LemmaStatement: AI-generatedProof: AI-generated

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A fixed outlier gap dominates the critical-source error

Statement

Let $d_{n} > 0$ , $β_{n} \geq 9 / 4$ and $p_{n} \geq 0$ be deterministic real sequences with $d_{n} \to \infty$ . For each $n$ , let $(Ω_{n}, F_{n}, P_{n})$ be a probability space, let $m_{n} \geq 1$ be a deterministic integer, and let $A_{n}$ assign a real symmetric $m_{n} \times m_{n}$ matrix to every outcome in $Ω_{n}$ . Let $E_{n} \in F_{n}$ satisfy $P_{n} (E_{n}) \to 1$ . Write $λ_{\min}$ for the least eigenvalue, $∥ \cdot ∥$ for the vertex Hilbert-space operator norm, $J$ for the all-ones matrix of order $m_{n}$ , and $Λ (t) = t / \sqrt{t - 1}$ for $t \geq 2$ .

Let $C, c, θ > 0$ be fixed. Suppose that at every outcome in $E_{n}$ , $∣ λ_{\min} (A_{n}) / \sqrt{d_{n}} + Λ (β_{n}) ∣ \leq C (d_{n}^{- c (Λ (β_{n}) - 2)} + d_{n}^{- θ / 2})$ . Put $Δ_{n} = A_{n} - p_{n} J$ and $g = 9 / (2 \sqrt{5}) - 2 > 0$ . Assume that the set $H_{n} = {ω \in Ω_{n} : ∥ Δ_{n} (ω) ∥ \geq (2 + g / 2) \sqrt{d_{n}}}$ belongs to $F_{n}$ . Then $P_{n} (H_{n}) \to 1$ .

Statement references

A negative adjacency eigenvalue survives all-ones centering

A rational entropy test above the critical degree ratio two

Asymptotic quantifiers and uniform parameter ranges

Real powers for positive bases, with the zero-base positive-exponent convention

The natural logarithm as the inverse of the exponential function

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

The exponential function is strictly increasing

Basic identities for a probability measure

Divergence to +∞ and to -∞

Probability measures and probability spaces

Finite vertex Hilbert space in the article convention

The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖

Facts & Assumptions

[F1]

The all-ones centering satisfies $∥ A - p J ∥ \geq - λ_{\min} (A)$ for every real symmetric matrix of positive finite order and $p \geq 0$ . A negative adjacency eigenvalue survives all-ones centering

[F2]

For $t \geq 9 / 4$ the function $Λ (t) = t / \sqrt{t - 1}$ is at least $2 + g$ , with $g > 0$ . A rational entropy test above the critical degree ratio two

[F3]

The hypothesis $d_{n} \to \infty$ means that for every real $M$ there is an $N$ such that $d_{n} > M$ for every $n \geq N$ . High probability means that the probabilities of the specified measurable events tend to one; limits and eventual inequalities use the ordinary real-sequence quantifiers. Divergence to +∞ and to -∞ Asymptotic quantifiers and uniform parameter ranges

[F4]

For $x > 0$ and $r \in R$ , $x^{r} = \exp (r \log x)$ . The exponential is strictly increasing, the logarithm is strictly increasing on $(0, \infty)$ , $\log 1 = 0$ , and $\log (\exp u) = u$ and $\exp (\log x) = x$ on their stated domains. Real powers for positive bases, with the zero-base positive-exponent convention The natural logarithm as the inverse of the exponential function Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm The exponential function is strictly increasing

[F5]

In a probability space, measurable events $B \subseteq D$ satisfy $P (B) \leq P (D)$ ; in particular their probabilities lie in $[0, 1]$ . Basic identities for a probability measure Probability measures and probability spaces

[F6]

The finite vertex Hilbert space has norm $∥ v ∥_{2} = (\sum_{i} ∣ v_{i} ∣^{2})^{1 / 2}$ . On this nonzero space, a matrix $T$ has induced operator norm $∥ T ∥ = \max_{∥ v ∥_{2} = 1} ∥ T v ∥_{2}$ . Finite vertex Hilbert space in the article convention The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖

Proof

Given: The data and hypotheses in the Statement.

Goal: Prove $P_{n} (H_{n}) \to 1$ .

1.1F3F4givenalgebra

Fix $a > 0$ and $ε > 0$ . Set $M = \max {1, \exp (- \log (ε) / a)}$ . Since $d_{n} \to \infty$ , eventually $d_{n} > M$ . By [F4], $\log d_{n} > - \log (ε) / a$ , hence $0 < d_{n}^{- a} = \exp (- a \log d_{n}) < \exp (\log ε) = ε$ . Thus $d_{n}^{- a} \to 0$ for each fixed $a > 0$ .

2.1F2F3F4givenstep 1.1algebra

By [F2], $Λ (β_{n}) - 2 \geq g > 0$ . Eventually $d_{n} \geq 1$ ; then $\log d_{n} \geq 0$ by [F4], so $d_{n}^{- c (Λ (β_{n}) - 2)} \leq d_{n}^{- c g}$ . Apply step 1.1 to the positive exponents $c g$ and $θ / 2$ , with $ε = g / (4 C)$ for each. Eventually $C (d_{n}^{- c g} + d_{n}^{- θ / 2}) < g / 2$ . At every outcome in $E_{n}$ , the assumed error inequality therefore gives $- λ_{\min} (A_{n}) / \sqrt{d_{n}} \geq Λ (β_{n}) - g / 2 \geq 2 + g / 2$ .

3.1F1F3F5F6givenstep 2.1algebra∎

Using the norm convention in [F6], apply [F1] at every outcome in $E_{n}$ to obtain $∥ Δ_{n} ∥ \geq (2 + g / 2) \sqrt{d_{n}}$ for all sufficiently large $n$ . Thus $E_{n} \subseteq H_{n}$ . Both sets are measurable by hypothesis, so [F5] gives $P_{n} (E_{n}) \leq P_{n} (H_{n}) \leq 1$ . For each $ε > 0$ , eventually $P_{n} (E_{n}) > 1 - ε$ by [F3]; consequently $1 - ε < P_{n} (H_{n}) \leq 1$ . This proves the claimed limit.

Depends on

Used by · 1

A near-threshold counterexample to the reported constant-two spectral bound

Dependency tree · 2 levels

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