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REVIEW 4 major objections 4 minor 19 references

Spectral radii of sparse non-Hermitian random matrices

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For a sparse non-Hermitian random matrix with fixed mean degree $d<1$, the spectral radius is zero with positive probability and of order one otherwise.

desk verdict Fresh upper-bound idea, but the nilpotency probability is computed in the wrong graph model and Lemma 3.1's algebra is wrong; not publishable as is. read the letter →

arxiv 2412.01086 v1 pith:4UHAP6XI submitted 2024-12-02 math.PR math.CO

Hyungwon Han This is my paper · ORCID
classification math.PRmath.CO MSC 60B2005C8015A18
keywords spectralradiussparsenon-HermitianrandommatrixErdős-RényidigraphdirectedcyclesphasetransitionnilpotentGelfandformulaconstantaveragedegree
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the spectral radius $\rho(Z)$ of an $n\times n$ matrix whose nonzero entries sit on the directed edges of an Erdős-Rényi digraph with edge probability $d/n$, for fixed $0

What carries the argument

The machinery is the structural phase transition of the Erdős-Rényi digraph $G^d(n,d/n)$: for $d<1$, with high probability every nontrivial strongly connected component is a directed cycle of length smaller than any prescribed slow-growing function. This turns spectral-radius control into cycle counting: forest edges outside cycles can be traversed at most once, while each cycle can be looped many times, so the entries of high powers of $Z$ factor into at-most-once forest weights and repeated cycle weights. The companion pieces are Gelfand's formula $\rho(A)=\lim_k\|A^k\|^{1/k}$, an explicit quantitative version that bounds how fast this limit is approached, and the known limiting probability that the random digraph is acyclic.

What would settle it

Simulate the model at $n=10^4$, $d=0.5$: generate many independent matrices $Z$, record the fraction with $\rho(Z)=0$ exactly, and separately record whether the skeleton contains any directed cycle. If the zero-radius fraction does not converge to $(1-d)e^{d+d^2/2}$ while the cycle/no-cycle event tracks it, then the identification of the acyclicity event with nilpotency is the step to revisit.

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Extended reading notes

Core claim

The central discovery, stated as Corollary 1.4, is that for fixed $0<d<1$ the spectral radius of the sparse non-Hermitian matrix $Z$ has an atomic limit: $\rho(Z)=0$ with probability converging to $(1-d)e^{d+d^2/2}$, and $\rho(Z)=\Theta_p(1)$ with the remaining probability. The upper bound $\rho(Z)=O_p(1)$ is proved by conditioning on the high-probability event that every nontrivial strongly connected component of the underlying digraph is a directed cycle of length smaller than any prescribed slow-growing function; paths in $Z^k$ can then be enumerated crudely and the product of weights along a path is bounded by a factor $t(n)^k$ with $t(n)\to\infty$. The lower bound $\rho(Z)=\Omega_p(1)$, conditioned on the existence of a cycle, follows by choosing a short cycle, using the assumption $\mathbb{P}(Y=0)=0$ so that all its weights are bounded away from zero with high probability, and applying an explicit Gelfand-formula lower bound to a high power of $Z$.

Load-bearing premise

The proof relies on identifying the event from the cited acyclicity theorem with the event that $Z$ is nilpotent; if the model's notion of a directed cycle includes short cycles such as 2-cycles, every such cycle makes $\rho(Z)>0$, so the numerical probability of the zero event would not be the same as the acyclicity probability used.

Editorial extensions

If this is right

  • For any fixed $0<d<1$, the spectral radius has a nonzero point mass at zero whose stated value is $(1-d)e^{d+d^2/2}$; this mass is absent in the supercritical regime where $\rho/\sqrt{np_n}\to 1$.
  • Conditioned on the existence of a directed cycle, the spectral radius is asymptotically neither tiny nor huge: $1/\varphi(n)\le\rho(Z)\le\varphi(n)$ with high probability for every $\varphi(n)\to\infty$.
  • In the subcritical case $np\to 0$, every component is a tree with high probability, so $Z$ is nilpotent and $\rho(Z)=0$ with high probability.
  • In the supercritical case $p\gg 1/n$, for Weibull weights with shape $\alpha$ and $pn\gg(\log n)^{2/\alpha}$, the normalized radius $\rho(Z)/\sqrt{np_n}$ converges to $1$ in probability.
  • The proof stops at $d=1$: for $d\ge 1$ a giant strongly connected component appears, and the short-cycle decomposition that carries the argument no longer holds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The cycle-decomposition picture suggests that below the critical mean degree, all spectral statistics of sparse non-Hermitian matrices should be expressible in terms of the Poissonian cycle counts of the skeleton, rather than in terms of a global eigenvalue density.
  • When the weights are allowed to be zero with positive probability, the lower bound on cycle-carrying components can fail because a cycle may carry a zero product; one would expect the zero-radius probability to increase and to depend on the weight distribution, not only on $d$.
  • The supercritical theorem's threshold $(\log n)^{2/\alpha}/n$ is set by the Weibull tail; replacing Weibull by bounded or heavy-tailed weights should shift the threshold, and the same Gelfand-based argument could be tested there.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies the spectral radius of an n×n sparse non-Hermitian random matrix Z = X⊙Y, where X has i.i.d. Bernoulli(d/n) entries (including the diagonal) and Y has i.i.d. entries with zero mean, unit second moment, and P(Y_ij=0)=0. For fixed 0<d<1, the main results assert that ρ(Z)=0 with probability converging to (1-d)e^{d+d^2/2}, and with the remaining probability ρ(Z)=Θ_p(1). The proofs use structural results on Erdős-Rényi digraphs (Łuczak 1990), a probability of acyclicity from Ralaivaosaona et al. (2020), and Gelfand-type spectral radius bounds.

Significance. The critical regime p=d/n for non-Hermitian sparse random matrices is largely open, and a correct result here would be valuable. However, the central claim is not defensible: the probability of nilpotency is misidentified, and the lower-bound proof contains a false limit. The actual probability of ρ(Z)=0 for the stated matrix model appears to be 1-d, not (1-d)e^{d+d^2/2}, because the model includes self-loops and 2-cycles with non-vanishing probability. The upper-bound argument, while likely repairable, is also incomplete as written because it assumes self-loops are negligible. These issues affect the main theorem and its corollary, so the paper in its current form does not establish its principal assertions.

major comments (4)
  1. [Section 2.3 and Theorem 1.3/Corollary 1.4] The assertion in Section 2.3 that 'with high probability, we may assume that G_d(n,d/n) has no self-loops' is false: P(no self-loop at a given vertex)=1-d/n, so P(no self-loops at all)=(1-d/n)^n → e^{-d}, which is not 1. Since the matrix model includes diagonal entries X_ii with P(Y_ii=0)=0, any self-loop forces ρ(Z) ≥ |Z_ii| > 0. Moreover, 2-cycles also occur with probability bounded away from zero (each unordered pair {i,j} has both directed edges with probability (d/n)^2, giving expected number d^2/2). The event {ρ(Z)=0} therefore requires the absence of all directed cycles, including self-loops and 2-cycles. The limit of this probability for the stated model is 1-d, not (1-d)e^{d+d^2/2}, which is the acyclicity probability for a digraph model without loops. Consequently the probability in Theorem 1.3 and Corollary 1.4 is overestimated.
  2. [Theorem 2.4 and its application in Section 3.3] Theorem 2.4 is cited from [15] for the probability that G_d(n,d/n) is acyclic, but the model in [15] is a random digraph without loops, where acyclicity means no directed cycles of length at least 2. The present matrix model has independent Bernoulli entries for all ordered pairs including i=j, so the cited event is not the event that Z is nilpotent. In fact, the claimed limiting probability (1-d)e^{d+d^2/2} is strictly larger than e^{-d}, while P(ρ(Z)=0) cannot exceed e^{-d} because any self-loop already gives a non-zero spectral radius. Thus the identification of acyclicity with nilpotency is invalid, and the proof of Theorem 1.3 collapses at this step.
  3. [Lemma 3.1] The displayed limit in Lemma 3.1 is algebraically wrong. From Theorem 2.6, σ_n(k)/k = ((n-1)^3/(n-2)^2) · (log(n-1)/log n) = O(n), so C_n^{-σ_n(k)/k} = (n^{3n/2})^{-O(n)} = exp(-O(n^2 log n)) → 0, not 1. The proof inserts an extra factor of k in the exponent of the first displayed limit. Since the subsequent application of Theorem 2.6 requires this prefactor to tend to 1 to obtain ρ(Z) ≥ θ(n) with high probability, the lower-bound proof of Theorem 1.3 fails.
  4. [Section 3.2, upper bound path counting] The upper-bound proof conditions on an event that excludes self-loops, but as noted in the first major comment, P(no self-loops) → e^{-d}, so this is not a high-probability event under the stated model. The path-count estimate (8) counts only cycles of length at least m and does not account for self-loops, which are directed cycles of length 1 that can be traversed arbitrarily many times. Thus the combinatorial bound on the number of paths contributing to (Z^k)_ij is incomplete for graphs with loops. This leaves a gap in the proof of Theorem 1.2, even though the conclusion ρ(Z)=O_p(1) may be salvageable by treating loops as cycles in the counting argument.
minor comments (4)
  1. [Title] The title contains a typo: 'SP ARSE' should be 'SPARSE'.
  2. [References] References [13] and [14] contain stray LaTeX control sequences '\suppress' that should be removed.
  3. [Definition 1.1] In Definition 1.1, the notation 'lim_{n→∞} φ(n) → ∞' is imprecise; the intended statement is simply φ(n) → ∞ as n → ∞.
  4. [Section 3.3, fact (i)] In the proof of fact (i), the phrase 'with the understanding that i = j_0' is awkward because j_0 is never defined; this phrase could be clarified or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main estimates are derived from external structural theorems and an independent spectral-norm estimate, while the identified self-loop issue is a correctness concern, not a circular reduction.

full rationale

The derivation chain does not reduce to its own inputs. Theorem 1.2 is proved directly from the external structural result of Łuczak [13] via path counting and Gelfand's formula; no parameter is fitted and no target quantity is defined in terms of the conclusion. Theorem 1.3 conditions on the existence of a directed cycle and applies the external lower-bound theorem [11] together with a spectral-norm estimate quoted from [9]. Although [9] is the present author's own prior work, it estimates ||Z||, the largest singular value, and does not assert the target spectral-radius result; it is an externally checkable statement and not defined in terms of rho(Z). The probability (1-d)e^{d+d^2/2} comes from the external acyclicity result [15], not from the paper's own assumptions, and no equation defines rho(Z) through that constant. The manuscript's Section 2.3 assertion that one may assume Gd(n,d/n) has no self-loops with high probability is false—P(no self-loop) converges to e^{-d}, not 1—but this is a substantive probabilistic/soundness concern affecting the nilpotency probability, not a circularity pattern. There is no fitted input called a prediction, no uniqueness theorem imported from the same authors, and no ansatz smuggled in via citation. Thus the appropriate circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on four imported results. Two are standard or plausible (Luczak structure, Kozyakin inequality), one is a self-cited preprint estimate with unstated tail assumptions, and one (the acyclicity constant) is misapplied to the nilpotency event. No new entities are postulated.

free parameters (1)
  • auxiliary sequences omega(n), eta(n), t(n), a(n), theta(n) = chosen as slowly growing or decaying functions
    Section 3.2-3.3 introduces these functions by hand: t(n)^2 >> eta(n) omega(n) for the upper bound, theta(n) -> 0 with P(|Y|>theta(n))^{a(n)} -> 1 for the lower bound. They are not fitted to data but are free choices that shape the proof.
assumptions (4)
  • domain assumption Luczak's structural theorem: for d<1, all nontrivial SCCs of G_d(n,d/n) are directed cycles of length < omega(n) w.h.p.
    Theorem 2.3, cited from [13]; the upper bound relies on this characterization of the subcritical random digraph.
  • standard math Kozyakin's Gelfand lower bound inequality (Theorem 2.6)
    External matrix inequality used in Section 3.3; the prefactors are claimed to tend to 1, but this is incompatible with the stated sigma_n(k), nu_n(k) definitions.
  • ad hoc to paper Spectral norm bound on Z from the author's preprint [9], depending on a tail parameter alpha
    Eq (16)-(18) in Section 3.3 import bounds on ||Z|| that require a tail condition (alpha), but the theorem assumptions on Y are only E|Y|^2=1 and P(Y=0)=0; alpha is not defined for arbitrary Y.
  • domain assumption Acyclicity probability limit from [15]: P(G_d(n,d/n) acyclic) -> (1-d)e^{d+d^2/2}
    Theorem 2.4 is used to claim rho(Z)=0 with that probability. In the paper's model with self-loops and 2-cycles allowed, the event of nilpotency is no directed cycles at all, with probability 1-d; the cited theorem appears to count only cycles of length at least 3.

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Pith. "Pith review of Spectral radii of sparse non-Hermitian random matrices." pith.science (2026). https://pith.science/paper/4UHAP6XI

@misc{pith2026241201086,
  author       = {Pith},
  title        = {Pith review of: Spectral radii of sparse non-Hermitian random matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4UHAP6XI}},
  note         = {Machine review of arXiv:2412.01086}
}
abstract

We provide estimates for the spectral radii of an $n\times n$ sparse non-Hermitian random matrix $Z$ with general entries in the regime $p=d/n$ where $0<d<1$ is fixed. Utilizing the structural results of ({\L}uczak, '90), we show that the spectral radius $\rho (Z)$ is $0$ with probability converging to some nonzero value, and satisfies the inequality $(\phi (n))^{-1}\leq \rho (Z)\leq \phi (n)$ in the asymptotic sense for any function $\phi$ satisfying $\lim_{n\to\infty}\phi (n)=\infty$ with the remaining probability.

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Reference graph

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