{"id":"4d343201-5e44-45c9-b4c3-18fbd3074084","arxiv_id":"2508.03631","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bulk local eigenvalue statistics of sparse complex non-Hermitian random matrices are universal and match the complex Ginibre ensemble.","lead":"This paper proves that the small-scale eigenvalue patterns of sparse complex random matrices match the universal Ginibre ensemble. The result covers matrices with only slightly more than four moments and relies on new multi-resolvent local laws.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The strengthened bulk single-resolvent law asserted in Theorem 3.1 is not proved in the text; since the multi-resolvent law and the final universality argument rest on it, the proof is conditional on that missing derivation.","rationale":"The reader's weakest assumption and my independent reading converge on the same point: Theorem 3.1 contains an error rate strictly stronger than the cited source [17, Theorem 3.1], and the text supplies only a promise that the bulk analogue of [17, Theorem 3.4] can be proved. This is load-bearing because the subsequent sparse multi-resolvent local law, Theorem 1.3, uses (3.18)-(3.19) as its m=0 input and propagates these rates through the zigzag and cumulant arguments. The final claim, Theorem 1.1, then needs N^{-omega} margins in the comparison of resolvent traces in Section 4. A weaker single-resolvent error would not by itself refute universality, but it would invalidate the proof as written unless other steps recover the missing powers. I did not find a circular derivation or a fitted parameter: the local laws are derived from stated inputs and applied to membership in M_N, and the core multi-resolvent argument is substantial and largely self-contained. The appropriate verdict remains CONDITIONAL, as the reader already gave it, so I recommend UNCHANGED rather than a new verdict category. The one concrete check that would settle the concern is to write out the bulk upgrade of the single-resolvent law and trace the resulting rates through the base case of Section 3.4 and the moment-matching step in Section 4.","tokens_in":40023,"tokens_out":4946,"duration_ms":58860,"concrete_test":"Derive the claimed bulk upgrade of the single-resolvent law by carrying out the 'similar arguments' from the edge strengthening in [17, Theorem 3.4] for all w in D(delta, tau), eta >= N^{-1+tau}, and compare the resulting rate with (3.18)-(3.19). If the derivation yields only (N eta)^(-1/6) + q^(-1/3), rerun the base case of the zigzag induction in Section 3.4 (m=0 in Lemmas 3.3 and 3.5) and the moment-matching bound (4.4)-(4.5); if the final error is not N^{-omega} for some omega > 0, Theorem 1.1 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.1) is reached through Theorem 1.3, and the proof of Theorem 1.3 uses as its base input the strengthened single-resolvent law (3.18)-(3.19), with E_av = 1/(N eta) + 1/q and E_iso = 1/sqrt(N eta) + 1/q. The text immediately after Theorem 3.1 concedes that the cited [17, Theorem 3.1] only gives (N eta)^(-1/6) + q^(-1/3) and says that 'similar arguments can be made in the bulk' to upgrade it. That upgrade is not supplied here: Section 3 uses it as the m=0 base case and in Lemmas 3.3-3.6 and Lemma 3.10, and Appendix B is explicitly only a sketch and is titled 'Proof of Proposition 3.1' rather than a derivation of the bulk single-resolvent law. If the best available bulk single-resolvent error were the weaker one, then at eta = N^{-1+tau} the base error would be N^{-tau/6} + N^{-epsilon/3} instead of N^{-tau} + N^{-epsilon}; all subsequent Psi_av/Psi_iso error bounds in Section 3 would inherit the weaker rate unless the zigzag/cumulant steps independently regain the missing powers. Since the moment-matching argument in Section 4 uses local-law bounds with margins N^{-omega}, the full universality theorem is currently conditional on an unverified strengthening. This is a correctness risk, not an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims bulk local universality for complex non-Hermitian random matrices with independent entries satisfying the sparse moment condition in Definition 1.1, i.e., E|X_ij|^r <= C_r/(N q^{r-2}) with q >= N^epsilon. This includes sparse Bernoulli-type matrices and, by a truncation argument, matrices with 4+epsilon moments. The main theorem (Theorem 1.1) asserts that the k-point eigenvalue correlation functions at scale N^{1/2} around a fixed point z in the unit disk converge to the GinUE correlation functions. The proof is organized around a deterministic membership class M_N (Definition 1.2), a result for Gaussian-divisible matrices (Theorem 1.2), and a sparse multi-resolvent local law (Theorem 1.3) that shows matrices in S_N(epsilon) lie in M_N with high probability. The bulk of the paper develops the multi-resolvent local law using the zigzag method combined with iterated cumulant expansions, and the final section sketches a moment-matching argument for Theorem 1.1.","tokens_in":40301,"tokens_out":6174,"duration_ms":67551,"significance":"If the proof is completed, the result is a substantial advance: it extends bulk universality for non-Hermitian random matrices to sparse models and to moment conditions weaker than bounded fourth moments, and it develops multi-resolvent local laws of arbitrary finite order for sparse non-Hermitian matrices. The paper is clearly written, with a useful reduction to the deterministic conditions of M_N and a detailed account of the zigzag method in the sparse setting. However, as written the main theorem is conditional on a strengthened bulk single-resolvent law (Theorem 3.1) whose proof is not supplied. Several other load-bearing steps are only sketched. The central idea is defensible, but the current version does not yet constitute a complete proof.","major_comments":[{"comment":"The strengthened bulk single-resolvent law with error E_av = 1/(N eta) + 1/q and E_iso = 1/sqrt(N eta) + 1/q is the base input for the entire multi-resolvent proof, but it is not proved in this manuscript. The text acknowledges that [17, Theorem 3.1] only gives (N eta)^{-1/6} + q^{-1/3} and then asserts that 'Similar arguments can be made in the bulk'; no derivation is supplied. Appendix B proves only Proposition 3.1 and is explicitly a sketch, not a derivation of the strengthened single-resolvent law. If the stronger law is unavailable, the error rates in Theorem 3.2 and all subsequent bounds (including the final universality theorem) are not justified. This is a load-bearing gap and must be fixed before the manuscript can be accepted.","section":"Section 3, Theorem 3.1, Eqs. (3.18)-(3.19)"},{"comment":"The global multi-resolvent law is used as the initial condition for the zigzag flow in Proposition 3.1 and Section 3.4, but its proof is only sketched ('we merely give a sketch'). The sketch does not fully present the induction for the averaged law, the detailed treatment of all cumulant expansion terms, or the verification that every term satisfies the claimed Psi_iso/Psi_av bounds. Since the flow propagation in Proposition 3.1 depends on the global law as input, this is a load-bearing step that needs a complete proof.","section":"Section 3.1, Lemma 3.2"},{"comment":"The transfer of membership in M_N from X^{(i-1)} to X^{(i)} is proved only for Re w_j = 0 and |Im w_j| approx t; the final paragraph states that the arguments 'can be easily extended' to |Im w_j| >= N^{-1+epsilon} and |Re w_j| <= delta |Im w_j|. This extension is necessary for the induction in Theorem 1.2 (the events E_i with n_i = n_{i-1}/2 - 1) and for the eventual conclusion that X in M_N holds uniformly on D(delta, tau). The details are not supplied and the claimed extension is not immediate, especially for the quadratic form bounds in Lemma 2.3.","section":"Section 2, Lemma 2.2"},{"comment":"The moment-matching argument for the main theorem is only sketched. In particular, the bound (4.4) is justified by a single line 'by the local law |G_ij| ≺ 1 ∨ (N eta)^{-1}' without exhibiting the expansion to the required power p, the control of the remainder, or the dependence on the truncation of the cumulant expansion. The restriction of the eta-integral to [N^{-1-xi}, T] uses a stated generalization of [2, Theorem 2.2] to complex matrices that is asserted but not proved. These steps are standard in the literature, but since they are used to establish Theorem 1.1, the paper should either provide the full argument or give precise references that cover the complex, non-centered-z case.","section":"Section 4, proof of Theorem 1.1"}],"minor_comments":[{"comment":"The word 'Bernouilli' is misspelled; it should be 'Bernoulli'.","section":"Abstract and Definition 1.1"},{"comment":"There is a typo: 'redudancy' should be 'redundancy'.","section":"Definition 1.2"},{"comment":"The word 'preceeding' should be 'preceding' in the sentence 'the displayed equation immediately preceeding [24, Lemma 7.3]'.","section":"Section 2, before Lemma 2.1"},{"comment":"There is a typo: 'multliple' should be 'multiple', and 'a mutliple' should be 'a multiple'.","section":"Section 3.3, proof of Lemma 3.5, case ii"},{"comment":"The domain D(delta, tau) is defined twice, once in the introduction and once in Section 3, with identical content; a single definition would avoid possible confusion. Also, the paper cites [2] for a result on least singular values but the stated generalization to complex matrices and fixed z deserves a remark in the text or a reference where it is proved.","section":"Notation and references"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well-structured and the main approach is promising, but the proof is currently conditional on an unproved strengthening of the single-resolvent local law (Theorem 3.1). This is not an internal inconsistency, but it is a serious completeness gap that the author should be able to close by writing out the bulk upgrade or by adjusting the argument to work with the weaker [17] input. The other sketched steps (global multi-resolvent law, Lemma 2.2 extension, and the moment-matching section) also need to be filled in. I would accept a revised version that supplies these details. The paper is within the scope of math.PR and would be of interest to the community."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my honest read of Osman's paper. The real news is that Theorem 1.1 closes the bulk universality gap for sparse complex non-Hermitian matrices, and the 4+epsilon moment corollary via truncation is stronger than anything in the cited literature. The main technical engine, a sparse multi-resolvent local law for products of resolvents of the Hermitisation with arbitrary finite length, is new and worked out in substantial detail. I think Theorem 1.1 is likely true, and I found no circularity or fitted parameters: the local law is derived from independent inputs in [17], [5], [8], [9], then applied to membership in M_N.\n\nThat said, the proof as written is not complete. The load-bearing issue is Theorem 3.1. The text explicitly says [17, Theorem 3.1] gives only (N eta)^(-1/6)+q^(-1/3), then asserts 'similar arguments can be made in the bulk' to obtain E_av = 1/(N eta)+1/q and E_iso = 1/sqrt(N eta)+1/q. That upgrade is not proved here; Appendix B is a sketch of a simpler argument using the weak law as input, not the required strengthening. Since the multi-resolvent bounds in Lemmas 3.3-3.10 and the final universality argument use the stronger error, the whole theorem is conditional on this missing derivation. This is a verification gap, not an internal inconsistency, but it is load-bearing.\n\nLesser soft spots: Lemma 3.2 (global multi-resolvent law) is only sketched; Theorem 1.1's moment-matching argument in Section 4 is a sketch; and Lemma 2.2 relies on 'easily extended' arguments for |Im w| >= N^{-1+epsilon}. These are repairable, but a referee would need to see them done.\n\nThe paper is honest about its gaps: the note after Theorem 3.1 is explicit, and the appendix structure is transparently labelled. The citation pattern is fair, with self-citations limited to the author's directly relevant earlier work. I would send this to a serious referee. It deserves the time, but it needs major revision before it is publishable. If the strengthened single-resolvent law cannot be supplied, the main theorem may still hold with a weaker error, but then the 4+epsilon corollary would need re-checking.\n\nBottom line: important, plausible, not yet verified. Worth engaging with, but not citeable in its current form.","headline":"A substantial and likely true universality result, but the proof is conditional on a strengthened single-resolvent law that is asserted, not proved.","tokens_in":40888,"tokens_out":1894,"would_cite":false,"duration_ms":19883,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Bulk universality of local eigenvalue statistics holds for sparse complex non-Hermitian random matrices, matching the Ginibre ensemble.","keywords":["bulk universality","non-Hermitian random matrices","sparse matrices","local eigenvalue statistics","multi-resolvent local law","Hermitisation","Ginibre ensemble","circular law"],"falsifier":"Measure $\\langle G_z(w)\\rangle - \\langle M_z(w)\\rangle$ for a sparse Bernoulli matrix with mean $N^{-1+\\epsilon}$ at $\\eta$ slightly above $N^{-1+\\epsilon}$: if the deviation is of order $(N\\eta)^{-1/6}+q^{-1/3}$ rather than $(N\\eta)^{-1}+q^{-1}$, the assumed input fails and Theorem 1.1 has no proof. A direct numerical check of the two-point correlation function at scale $N^{-1/2}$ against GinUE would settle the theorem itself.","tokens_in":39754,"feed_emoji":"🎲","tokens_out":6985,"duration_ms":76212,"temperature":0.7,"pith_summary":"This paper proves that in the bulk of the spectrum, the local eigenvalue statistics of sparse complex random matrices are universal. For matrices whose $r$-th absolute moments decay as $N^{-1-(r-2)\\epsilon}$, the $k$-point correlation functions at scale $N^{1/2}$ around any fixed $z$ in the unit disk converge to the corresponding Ginibre ensemble correlation functions with error $O(N^{-\\omega})$. The model includes sparse matrices built from Bernoulli variables with mean $N^{-1+\\epsilon}$, so on the order of $N^{1+\\epsilon}$ nonzero entries, and a standard truncation argument extends the result to matrices with only $4+\\epsilon$ moments. This matters because sparsity breaks the usual two-resolvent proofs: the paper shows that evaluating bulk statistics forces control of products of an arbitrary finite number of resolvents of the Hermitisation. The main new ingredient is a sparse multi-resolvent local law for such products, proved by combining the Zigzag flow with iterated cumulant expansions.","feed_headline":"Sparse random matrices match Ginibre's bulk eigenvalue law","feed_subtitle":"Bulk universality now covers matrices with about N^{1+epsilon} nonzero entries, not just dense ones.","key_machinery":"The load-bearing object is the Hermitised resolvent chain $G_z(w_1)B_1\\cdots G_z(w_m)B_m$, where $G_z(w)=(W_z-w)^{-1}$ for $W_z=\\begin{pmatrix}0&X-z\\\\X^*-\\bar z&0\\end{pmatrix}$, and each $B_j$ is one of four deterministic $2N\\times 2N$ matrices $E_\\pm,F,F^*$ whose $N\\times N$ blocks are multiples of the identity; these chains track how eigenvalue correlations respond to insertions. The paper proves a sparse multi-resolvent local law: the normalised trace deviates from the deterministic approximation $M_z$ by $O_\\prec\\big((\\frac{1}{N\\eta}+\\frac{1}{q})\\eta^{-(m-a/2-1)}\\big)$ and entries by $O_\\prec\\big((\\frac{1}{\\sqrt{N\\eta}}+\\frac{1}{q})\\eta^{-(m-a/2)}\\big)$, where $a$ counts $F,F^*$ insertions and $\\eta$ is the smallest imaginary part of the spectral parameters. The law is obtained by following the Zigzag flow, which jointly evolves the matrix and the spectral parameters, and using iterated cumulant expansions to close the equations when sparsity forbids the usual reduction inequality.","core_discovery":"The central claim is Theorem 1.1: for any fixed $z\\in\\mathbb{D}$, any $k>0$, and any $f\\in C_c^2(\\mathbb{C}^k)$, the expectation of the $k$-point eigenvalue statistic of $X\\in S_N(\\epsilon)$ at scale $N^{1/2}$ around $z$ equals the corresponding Ginibre correlation integral plus $O(N^{-\\omega})$. The proof proceeds in two stages. First, for deterministic matrices satisfying the multi-resolvent bounds of Definition 1.2, the Gaussian-divisible matrix $X+\\sqrt{t}Y$ is shown to have universal bulk correlation functions already at $t=N^{-1+\\epsilon}$ (Theorem 1.2), improving the earlier threshold $N^{-1/3+\\epsilon}$. Second, matrices in the sparse class are shown to satisfy those multi-resolvent bounds with very high probability (Theorem 1.3), after which a moment-matching argument based on Girko's formula removes the Gaussian component and yields the full theorem.","pith_inferences":["If the strengthened bulk single-resolvent input holds as assumed, the same machinery likely yields bulk universality for real sparse non-Hermitian matrices by adapting the partial Schur decomposition, extending known real edge results to the bulk.","The condition 1.2(e) can be read as uniform control of $\\log\\det(W_{z+N^{-1/2}x}-i\\eta)$; a direct Girko-formula proof might bypass the partial Schur decomposition altogether, at the cost of controlling the same truncation errors.","Pushing sparsity toward bounded average degree, where $q=N^\\epsilon$ is no longer available, would require new ideas since the cumulant expansion gains vanish in that regime.","The sparse multi-resolvent law also provides the input needed to prove asymptotic normality of linear spectral statistics for sparse non-Hermitian matrices at mesoscopic scales, paralleling dense-matrix results."],"forward_implications":["For any fixed $z$ in the unit disk, the $k$-point correlation functions of sparse Bernoulli-type matrices coincide with the Ginibre correlation functions at the $N^{-1/2}$ eigenvalue scale, up to $O(N^{-\\omega})$.","The Gaussian-divisible ensemble $X+\\sqrt{t}Y$ reaches universal bulk statistics already at $t=N^{-1+\\epsilon}$, matching the optimal time scale known for Hermitian matrices.","Matrices with a finite $4+\\epsilon$ moment, without the sparse moment decay, also satisfy the universality theorem by truncation.","The same sparse multi-resolvent law implies that bulk eigenvector distribution results for dense non-Hermitian matrices carry over to the sparse class, as noted in the paper.","The moment-matching comparison of log-determinants gives an explicit polynomial-rate error between the statistics of a sparse matrix and those of its Gaussian-divisible regularization."],"supporting_citations":[{"why":"Supplies the single-resolvent local law for sparse directed matrices that the paper strengthens to the bulk via 'similar arguments'.","marker":"[17]"},{"why":"Establishes bulk universality for dense iid complex non-Hermitian matrices at $t=N^{-1/3+\\epsilon}$; the Gaussian-divisible analysis here extends it.","marker":"[24]"},{"why":"Introduces the Zigzag method for propagating multi-resolvent laws, the backbone of the sparse local law.","marker":"[9]"},{"why":"Proves optimal multi-resolvent local laws for Wigner matrices; its reduction inequality is replaced here by iterated cumulant expansions.","marker":"[8]"},{"why":"Provides the recursive moment estimates used for the global law and the iterated cumulant expansion technology for sparse random matrices.","marker":"[23]"},{"why":"Defines the deterministic approximation $M_z$ for resolvent chains and the calculus of regular matrices used in the bounds.","marker":"[5]"},{"why":"Supplies the circular law and least singular value estimates used in the Girko-formula truncation for Theorem 1.1.","marker":"[31]"},{"why":"Gives the least singular value bound for small $\\eta$ in the Green's function comparison argument.","marker":"[2]"},{"why":"Provides the truncation lemma that converts $4+\\epsilon$ moment assumptions into the sparse moment condition.","marker":"[13]"}],"fun_headline_variants":["Sparse non-Hermitian matrices match Ginibre in the bulk","Bulk universality extends to sparse complex random matrices","Sparse matrices now covered by Ginibre bulk law","Thinning the spectrum: sparse matrices hit universal bulk law","Non-Hermitian sparse matrices: bulk stats are Ginibre's"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on a strengthened single-resolvent local law for the bulk that is asserted as an extension of a known edge result ('similar arguments can be made in the bulk') and only sketched; if that sharpened error rate fails, the multi-resolvent law and the main universality theorem do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Sparse non-Hermitian matrices match Ginibre in the bulk","Bulk universality extends to sparse complex random matrices","Sparse matrices now covered by Ginibre bulk law","Thinning the spectrum: sparse matrices hit universal bulk law","Non-Hermitian sparse matrices: bulk stats are Ginibre's"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1336,"prompt_tokens":882,"completion_tokens":454,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":371}},"tokens_in":498,"tokens_out":454,"duration_ms":5761,"temperature":1.0,"reasoning_tokens":371,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:19:04.296482+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure $\\langle G_z(w)\\rangle - \\langle M_z(w)\\rangle$ for a sparse Bernoulli matrix with mean $N^{-1+\\epsilon}$ at $\\eta$ slightly above $N^{-1+\\epsilon}$: if the deviation is of order $(N\\eta)^{-1/6}+q^{-1/3}$ rather than $(N\\eta)^{-1}+q^{-1}$, the assumed input fails and Theorem 1.1 has no proof. A direct numerical check of the two-point correlation function at scale $N^{-1/2}$ against GinUE would settle the theorem itself.","supporting_citations":[{"cited_title":"Theory Relat","cited_arxiv_id":null,"evidence_quote":"Establishes bulk universality for dense iid complex non-Hermitian matrices at $t=N^{-1/3+\\epsilon}$; the Gaussian-divisible analysis here extends it."},{"cited_title":"Theory Relat","cited_arxiv_id":null,"evidence_quote":"Introduces the Zigzag method for propagating multi-resolvent laws, the backbone of the sparse local law."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves optimal multi-resolvent local laws for Wigner matrices; its reduction inequality is replaced here by iterated cumulant expansions."},{"cited_title":"Theory Relat","cited_arxiv_id":null,"evidence_quote":"Provides the recursive moment estimates used for the global law and the iterated cumulant expansion technology for sparse random matrices."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the deterministic approximation $M_z$ for resolvent chains and the calculus of regular matrices used in the bounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the circular law and least singular value estimates used in the Girko-formula truncation for Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the least singular value bound for small $\\eta$ in the Green's function comparison argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the truncation lemma that converts $4+\\epsilon$ moment assumptions into the sparse moment condition."}],"review_version":1}