{"id":"40834b77-ba61-4897-8c03-e1dff765d197","arxiv_id":"2412.01086","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"For sparse non-Hermitian random matrices with constant average degree d<1, the paper claims spectral radius zero with probability (1-d)e^{d+d^2/2} and Theta_p(1) otherwise, but the zero probability and lower-bound proof have serious errors.","lead":"A mathematical study of sparse random matrices claims that when the average number of nonzero entries per row is below 1, the matrix often has all-zero eigenvalues, and otherwise has moderate largest eigenvalue. The paper's key probability is miscalculated because it ignores short loops that already break the all-zero property.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The nilpotency probability in Corollary 1.4 uses the wrong graph model: X has independent diagonal and bidirectional off-diagonal entries, so P(rho(Z)=0) tends to 1-d, not (1-d)e^{d+d^2/2}.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing flaw: the acyclicity event from Theorem 2.4 is not the event that Z is nilpotent in the paper's matrix model. The matrix X has independent entries on all n^2 positions, so self-loops and bidirectional arcs are present with non-vanishing probability. Under the assumption P(Y_ij=0)=0, any directed cycle almost surely makes rho(Z)>0, so the probability of rho(Z)=0 equals the probability that the digraph has no directed cycle at all, which converges to 1-d in this model. The cited value (1-d)e^{d+d^2/2} is the acyclicity probability for a random digraph without loops and without 2-cycles (or equivalently, counting only cycles of length at least 3), so the paper's central quantitative claim is internally inconsistent with its own model. The qualitative phenomenon (zero spectral radius with positive probability, and Theta_p(1) otherwise) may be salvageable with the corrected probability, but the paper as written states a wrong constant. The upper-bound proof in Section 3.2 has a related gap: it asserts that self-loops can be ignored with high probability, which is false, and its path-counting estimate (8) does not account for loops as repeatable cycles. These issues independently support rejection. The reader's verdict of REJECT is therefore appropriate; no change is needed.","tokens_in":8374,"tokens_out":11423,"duration_ms":111235,"concrete_test":"Monte Carlo check for d=0.5 and n=2000: generate many realizations of X with independent Bernoulli(d/n) entries on the full n by n grid, and record whether the digraph contains any directed cycle of any length, including self-loops and 2-cycles. Corollary 1.4 predicts a cycle-free fraction near (1-d)e^{d+d^2/2}=0.962, while the 1-d prediction is 0.500. If the empirical fraction is near 0.5, the cited acyclicity theorem is being applied to the wrong model. Equivalently, re-derive the limiting Poisson mean of total directed cycles as -log(1-d) and compare with the mean -log((1-d)e^{d+d^2/2}) implied by Theorem 2.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on identifying the event {rho(Z)=0} with the event that G_d(n,d/n) is acyclic in Theorem 2.4. But the paper's own model has X_{ij} ~ Bernoulli(d/n) for every 1<=i,j<=n, so self-loops and 2-cycles occur with probabilities 1-e^{-d} and 1-e^{-d^2/2}, neither of which is o(1). Since P(Y_{ij}=0)=0, a directed cycle of any length almost surely forces a nonzero closed-walk product and hence rho(Z)>0; conversely an acyclic directed graph makes Z nilpotent. Thus rho(Z)=0 iff there is no directed cycle at all. In this model, the expected number of directed cycles is sum_{k=1}^n (n)_k/k (d/n)^k -> -log(1-d), and the standard Poisson approximation gives P(no cycles) -> 1-d. The value (1-d)e^{d+d^2/2} cited from [15] is acyclicity in a convention without self-loops and without 2-cycles; it does not apply to the matrix model here. Section 2.3's statement that 'with high probability, we may assume G_d(n,d/n) has no self-loops' is false: P(no self-loop)=e^{-d}. This also affects the upper-bound path count (8), which treats loop edges as non-cycle edges traversable at most once, whereas a loop is a directed cycle that can be repeated arbitrarily many times.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8675,"tokens_out":9227,"duration_ms":80578,"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":[{"comment":"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.","section":"Section 2.3 and Theorem 1.3/Corollary 1.4"},{"comment":"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.","section":"Theorem 2.4 and its application in Section 3.3"},{"comment":"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.","section":"Lemma 3.1"},{"comment":"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.","section":"Section 3.2, upper bound path counting"}],"minor_comments":[{"comment":"The title contains a typo: 'SP ARSE' should be 'SPARSE'.","section":"Title"},{"comment":"References [13] and [14] contain stray LaTeX control sequences '\\suppress' that should be removed.","section":"References"},{"comment":"In Definition 1.1, the notation 'lim_{n→∞} φ(n) → ∞' is imprecise; the intended statement is simply φ(n) → ∞ as n → ∞.","section":"Definition 1.1"},{"comment":"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.","section":"Section 3.3, fact (i)"}],"recommendation":"reject","confidential_remarks":"The reader's central concern is fully borne out: the model includes diagonal entries and independent bidirectional edges, so the nilpotency probability is not the acyclicity probability of a loopless digraph. The false limit in Lemma 3.1 is a separate, independent error that invalidates the lower-bound proof. These are load-bearing problems in the main theorem, not local presentation issues. The paper may be salvageable by excluding self-loops from the model and re-deriving the statistics, but that would be a substantially different result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the upper-bound proof is a fresh and plausible way to use Luczak's digraph structure, but the paper's central probability is computed in the wrong graph model and the lower-bound lemma has an algebraic error. Not publishable as written.\n\nWhat is actually new: applying Luczak's structural theorem for subcritical digraphs to weighted sparse matrices is a new angle, and the upper bound rho(Z)=O_p(1) is a reasonable argument. The path-counting idea—traverse non-cycle edges at most once, allow looping inside the small cyclic components—is sound in spirit. The intended dichotomy (zero versus constant order) is plausible and would be a useful subfield result if fixed.\n\nSoft spots, in order of severity:\n1. The zero-probability claim in Theorem 1.3/Corollary 1.4 is for the wrong model. The matrix has independent Bernoulli(d/n) entries on every ordered pair (i,j), so self-loops and 2-cycles are present with non-vanishing probability. The statement 'with high probability we may assume Gd(n,d/n) has no self-loops' is false; P(no loop) -> e^{-d}. A loop or 2-cycle is a directed cycle, and with nonzero weights it gives rho>0. The correct probability of no directed cycle at all is 1-d, not (1-d)e^{d+d^2/2}. The cited theorem [15] is for a loopless (or 2-cycle-free) convention.\n2. Lemma 3.1 is algebraically wrong. sigma_n(k) and nu_n(k) are linear in k, so C_n^{-sigma_n(k)/k} = exp(-Theta(n^2 log n)), which tends to 0, not 1. The lower bound rho >= theta(n) collapses.\n3. The lower bound also imports an unproven spectral-norm estimate from the author's preprint [9], with '≈' probabilities; that is not a substitute for a proof.\n\nWho it is for: researchers working on sparse non-Hermitian spectral edges might find the upper-bound approach worth examining, but I would not rely on the stated results. The main claims are not defensible in the current form.\n\nRecommendation for peer review: yes, it deserves a serious referee—the approach is novel and the upper bound may be salvageable—but the referee should be pointed at the graph-model mismatch and the Gelfand-formula algebra. I expect a reject in the current form, with a path to revision if the author can fix the probability and the lower bound.","headline":"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.","tokens_in":9246,"tokens_out":6649,"would_cite":false,"duration_ms":59734,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","05C80","15A18"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["spectral radius","sparse non-Hermitian random matrix","Erdős-Rényi digraph","directed cycles","phase transition","nilpotent matrix","Gelfand formula","constant average degree"],"falsifier":"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.","tokens_in":8094,"feed_emoji":"🎲","tokens_out":15911,"duration_ms":134090,"temperature":0.7,"pith_summary":"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 $0<d<1$, and whose weights are independent, zero-mean, unit-variance random variables that are never exactly zero. The main claim is that $\\rho(Z)$ is exactly $0$ whenever the underlying digraph has no directed cycle, and that this happens with probability converging to $(1-d)e^{d+d^2/2}$. On the complementary event, the radius is $\\Theta_p(1)$, meaning it stays between $1/\\varphi(n)$ and $\\varphi(n)$ with high probability for any $\\varphi(n)\\to\\infty$. The point is that in this constant-average-degree regime the usual moment bounds fail, and the spectral radius is controlled by the directed-cycle phase transition of the underlying graph rather than by a smooth limit.","feed_headline":"For sparse random matrices, spectral radius can be exactly 0","feed_subtitle":"Below the percolation threshold, no directed cycle means radius 0; with one, radius stays order 1.","key_machinery":"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.","core_discovery":"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 $Z$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the structural theorem that for $d<1$ every nontrivial strongly connected component of the random digraph is a short directed cycle; this is the skeleton used for both bounds.","marker":"[13]"},{"why":"It provides the limiting probability that the random digraph is acyclic, which the paper uses as the probability that $\\rho(Z)=0$.","marker":"[15]"},{"why":"It gives the upper-tail estimate on the spectral radius of sparse random matrices used in the supercritical upper bound and as a preliminary weak bound.","marker":"[2]"},{"why":"It supplies the sparse circular law, whose lower edge gives the supercritical lower bound $\\rho(Z)\\gtrsim\\sqrt{np_n}$.","marker":"[16]"},{"why":"It gives the quantitative Gelfand-formula lower bound used to turn a lower bound on $\\|Z^k\\|$ into a lower bound on $\\rho(Z)$.","marker":"[11]"},{"why":"It supplies the norm estimate on $Z$ that controls the correction factor in the Gelfand lower bound.","marker":"[9]"},{"why":"It gives the forest lemma used to show nilpotency when $np\\to 0$.","marker":"[12]"}],"fun_headline_variants":["Sparse random matrices can have exactly zero spectral radius","Zero spectral radius occurs with positive probability","Sparse non-Hermitian matrices: radius 0 or order 1","Sparse random matrix spectral radius: zero with positive chance","Zero spectral radius in sparse non-Hermitian random matrices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sparse random matrices can have exactly zero spectral radius","Zero spectral radius occurs with positive probability","Sparse non-Hermitian matrices: radius 0 or order 1","Sparse random matrix spectral radius: zero with positive chance","Zero spectral radius in sparse non-Hermitian random matrices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000797,"raw_usage":{"total_tokens":3482,"prompt_tokens":892,"completion_tokens":2590,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":2510}},"tokens_in":508,"tokens_out":2590,"duration_ms":16612,"temperature":1.0,"reasoning_tokens":2510,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:45:54.097261+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Journal of Graph Theory, 14:217-223, 1990","cited_arxiv_id":null,"evidence_quote":"It supplies the structural theorem that for $d<1$ every nontrivial strongly connected component of the random digraph is a short directed cycle; this is the skeleton used for both bounds."},{"cited_title":"On the probability that a random digraph is acyclic","cited_arxiv_id":null,"evidence_quote":"It provides the limiting probability that the random digraph is acyclic, which the paper uses as the probability that $\\rho(Z)=0$."},{"cited_title":"Spectral radii of sparse random matrices","cited_arxiv_id":null,"evidence_quote":"It gives the upper-tail estimate on the spectral radius of sparse random matrices used in the supercritical upper bound and as a preliminary weak bound."},{"cited_title":"The sparse ci rcular law under minimal assumptions","cited_arxiv_id":null,"evidence_quote":"It supplies the sparse circular law, whose lower edge gives the supercritical lower bound $\\rho(Z)\\gtrsim\\sqrt{np_n}$."},{"cited_title":"Linear Algebra and its Applications, 431(11):2134-2141, 2009","cited_arxiv_id":null,"evidence_quote":"It gives the quantitative Gelfand-formula lower bound used to turn a lower bound on $\\|Z^k\\|$ into a lower bound on $\\rho(Z)$."},{"cited_title":"Large deviations for the largest singular value of sparse non-Hermitian matrices","cited_arxiv_id":"2406.09851","evidence_quote":"It supplies the norm estimate on $Z$ that controls the correction factor in the Gelfand lower bound."},{"cited_title":"The largest eig envalue of sparse random graphs","cited_arxiv_id":null,"evidence_quote":"It gives the forest lemma used to show nilpotency when $np\\to 0$."}],"review_version":1}