}$ , $b = \max_{i j} ∣ H_{i j} ∣$ , and $f (x) = 2$ on $[0, 1]$ , $f (x) = x + x^{- 1}$ on $[1, \infty)$ . Its $B$ is Definition 2.1. The formula is used only for $r > 0$ ; $H = 0$ is handled separately. This deterministic theorem is imported, while its graph application and the uniform probability argument are local.

Proof

Given: The objects and hypotheses in the Statement.

Goal: Prove all stated conclusions.

1.1F1F2givenalgebra

Put $u = \min {e / 8, 1 / 8}$ , $δ = u$ , $t = u$ , and choose $g_{e} > \max {1, 3 / u^{2}}$ . Off a union of two events whose probabilities tend uniformly to zero by [F1] and [F2], $r \leq \sqrt{1 + u} \leq 1 + u$ and $ρ (B) < 1 + u$ . For $r > 0$ , the imported deterministic inequality gives $∥ H ∥ \leq r f (ρ (B) / r) + 7 / \sqrt{μ} \leq 2 \max {r, ρ (B)} + 7 / \sqrt{μ} \leq 2 + 2 u + 7 / \sqrt{μ}$ . Indeed for $ρ \leq r$ its first term is $2 r$ , and for $ρ \geq r$ it is $ρ + r^{2} / ρ \leq 2 ρ$ . When $r = 0$ , $H = 0$ and the same bound holds directly.

2.1F1F3F4step 1.1algebra

For any two operators $S, T$ on the same nonzero Euclidean space and any unit vector $v$ , [F4] gives $∥ (S + T) v ∥ \leq ∥ S v ∥ + ∥ T v ∥ \leq ∥ S ∥ + ∥ T ∥$ . Taking the maximum yields $∥ S + T ∥ \leq ∥ S ∥ + ∥ T ∥$ . Vector homogeneity and the same maximum formula give $∥ a T ∥ = ∣ a ∣ ∥ T ∥$ and $∥ I ∥ = 1$ . Apply these facts to the centering identity $A - p J = \sqrt{μ} H - p I$ from [F1], using $0 < p \leq 1$ and step 1.1, to obtain $∥ A - p J ∥ / \sqrt{μ} \leq ∥ H ∥ + p / \sqrt{μ} \leq 2 + 2 u + 8 / \sqrt{μ}$ .

3.1F2F3F5step 1.1step 2.1algebra∎

For any real $M$ , [F5] gives $b > 0$ with $L (b) > M$ and a natural $N > b$ . For $n \geq N$ , $\log n = L (n) \geq L (N) > M$ , so $\log n \to \infty$ . Thus $μ \geq g_{e} \log n \to \infty$ with a lower bound independent of $p$ . Since $2 u \leq e / 4$ , the bound in step 2.1 is eventually at most $2 + e$ , with one common cutoff for all $p$ . The only exceptional events are those in step 1.1; [F2] bounds their union probability by the sum of two probabilities tending uniformly to zero. The uniform quantifiers in [F3] therefore give the stated supremum limit, including every allowed varying sequence $p = p (n)$ .

Depends on

Used by · 1

Synchronization for sufficiently large logarithmic density constants

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.

External dependencies

Florent Benaych-Georges, Charles Bordenave and Antti Knowles, Spectral radii of sparse random matrices (19 September 2019) · Section 2, PDF p. 3 Definition 2.1 and (2.2); PDF p. 4 Theorem 2.2

For a finite Hermitian H and its associated nonbacktracking B, with r=max_i(sum_j |H_ij|^2)^(1/2)>0 and b=max_ij |H_ij|, ||H||<=r f(rho(B)/r)+7b, where f(x)=2 for 0<=x<=1 and f(x)=x+x^(-1) for x>=1.

Hypotheses: Finite Hermitian H, the specified ordered-pair nonbacktracking matrix; r>0 for the displayed ratio. The zero matrix case is handled locally.

Reading limits: Read the extracted PDF p. 3 Definition 2.1 and equation (2.2), and p. 4 Theorem 2.2, checking the ordered-pair matrix definition and the exact deterministic inequality against this application. The theorem is imported as stated; its proof was not read, reconstructed or independently audited. The graph normalization, probability estimates and local composition are supplied separately.

Sources

Florent Benaych-Georges, Charles Bordenave and Antti Knowles, Spectral radii of sparse random matrices (19 September 2019) · Section 2, PDF p. 3 Definition 2.1 and (2.2); PDF p. 4 Theorem 2.2