TheoremStatement: AI-adaptedProof: AI-generated

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A fixed-law Wigner spectral-edge input

Statement

Let $μ$ be one fixed compactly supported probability distribution on $R$ with $\int x d μ (x) = 0$ , $\int x^{2} d μ (x) = σ^{2}$ , and $\int x^{3} d μ (x) \neq 0$ , where $σ \geq 0$ . Here compactly supported means that $μ ([- L, L]) = 1$ for some real $L > 0$ . Suppose that a constant $K > 0$ satisfies $\int ∣ x ∣^{k} d μ (x) \leq K^{k}$ for every integer $k \geq 4$ .

For each positive integer $N$ , let the real random variables $a_{i j}$ , $1 \leq i \leq j \leq N$ , be independent and identically distributed with law $μ$ , and put $a_{j i} = a_{i j}$ for $i < j$ . Set $M_{N} = (a_{i j})$ and $A_{N} = N^{- 1 / 2} M_{N}$ . The spectral norm here is $∥ A_{N} ∥ = \sup {∥ A_{N} v ∥_{2} : v \in C^{N}, ∥ v ∥_{2} = 1}$ .

For every $ε > 0$ there is a deterministic real sequence $r_{N} = o (N^{- 6 / 11 + ε})$ such that $P (∥ A_{N} ∥ \leq 2 σ + r_{N}) \to 1$ as $N \to \infty$ . The probability spaces may vary with $N$ , but the law $μ$ and its moment bound are fixed.

Statement references

Asymptotic quantifiers and uniform parameter ranges

Finite vertex Hilbert space in the article convention

Independent random elements

Moments, variance, and covariance on a probability space

Probability measures and probability spaces

Random elements and real random variables

Law or distribution of a random element

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

External input

Péché and Soshnikov, Section 1, equation (1), Theorem 1.1 and Remark 1.1, supply the fixed-law bound for $\max_{i} ∣ λ_{i} (A_{N}) ∣$ . No local proof of the imported estimate is claimed. This input alone supplies no uniform estimate for a law varying with $N$ . The proof below only identifies this eigenvalue bound with the norm in the Statement.

Facts & Assumptions

[F1]

The vertex space $C^{N}$ is finite-dimensional, has the displayed Euclidean norm, and represents the adjoint by conjugate transpose; real symmetric matrices are self-adjoint. The library and vertex inner-product conventions have the same norm and orthogonality. Finite vertex Hilbert space in the article convention

[F2]

A normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis. Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

[F3]

On a nonzero finite-dimensional inner product space, the operator norm is the maximum of $∥ T v ∥_{2}$ over unit vectors $v$ . The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖

[F4]

For the fixed law and the independent identically distributed entries including the diagonal specified in the Statement, the imported Theorem 1.1 and Remark 1.1 give, for every $ε > 0$ , a deterministic $r_{N} = o (N^{- 6 / 11 + ε})$ with $P (\max_{j} ∣ λ_{j} (A_{N}) ∣ \leq 2 σ + r_{N}) \to 1$ .

Proof

Given: The law, matrices and hypotheses in the Statement, and $ε > 0$ .

Goal: Establish the conclusion in the statement.

1.1F1F2F3algebra

Fix $N \geq 1$ and a realization of $A_{N}$ . By [F1], $A_{N} = A_{N}^{*}$ , so $A_{N} A_{N}^{*} = A_{N}^{*} A_{N}$ : the endomorphism is normal, meaning that it commutes with its adjoint. Thus [F2] applies and gives an orthonormal eigenbasis $u_{1}, \dots, u_{N}$ , unchanged by the convention transfer in [F1]. Put $m_{N} = \max_{j} ∣ λ_{j} (A_{N}) ∣$ . For $v = \sum_{j} c_{j} u_{j}$ with $∥ v ∥_{2} = 1$ , orthonormality gives $\sum_{j} ∣ c_{j} ∣^{2} = 1$ and $∥ A_{N} v ∥_{2}^{2} = \sum_{j} ∣ λ_{j} (A_{N}) ∣^{2} ∣ c_{j} ∣^{2} \leq m_{N}^{2}$ . A unit eigenvector attaining this finite maximum gives equality. Consequently [F3] gives $∥ A_{N} ∥ = m_{N}$ ; the same attained maximum equals the supremum used in the Statement.

2.1givenF4step 1.1∎

The law, its moments and its moment bound are fixed independently of $N$ , and all upper-triangular entries, including the diagonal, have that law independently. These are exactly the model hypotheses of [F4]. Choose its sequence $r_{N}$ . The pointwise identity in step 1.1 makes the events ${∥ A_{N} ∥ \leq 2 σ + r_{N}}$ and ${m_{N} \leq 2 σ + r_{N}}$ equal for every $N$ . Their probabilities therefore have the limit in [F4], proving the Statement with its fixed-law scope.

Depends on

Used by · 1

The reported centered-adjacency estimate and its sparse-law limit

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

Wigner random matrices with non-symmetrically distributed entries · Section 1, equation (1), Theorem 1.1 and Remark 1.1, PDF pages 1–2

For $A_N=N^{-1/2}(a_{ij})$ in the fixed-law model of equation (1), Theorem 1.1 gives $\lambda_{\max}(A_N)\le2\sigma+o(N^{-6/11+\varepsilon})$ with probability tending to one for every $\varepsilon>0$. Remark 1.1 gives the same bound for $\max_i|\lambda_i(A_N)|$.

Hypotheses: One compactly supported real law $\mu$, independent of $N$, with mean zero, variance $\sigma^2$ where $\sigma\ge0$, nonzero third moment, and $\int|x|^k\,d\mu(x)\le K^k$ for all integers $k\ge4$ with fixed $K>0$. The entries indexed by $i\le j$, including the diagonal, are independent with common law $\mu$; $a_{ji}=a_{ij}$.

Reading limits: Checked the extracted PDF text of Section 1, equation (1), Theorem 1.1 and Remark 1.1 (PDF pages 1–2). The source proof was not reconstructed. PDF screenshots were unavailable. No extension to laws depending on N is imported; the norm identification is derived locally below.

Sources

Wigner random matrices with non-symmetrically distributed entries · Section 1, equation (1), Theorem 1.1 and Remark 1.1, PDF pages 1–2