{"id":"4c21dd92-d745-437f-bc6b-b83a17000591","arxiv_id":"1908.01653","paper_version":6,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For shifted real or complex Ginibre matrices near the spectral edge, the least singular value has optimal tail probability of order x in the complex case and square-root x with a Gaussian imaginary-part damping in the real case.","lead":"This paper proves sharp lower-tail estimates for the smallest singular value of a Gaussian random matrix shifted by a complex constant near the spectral edge. The bounds are optimal up to logarithms in both real and complex cases, and the real case requires a new supersymmetric contour calculation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Complex-case Corollary 2.4 lacks a worked derivation from Theorem 2.1; error in the 1-point function is too weak to control the tail via Markov alone.","rationale":"The reader identified Appendix A's meromorphic superbosonization extension as the weakest assumption. That is a genuine analytic premise, and the appendix does sketch a three-step Laguerre approximation argument; it is not machine-checked but is plausible and is the kind of standard approximation that can usually be completed. However, the larger gap in the paper, from the point of view of the central claim, is that Corollary 2.4 (the headline tail result) is asserted to follow from Theorems 2.1 and 2.3 by a Markov inequality, but the paper never exhibits the required estimate on the imaginary part of the resolvent at the scale E = c(N,z)x. Theorem 2.1 is for E Tr(Y - w)^{-1} with w = E + i0, and Theorem 2.3 in the real case bounds the absolute value at E > 0; neither statement directly yields the positive, log-corrected tail probability uniform in x → 0 and in x ≤ C. The constants in (13c) are for the trace itself, and the switch from the trace to an eigenvalue tail needs an explicit integration over the density of states with a lower bound for the contribution of eigenvalues below E. This is a standard but nontrivial step, and it is precisely where hidden constants or missing log factors could invalidate the claimed uniformity. The concern is testable by writing out the derivation; if the derivation goes through with existing error bounds, the paper is fine and deserves ACCEPT. Since the proof is absent rather than demonstrably wrong, CONDITIONAL is the appropriate adjustment: the verdict should be ACCEPT contingent on the corollary being explicitly derived from the stated theorems. This is the single most load-bearing concern because Corollary 2.4 is what the abstract and introduction present as the main result, and every downstream application ('edge universality', circular-law-type uses) relies on that tail bound, not on the unproved appendix.","tokens_in":42715,"tokens_out":2554,"duration_ms":24734,"concrete_test":"Write out the full Markov-chain argument for Corollary 2.4: start from ℱ Tr(Y - (E+i0))^{-1} = ∫ (E' - E)^{-1} ρ_Y(E') dE' with ρ_Y the empirical density, integrate the bound of Theorem 2.1 (complex) and the ℱ-part version of Theorem 2.3 (real) over E in [0, c(N,z)x], and verify that the resulting tail estimate matches (16)-(17) without any extra hidden factor. In particular, check the x → 0 (λ → 0) regime where (13c) only logs while (16) needs (1+|log x|)x, and the δ < 0 regime where Theorem 2.1 is stated for |δ| ≤ C N^{-1/2} but the corollary allows 1-|z|^2 > -C N^{-1/2}; re-derive the δ < 0 case of (16) directly from (13a) with the stated error term. If any step requires an unproven lower bound on the imaginary part or a new integration lemma, the corollary is not a 'straightforward' consequence and should be proved explicitly.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Corollary 2.4 calls the tail bound a 'straightforward Markov inequality' from Theorems 2.1 and 2.3, but no detailed derivation is given. The complex bound is P(lambda_1 <= x c(N,z)) <= (1+|log x|) x. Naively, Markov with Theorem 2.1 requires controlling ℱ Tr(Y - E + i0)^{-1} times E, and plugging values such as E = c(N,z) x/2 yields only an O((1+|log x|) x) bound if the error term O(N(1∨δ)(1+|log λ|)) is genuinely subleading on that scale. Yet the leading double integral in (13a) and the displayed RHS bounds (13c) are both for the trace itself, not for its imaginary part after multiplying by E/c(N,z), and the paper does not prove that the imaginary part has the required sign/positivity or that the error term remains small after taking the imaginary part and multiplying by the scale. In the real case the analogue is even starker: Theorem 2.3 bounds |E Tr(Y+E)^{-1}|, while ℱ Tr(Y - E + i0)^{-1} (which is what feeds the edge tail via the density of states) is not directly bounded by it; the displayed bound (15) has an e^{-N(ℱz)^2/2} N^{3/4} √E term that, after Markov, supplies only √x e^{-N(ℱz)^2/2}, matching the real-case claim (17) only if the E^{1/2} factor is taken at E = x c(N,z). But no lemma in Sections 5-6 states the imaginary-part bound at the scale E = c(N,z)x with explicit (1+|log x|) x control. The gap is thus the missing quantitative statement that ℱ E Tr[...] ≳ P(lambda_1 <= E)-type eigenvalue-counting identity holds uniformly through the transition window including x near 0 and δ negative down to -C N^{-1/2}.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the smallest eigenvalue λ1(Y_z) of Y_z=(X-z)(X-z)^* for N×N real or complex Ginibre matrices X, in the edge regime |z|≤1+CN^{-1/2}. The main result, Corollary 2.4, asserts the optimal lower tail bound P(λ1(Y_z)≤c(N,z)x)≲(1+|log x|)x in the complex case and P(...)≲e^{-N(Im z)^2/2}√x+(1+|log x|)x in the real case, with c(N,z)=min{N^{-3/2},N^{-2}|1-|z|^2|}. The proof uses the superbosonization formula to reduce the averaged resolvent trace to low-dimensional contour integrals: Theorem 2.1 gives an asymptotic formula in the complex case and Theorem 2.3 gives a bound on E Tr(Y+E)^{-1} in the real case; Corollary 2.4 is stated to follow by a Markov inequality. The complex-case derivation is corroborated against the Ben Arous–Péché kernel in Appendix C.","tokens_in":43141,"tokens_out":30322,"duration_ms":298622,"significance":"If correct, the result is a substantial improvement over the classical Sankar–Spielman–Teng bound in the previously unexplored edge regime, and it identifies a sharp real/complex transition controlled by Im z. The paper's strengths are its explicit use of the superbosonization method, the absence of fitted parameters, and the independent confirmation of the complex one-point function against the known contour-integral kernel in Appendix C. The real-case result is novel and goes beyond available Brézin–Hikami formulas.","major_comments":[{"comment":"The first term e^{-N(Im z)^2/2}(N^{3/4}∨N√|δ|)√E in the bound on |E Tr(Y+E)^{-1}| has the wrong E-scaling to imply the √x term in Corollary 2.4 via the stated Markov step. For E=xc(N,z), at δ=0 this term is O(√x), and after multiplying by the Markov factor 2E it becomes O(x^{3/2}N^{-3/2}), which is far smaller than √x; at δ=1 it is O(√x), and the same problem occurs. Moreover, for z=0 (δ=1), the known real Ginibre result (3) implies E Tr(XX*+xN^{-2})^{-1}∼N^2/√x for fixed small x, which contradicts the displayed √x term of (15). The lemmas in Section 6 suggest the intended first term is proportional to E^{-1/2}, not √E; if so, Eq. (15) must be corrected and the proof of Theorem 2.3 must be checked to confirm the E^{-1/2} factor.","section":"Theorem 2.3, Eq. (15)"},{"comment":"The derivation of Corollary 2.4 from Theorems 2.1 and 2.3 is not written out, and the advertised 'straightforward Markov inequality' is not straightforward in the stated form. In the complex case the needed step is an eigenvalue-counting inequality such as P(λ1≤E)≤π^{-1}∫_0^E Im E Tr(Y-t+i0)^{-1} dt, which should be stated explicitly because Theorem 2.1 bounds the trace, not the counting function. In the real case, after the correction of (15), the step P(λ1≤E)≤2E E Tr(Y+E)^{-1} should be displayed; without it, the claimed √x term has no visible proof. This omission is load-bearing because Corollary 2.4 is the main result of the paper.","section":"Corollary 2.4"},{"comment":"The proof of Theorem 2.3 in the case δ≥0 consists of a single sentence referring to Lemma 6.4 and the expansions (76)–(77). Given that the statement of (15) appears to contain a scaling error, the real-case proof needs to be expanded to show how the e^{-N(Im z)^2/2}E^{-1/2} (or corrected) term, the N^{3/2}(1+|log(NE^{2/3})|) term, and the δ<0 case are obtained from Lemmas 5.2, 6.2, 6.3 and 6.4. As written, the proof is not sufficiently checkable for a result of this specificity.","section":"Section 6, proof of Theorem 2.3"}],"minor_comments":[{"comment":"The statement 'uniformly in E≥0' is incompatible with the factor 1+|log(NE^{2/3})|, which is singular at E=0; the statement should restrict to E>0 or clarify the interpretation of the logarithmic factor at E=0.","section":"Theorem 2.3"},{"comment":"The justification of the superbosonization identity for the meromorphic function F in (26)/(27) is an outlined approximation argument. Since this is a load-bearing analytic premise, the approximation steps in Appendix A should be presented in complete detail, in particular the Laguerre-polynomial approximation for coefficient functions with a pole at −1 and the control of the truncation order in the y-integration.","section":"Appendix A"},{"comment":"The caption of Figure 1 states that the difference between x- and √x-scaling is 'observable' for real z=±1, but it does not specify the x-axis scaling used in the second plot; this should be stated so that the reader can verify the claimed exponent against the theoretical c(N,z).","section":"Figure 1"},{"comment":"The notation ≲ in (4a) should specify that the constants are uniform in the stated x-range and in N→∞; currently the uniformity is only described in Corollary 2.4.","section":"Introduction, Eq. (4a)"}],"recommendation":"major_revision","confidential_remarks":"The scaling mismatch in Eq. (15) appears to be a correctable error — the proofs in Section 6 contain E^{-1/2} factors that are absent from the displayed statement — rather than a fatal flaw. However, it currently undermines the main real-case corollary, so the authors must correct the theorem statement and expand the derivation of Corollary 2.4. The complex-case part and the superbosonization setup are convincing and well corroborated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. The real-case bound is the real contribution; the complex case is honestly labeled as recoverable from Ben Arous–Peché and Shcherbina, and the authors don't oversell it. And the allegedly hand-wavy Markov step in Corollary 2.4 is actually fine; the proof is shorter than the paper makes it look.\n\nThe paper proves the sharp lower tail on the least singular value of (X−z)(X−z)* in the transitional regime |z| ≤ 1 + C N^{−1/2}, on the scale c(N,z) = min{N^{−3/2}, N^{−2}|1−|z|^2|}. Complex case: tail ≲ x(1+|log x|). Real case: tail ≲ e^{−N(Im z)^2/2}√x + x(1+|log x|), giving the real/complex transition controlled by Im z. The real-case bound is new and required the full superbosonization machinery; the complex-case formula is cross-checked against the Ben Arous–Peché kernel in Appendix C. That cross-check is good citizenship.\n\nThe proof is long but credible. The extension of superbosonization to the meromorphic integrand in Appendix A is a genuine technical point, handled with a truncation plus Laguerre approximation argument rather than glossed over. The contour estimates in Sections 5–6 are the meat, done carefully enough for a serious analyst to check. No circularity: the self-citations [20,21,22] are applications of the result, not inputs to it.\n\nSoft spots. Corollary 2.4 is stated as a 'straightforward Markov inequality' with no derivation. The worry that the imaginary part of the resolvent is not controlled does not land. In the real case, Theorem 2.3 bounds |E Tr(Y+E)^{−1}|, and since Tr(Y+E)^{−1} ≥ 1/(λ1+E), Markov with E = x c(N,z) gives exactly (17). In the complex case, the expected number of eigenvalues ≤ E equals (1/π)∫_0^E Im E Tr(Y−s−i0)^{−1} ds, and |Im| ≤ |Tr| combined with (13a)–(13c) yields the integrated x(1+|log x|) bound; the error term integrates to a lower order in the transitional window. The corollary is right, but it should have been proved in a few lines. A referee may reasonably ask for those lines.\n\nThe real-case Theorem 2.3 rests on a long chain of estimates with implicit constants; it is not machine-checked and would benefit from independent verification. That is a normal caveat for this kind of paper, not a defect.\n\nThis paper is for random-matrix theorists working on circular-law universality, edge universality for non-Hermitian matrices, and for anyone wanting a careful demonstration of superbosonization as a rigorous tool. Deserves a serious referee; accept after the short Markov derivation is added.","headline":"Real-case optimal tail bound is new and the complex case is honestly checked; the short Markov step in Corollary 2.4 is sound, and the paper deserves a serious referee.","tokens_in":43692,"tokens_out":7900,"would_cite":true,"duration_ms":73413,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a shifted Ginibre matrix, the least singular value's lower tail is optimal near the spectral edge, and in the real case a nonzero imaginary part of the shift suppresses the sqrt{x} term.","keywords":["least singular value","Ginibre ensemble","shifted random matrix","lower tail estimate","superbosonization","spectral edge","real versus complex symmetry","circular law"],"falsifier":"For real Ginibre matrices of size $N=200$, sample $\\mathrm{P}(\\lambda_1(Y_z)\\le c(N,z)x)$ for $z=1$ and $z=i$ at several small $x$; the theorem predicts a factor-$e^{-N/2}$ suppression of the $\\sqrt{x}$ term at $z=i$ but not at $z=1$. Numerically observing the same tail exponent for the two shifts, or a violation of the bound (17) at any fixed $x$, would contradict the central claim. A more direct check is to evaluate the real-case integral (34) at $z=i$, $\\delta=0$, $E=c(N)x$ and verify the claimed upper bound (15).","tokens_in":2095,"feed_emoji":"📉","tokens_out":2953,"duration_ms":89171,"temperature":0.7,"pith_summary":"This paper studies the smallest singular value of a large random matrix $X-z$, where $X$ has i.i.d. Gaussian entries and $z$ is a fixed complex shift. It proves that when $z$ lies within $N^{-1/2}$ of the spectral edge $|z|=1$, the probability that the least singular value falls below the natural scale $c(N,z)x$ is at most order $x$, up to logarithms, in the complex case. In the real case the same bound holds with an additional $\\sqrt{x}$ term that is suppressed by $e^{-N(\\Im z)^2/2}$, so real matrices behave like complex ones once the shift has a nonzero imaginary part. The estimate is optimal up to logarithmic factors because at $z=0$ it reproduces Edelman's exact tails. This matters because least-singular-value tails control condition numbers, the circular law, and edge universality for non-Hermitian matrices.","feed_headline":"Shifted Ginibre least singular value gets optimal tail bound","feed_subtitle":"The lower tail runs as x times logs, except near real shifts where a sqrt-x term survives.","key_machinery":"The load-bearing object is the superbosonization formula of Littelmann, Sommers and Zirnbauer, an identity that rewrites an integral over $N$ bosonic and $N$ fermionic variables as an integral over a small supermatrix with two contour variables in the complex case and three in the real case. Applied to the resolvent trace $\\mathrm{E}\\operatorname{Tr}(Y-w)^{-1}$, it yields the exact double-integral representation (28) for complex $X$ and the triple-integral representation (34) for real $X$, with meromorphic phase functions $f(x)=\\log\\frac{1+x}{x}-\\frac{|z|^2}{1+x}-wx$ and $g(a,\\tau,\\eta)$. These representations reduce the problem to low-dimensional contour integrals whose phase is governed by the cubic equation (11) of the matrix Dyson equation; outside the critical scale $c(N,z)$ a saddle-point analysis applies, while inside it the paper rescales the integral by $|z_*|$ and extracts a universal double-integral limit. Appendix A extends the superbosonization identity from holomorphic to the meromorphic functions (26)--(27), which is the step that makes the whole representation valid.","core_discovery":"The paper's central claim is that for the shifted Ginibre ensemble $Y_z=(X-z)(X-z)^*$, the least singular value $\\lambda_1(Y_z)$ has, uniformly for $1-|z|^2 > -C N^{-1/2}$, the lower tail\n$$\\mathrm{P}\\big(\\lambda_1(Y_z)\\le c(N,z)x\\big)\\lesssim (1+|\\log x|)\\,x$$\nin the complex case, and\n$$\\mathrm{P}\\big(\\lambda_1(Y_z)\\le c(N,z)x\\big)\\lesssim $e^{{-\\frac12 N(\\Im z)^2}}$\\sqrt{x} + (1+|\\log x|)\\,x$$\nin the real case, where $c(N,z)=\\min\\{N^{-3/2},\\, N^{-2}|1-|z|^2|\\}$. The bound is optimal up to logarithmic corrections: at $z=0$ it reproduces Edelman's exact tails, and it improves the classical uniform bound $\\mathrm{P}(\\lambda_1\\le x N^{-2})\\lesssim \\sqrt{x}$ of Sankar, Spielman and Teng in precisely the transitional regime where the spectral edge sits at a distance comparable to the eigenvalue spacing. The real-case formula exhibits a transition: when $\\Im z$ is of order one, the exponential factor suppresses the $\\sqrt{x}$ term, so real matrices behave like complex ones, whereas near $\\Im z=0$ the real $\\sqrt{x}$ behavior survives.","pith_inferences":["If the bound is right, the same optimal order should extend to smoothed-analysis condition numbers for rank-one perturbations $A_0+X$ when the determinant of $A_0$ is near the spectral edge, since the shift $z$ is the rank-one case.","The explicit real/complex transition suggests that local edge statistics of real non-Hermitian matrices should interpolate between symmetry classes depending on $\\Im z$, a phenomenon not established in the paper itself.","One can test the predicted $e^{-N(\\Im z)^2/2}$ suppression numerically by sampling real Ginibre matrices at $z=1$ and $z=i$ and comparing the tail exponents, which would give a direct check of the transition."],"forward_implications":["At $z=0$ the new bound matches Edelman's exact tails ($x$ in the complex case, $\\sqrt{x}$ in the real case), showing that the improvement is sharp up to logarithms.","For shifts with $1-|z|^2$ between $-C N^{-1/2}$ and $1$, the old $N^{-2}$-scale universal bound is replaced by the smaller scale $c(N,z)$, and the tail at that scale has power-law order $x$ up to logs.","In the real case, a shift with nonzero imaginary part removes the $\\sqrt{x}$ behavior, so real matrices inherit the complex tail away from the real axis.","The paper states that this tail bound supplies the missing input used in a companion work to prove edge universality for non-Hermitian random matrices with i.i.d. entries.","The bound also feeds into central limit theorems for linear eigenvalue statistics of complex and real i.i.d. matrices, as the paper notes."],"supporting_citations":[{"why":"Supplies the classical uniform bound $\\mathrm{P}(\\lambda_1(AA^*)\\le xN^{-2})\\lesssim\\sqrt{x}$ that the paper improves in the edge regime.","marker":"[53]"},{"why":"States the superbosonization formula that reduces the $N$-variable integrals to low-dimensional contour integrals in both symmetry classes.","marker":"[45]"},{"why":"Gives Edelman's exact $z=0$ tails for real and complex Ginibre, the benchmark showing the new bound is optimal up to logs.","marker":"[29]"},{"why":"Provides the Brézin-Hikami contour-integral correlation kernel for complex Ginibre, used in Appendix C to verify the complex asymptotic formula.","marker":"[10]"},{"why":"Gives precise conditions for superbosonization which Appendix A invokes when extending the identity to meromorphic functions.","marker":"[4]"},{"why":"Establishes that no eigenvalues lie outside the limiting support, used to isolate the edge regime $|z|\\ge 1+\\epsilon$ as having a trivial lower bound.","marker":"[8]"},{"why":"Supplies the matrix Dyson equation framework whose solution gives the limiting Stieltjes transform and the edge locations $e_\\pm$.","marker":"[2]"},{"why":"Companion work announced in the paper to rely on the present tail bound for edge universality of non-Hermitian matrices.","marker":"[21]"}],"fun_headline_variants":["Optimal lower tail for shifted Ginibre least singular value","Ginibre least singular value gets optimal edge tail","Optimal lower tail for Ginibre with shifts at the edge","Real Ginibre tail: sqrt(x) survives only near real shifts","Improved edge tail for shifted Ginibre least singular value"],"cache_read_input_tokens":45696,"weakest_assumption_plain":"The argument depends on extending the superbosonization identity to the meromorphic function in (26)/(27), whose pole at $\\langle s,s\\rangle=iN$ is away from the integration domain; if that extension is not valid, the integral representations (28) and (34), and hence every later bound, would lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["Optimal lower tail for shifted Ginibre least singular value","Ginibre least singular value gets optimal edge tail","Optimal lower tail for Ginibre with shifts at the edge","Real Ginibre tail: sqrt(x) survives only near real shifts","Improved edge tail for shifted Ginibre least singular value"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000698,"raw_usage":{"total_tokens":3151,"prompt_tokens":940,"completion_tokens":2211,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":2129}},"tokens_in":556,"tokens_out":2211,"duration_ms":16695,"temperature":1.0,"reasoning_tokens":2129,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:06:32.157719+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For real Ginibre matrices of size $N=200$, sample $\\mathrm{P}(\\lambda_1(Y_z)\\le c(N,z)x)$ for $z=1$ and $z=i$ at several small $x$; the theorem predicts a factor-$e^{-N/2}$ suppression of the $\\sqrt{x}$ term at $z=i$ but not at $z=1$. Numerically observing the same tail exponent for the two shifts, or a violation of the bound (17) at any fixed $x$, would contradict the central claim. A more direct check is to evaluate the real-case integral (34) at $z=i$, $\\delta=0$, $E=c(N)x$ and verify the claimed upper bound (15).","supporting_citations":[{"cited_title":"Universality of the least singular value for the sum of random matrices","cited_arxiv_id":"1908.04060","evidence_quote":"Supplies the matrix Dyson equation framework whose solution gives the limiting Stieltjes transform and the edge locations $e_\\pm$."}],"review_version":1}