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A parameter-uniform nonbacktracking spectral tail

Statement

For fixed $g > 0$ and $t > 0$ , the normalized matrices and ordered-pair $B$ of Uniform centered normalization across the full Bernoulli density range satisfy, for all sufficiently large $n$ and every $p$ with $g \log n \leq n p \leq n$ , $P (ρ (B) \geq 1 + t) \leq C n^{3 - c \sqrt{g \log n} \log (1 + t)} .$ Here $C, c > 0$ are universal constants; the displayed deterministic bound tends to zero. This is uniform over the probability parameter, not a simultaneous coupling assertion for all graphs.

Facts & Assumptions

[F1]

The independent centered entries satisfy all three variance/entry bounds with $κ = 1$ and $q = \min {\sqrt{n p}, n^{1 / 10}} \geq \sqrt{g \log n}$ eventually (Uniform centered normalization across the full Bernoulli density range).

[F2]

For every $x > 0$ , the integral logarithm satisfies $L (x) = \log x$ , where $\log$ is the natural logarithm used in [F1] and the imported estimate (The integral logarithm L is the published natural logarithm). Hence this same logarithm is unbounded above (The integral logarithm is unbounded above and below), strictly increasing on $(0, \infty)$ (The integral logarithm is continuous and strictly increasing on (0,∞)), and satisfies $\log 1 = 0$ (The integral logarithm satisfies L'(x)=1/x for x>0 and L(1)=0).

[F3]

The natural numbers are cofinal in the reals (The Cauchy-sequence reals are Archimedean).

External input

BBK Theorem 2.5 under Assumption 2.4 supplies universal $C, c > 0$ and the finite- $n$ inequality $P (ρ (B) \geq 1 + t) \leq C n^{3 - c q \log (1 + t)}$ for $t \geq 0$ , $κ \geq 1$ , $q > 0$ and $1 \lor q \leq n^{1 / 10} κ^{- 1 / 9}$ . The assumptions are independent centered Hermitian upper entries, row variance at most $1$ , entry variance at most $κ / n$ , and entries bounded by $1 / q$ . Definition 2.1 is exactly the ordered-pair formula in Uniform centered normalization across the full Bernoulli density range. The logarithm in this estimate is the natural logarithm identified with $L$ in [F2]. Remark 2.6 permits decreasing $q$ . This is the only imported probability estimate here.

Proof

Given: The objects and hypotheses in the Statement.

Goal: Prove all stated conclusions.

1.1F1F2givenalgebra

By the identification and properties in [F2], the natural logarithm in [F1] and the imported estimate satisfies $\log (1 + t) = L (1 + t) > L (1) = \log 1 = 0$ . Apply the imported BBK Theorem 2.5 to [F1]. Its cutoff $1 \lor q \leq n^{1 / 10} κ^{- 1 / 9}$ holds with $κ = 1$ . Remark 2.6 expressly permits the smaller capped $q$ , even if $n p$ is of order $n$ . Thus the theorem bounds the tail by $C n^{3 - c q \log (1 + t)}$ , at most the bound stated above. Every parameter in the source inequality is finite- $n$ and its constants are universal.

2.1F1F2F3step 1.1algebrachoose∎

Set $M = 25 / (c^{2} g (\log (1 + t))^{2}) > 0$ . Unboundedness in [F2] provides $b > 0$ with $\log b > M$ . By [F3] choose an integer $N > \max {b, 1}$ . For every integer $n \geq N$ , monotonicity in [F2] gives $\log n > \log b > M$ , so $c \sqrt{g \log n} \log (1 + t) > 5$ . Thus the bound in step 1.1 is at most $C n^{- 2}$ . For any $ε > 0$ , [F3] also gives an integer $N_{ε} > \sqrt{C / ε}$ ; then $C n^{- 2} < ε$ for $n \geq N_{ε}$ . Combining these cutoffs with the one in [F1] proves convergence with cutoffs depending only on $g, t, C, c, ε$ , uniformly in $p$ , including the deterministic centered matrix at $p = 1$ .

Depends on

Used by · 1

Uniform constant-two adjacency control above a large logarithmic density

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; PDF p. 4 Assumption 2.4, Theorem 2.5 and Remark 2.6

For an n by n Hermitian H with independent centered upper entries, row variance sum at most 1, entry variance at most kappa/n and entries bounded by 1/q, there are universal C,c>0 such that P(rho(B)>=1+t)<=C n^(3-c q log(1+t)) whenever 1 vee q<=n^(1/10) kappa^(-1/9), kappa>=1 and t>=0. Here B_(i,j),(k,l)=H_kl 1_(j=k) 1_(i!=l). Remark 2.6 explicitly permits replacing q by any smaller positive value.

Hypotheses: All Assumption 2.4 hypotheses just listed; finite n, q>0, kappa>=1; full upper triangular independence includes deterministic zero diagonal. No fixed entry distribution as n varies is imposed.

Reading limits: Read the extracted PDF statement of Definition 2.1 on p. 3 and Assumption 2.4, Theorem 2.5 and Remark 2.6 on p. 4. The probability estimate is imported as stated; its proof is not reconstructed or independently audited. The normalization and uniform-limit application are local. No reading of the remaining paper is claimed.

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; PDF p. 4 Assumption 2.4, Theorem 2.5 and Remark 2.6