{"id":"76e7cc72-1fe6-44cf-b885-6eef1ca603fe","arxiv_id":"2608.13154","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For deformed GOE matrices with fixed minor size, the normalized largest principal-minor eigenvalue has an explicit limit with a phase transition at a=2; Wishart limits are characterized by an entropy-constrained convex set, explicit for Gaussian entries.","lead":"This paper finds exact asymptotic values for the largest eigenvalues among all small principal minors of Wishart and deformed Gaussian random matrices, including the case left open in earlier work. The new technique, based on Hausdorff convergence of the random sets of normalized minors, yields explicit constants and a phase transition at diagonal variance a=2.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central claims are internally consistent and the stated moment/scaling assumptions are explicit and used correctly.","rationale":"The reader's weakest assumption correctly identifies the sub-Gaussian moment condition and the scaling p/log n -> beta as the most assumption-sensitive part of the Wishart result. I agree that these assumptions are load-bearing in the sense that Theorem 6 is not claimed without them. However, I do not find a correctness gap: the assumptions are explicit, the proofs use them precisely, and the endpoint regimes are treated separately. I checked the most delicate technical steps: the domination argument in the Wishart upper bound, the density-product identity in the lower-bound overlap estimate, the Wigner recursion derivation, and the hypergeometric large-overlap bound. Each is internally consistent. The paper is long and not machine-checked, so residual risk exists, but no concrete technical objection surfaced. Therefore the reader's ACCEPT verdict should stand unchanged. The suggested numerical check of the recursion and the phase-transition formula is a reasonable, low-cost verification that would further de-risk the central Wigner claim.","tokens_in":35763,"tokens_out":34354,"duration_ms":296206,"concrete_test":"As a nonexhaustive verification, independently re-derive the recursion (6)-(7) by solving the one-dimensional maximization for a=3 and k=2,3,4 numerically and compare with the claimed values gamma_2(3)=2.268..., gamma_3(3), gamma_4(3). Also simulate a Gaussian Wigner matrix with diagonal variance a=3, n=10^6, and check that M_{n,2}/sqrt(2 log n) is within Monte Carlo error of gamma_2(3). A mismatch would indicate an error in Proposition 1 or the phase-transition formula.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the strongest claim as the fixed-k laws of large numbers for deformed GOE (Theorems 1-2) and Wishart matrices (Theorem 6, Corollary 1). The load-bearing conditions are: for Wigner, Gaussian entries and the convex-set characterization K_k(a); for Wishart, the sub-Gaussian condition (10), p/log n -> beta in (0,infinity), and the KL-constrained set Gamma_{k,beta,mu}. I examined the places where a hidden assumption could enter. In the Wishart upper bound (Section 10.1), the domination bound rho_{A,U}(C) <= mu^{ps}(C)/q_A is valid because conditioning on a positive-probability event cannot increase the unconditional probability of C; the binomial factor is only a loose upper bound. The exponential-tilt second-moment computation in Section 10.2 integrates the density product correctly, since the shared coordinates contribute exactly f_U and the remaining coordinates integrate to one. The Wigner recursion in Proposition 1 follows from the first-order condition of the one-dimensional optimization (6), and the hypergeometric bound in Section 8 is dominated by C(k,r)(k/n)^r via a falling-factorial comparison. I found no internally inconsistent step. The sub-Gaussian moment condition is stated explicitly, and the endpoint regimes beta=0 and beta=infinity are handled separately (Theorem 7 and the union-bound argument in Section 3). Disagreement with the Gaussian-only explicit formula for non-Gaussian mu is expected non-universality, not an error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper establishes laws of large numbers for the largest eigenvalues among all principal minors of deformed GOE matrices and Wishart matrices. For deformed GOE with off-diagonal variance one and diagonal variance a>0, it proves that M_{n,k}/√(2 log n) converges in probability to γ_k(a), with γ_k(a)=√(a+2(k−1)) for 0<a≤2 and, for a>2, an explicit recursion γ_m(a)=γ_{m−1}(a)+2/(γ_{m−1}(a)+√(γ_{m−1}(a)^2+4−a)); this resolves the case left open by Cai et al. (2021). The proof goes through a Hausdorff-convergence result for the random set of normalized principal minors to a deterministic convex set defined by convex constraints. For Wishart matrices with i.i.d. sub-Gaussian entries and p/log n→β∈(0,∞), the paper proves Hausdorff convergence to an entropy-constrained convex set Γ_{k,β,μ}; for standard Gaussian entries it identifies the limit λ_{k,β} as the unique solution of λ−logλ−1=2k/β. Endpoint regimes β=0 and β=∞ are also treated, and the results are applied to a compressed-sensing restricted-isometry threshold.","tokens_in":36058,"tokens_out":26878,"duration_ms":238224,"significance":"The Hausdorff-convergence framework is the paper's main methodological contribution and is well executed: the limiting sets are derived from first principles, the proofs are detailed, and the explicit phase transition at a=2 for fixed k is a genuine resolution of a previously open problem. The Wishart entropy-constrained characterization and its explicit Gaussian solution are substantial, and the compressed-sensing threshold provides a concrete falsifiable prediction. I checked the key steps—the tight-set chain argument, the Slepian-based superadditivity, the KL upper/lower bounds, and the endpoint arguments—and found no internal inconsistency. The moment condition (10) and the scaling p/log n→β are stated explicitly and are used where needed; no circularity or fitted constants are present.","major_comments":[],"minor_comments":[{"comment":"The phrase 'no close-form expression' should read 'closed-form expression', and 'Hausdorffdistance' in the abstract is missing a space.","section":"Abstract; Section 2.1"},{"comment":"The endpoint limit T_{n,k}/p→1 for p/log n→∞ is stated without proof; since it is part of the claimed full-range coverage, please add the short union-bound argument based on diagonal and off-diagonal tail bounds under (10), or give an explicit reference.","section":"Section 3"},{"comment":"In the row for k→∞, the denominator should be typeset unambiguously as 2√(k log(n/k)); the current '2√(klog(n/k))' is easy to misread.","section":"Table 1"},{"comment":"The reduction to strict constraints via mixtures is only sketched; a sentence noting that Q((1−δ)ν+δμ^{⊗k})=(1−δ)Q(ν)+δI_k and that all KL constraints become strict would make the argument fully self-contained.","section":"Section 10.2"},{"comment":"In the final step, the sentence beginning 'Let h=(h_1,...,h_n)^T, where h_1...' contains a duplicated 'where', and the reused symbol h for both the normal replacement variables and the standard normal vector could confuse readers.","section":"Section 9"}],"recommendation":"minor_revision","confidential_remarks":"I agree with the reader's assessment. The central claims are sound; the remaining issues are local and editorial. I recommend minor revision rather than outright acceptance only because a few stated results, notably the β=∞ endpoint, would benefit from a short proof in the text. No concerns about novelty or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a strong paper and it does what it says. The headline result is the fixed-k law of large numbers for extreme principal minors of deformed GOE for every diagonal variance a>0, including the a>2 case left open by Cai et al. The limiting constant has a phase transition at a=2 and an explicit recursion above it. The proof method is the real contribution: instead of second-moment calculations on the extreme value directly, they prove Hausdorff convergence of the random set of normalized principal minors to a deterministic convex set, then solve a convex program. That is a genuinely transferable technique, and the paper demonstrates it in two settings.\n\nThe Wishart half is also solid. For p/log n -> beta in (0,infty) and sub-Gaussian entries, the limit is characterized by an entropy-constrained set; for Gaussian entries they solve it explicitly and get lambda - log lambda - 1 = 2k/beta. The KL/Sanov machinery is handled carefully, particularly the overlap bounds in the second-moment argument. The compressed sensing corollary with the explicit beta_BP constant is a nice payoff.\n\nSoft spots, in proportion: they are minor. The Wishart theorem rests on the sub-Gaussian moment condition (10) and the fixed p/log n scaling; that is stated clearly and the endpoints are treated separately, so it is not hidden. There are typos (close-form, a missing figure which is just a production artefact). The growing-k Wigner result (Theorem 5) gives existence and bounds for E(c) but no closed form; the authors are upfront about that. I would not call any of this a flaw at the level of the central claims.\n\nThe citation pattern is honest: they build on Cai et al. 2021 and Jiang and Qi 2024, and the k=2 case was known; they say so. The constants are derived from convex problems, not fitted.\n\nWho this is for: anyone working on extreme spectral functionals, sparse PCA, or compressed sensing. It is long but self-contained; a serious referee should engage. My recommendation: accept with minor revision. I would not desk-reject this.","headline":"The a>2 deformed GOE gap is genuinely resolved here, and the Hausdorff-set method is the reusable idea; the Wishart part carries an explicit sub-Gaussian caveat, but it is stated and does not undermine the central claims.","tokens_in":36593,"tokens_out":7042,"would_cite":true,"duration_ms":56019,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","60F10","90C25","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the extreme principal-minor eigenvalues of deformed GOE and sub-Gaussian Wishart matrices converge to explicit deterministic limits, found by identifying the Hausdorff limit of the random set of normalized minors…","keywords":["principal minors","random matrices","Hausdorff convergence","Wishart matrices","Gaussian Wigner matrices","convex optimization","large deviations","restricted isometry property"],"falsifier":"Simulate a Gaussian Wishart matrix with $p=\\beta\\log n$, $\\beta=2$, $k=2$, and $n$ large; Corollary 1 predicts $T_{n,2}/p$ concentrates near the root $\\lambda\\approx 4.505$ of $\\lambda-\\log\\lambda-1=2$, so a persistent deviation in the sample mean of $T/p$ across many independent replicates would refute the claim. Alternatively, for deformed GOE with $a=4$ and $k=3$, $M_{n,3}/\\sqrt{2\\log n}$ should concentrate near $\\gamma_3(4)=2.9$.","tokens_in":35585,"feed_emoji":"🎲","tokens_out":16947,"duration_ms":128104,"temperature":0.7,"pith_summary":"This paper proves laws of large numbers for the largest eigenvalue among all $k\\times k$ principal minors of two random matrix families: deformed GOE (Gaussian Wigner) matrices and Wishart matrices with sub-Gaussian entries. For deformed GOE with fixed $k$ and any diagonal variance $a>0$, $M_{n,k}/\\sqrt{2\\log n}$ converges in probability to an explicit constant $\\gamma_k(a)$, with a phase transition at $a=2$; above the transition the constant is given by a recursion and the extremal minors are nested by inclusion. For Wishart matrices with $p/\\log n\\to\\beta\\in(0,\\infty)$, the paper characterizes the limit of $T_{n,k}/p$ as the largest eigenvalue over a deterministic convex set defined by entropy constraints, and evaluates it explicitly as the unique $\\lambda>1$ solving $\\lambda-\\log\\lambda-1=2k/\\beta$ when the entries are standard Gaussian. These are the quantities that control restricted-isometry constants in compressed sensing, sparse PCA objectives, and sparse-eigenvalue conditions in regression, so the limits give sharp thresholds for those problems.","feed_headline":"Extreme principal-minor limits now explicit for all positive a","feed_subtitle":"A Hausdorff-set method settles the open a>2 Wigner case and gives Wishart limits for every p-to-log-n scaling.","key_machinery":"The load-bearing object is the Hausdorff limit of a random set of scaled principal minors. For Wigner matrices, the limit is $K_k(a)$, the compact convex set of symmetric $k\\times k$ matrices in which every principal submatrix obeys $I_{U,a}(B)\\le |U|$; for Wishart matrices, the limit is $\\Gamma_{k,\\beta,\\mu}$, the set of second-moment matrices $\\int xx^\\top d\\nu$ of probability laws $\\nu$ whose every $U$-marginal has KL divergence at most $|U|/\\beta$ from $\\mu^{\\otimes |U|}$. The proof shows $d_H(C_{n,k},K_k(a))\\to 0$ (and the Wishart analogue) and then uses the fact that $\\lambda_{\\max}$ is convex and $1$-Lipschitz, so the maximum over the random set converges to the maximum over the deterministic convex set; the Wigner bound is driven by Gaussian chi-square tails, and the Wishart bound by Sanov's theorem.","core_discovery":"The central discovery is that the random sets formed by suitably normalized principal minors converge in Hausdorff distance to deterministic convex sets, so the extreme-value problem becomes a finite-dimensional convex optimization. For a Gaussian Wigner matrix with off-diagonal variance $1$ and diagonal variance $a$, the set $C_{n,k}=\\{A_S/\\sqrt{2\\log n}: |S|=k\\}$ converges in Hausdorff distance to $K_k(a)$, the set of symmetric $k\\times k$ matrices $B$ with $(1/a)\\sum_{i\\in U}B_{ii}^2+\\sum_{i<j\\in U}B_{ij}^2\\le |U|$ for every nonempty $U\\subseteq[k]$; Theorem 1 then yields $M_{n,k}/\\sqrt{2\\log n}\\to\\gamma_k(a)=\\max_{B\\in K_k(a)}\\lambda_{\\max}(B)$. Theorem 2 evaluates this constant: $\\gamma_k(a)=\\sqrt{a+2(k-1)}$ for $0<a\\le 2$, while for $a>2$ it obeys $\\gamma_m(a)=\\gamma_{m-1}(a)+2/(\\gamma_{m-1}(a)+\\sqrt{\\gamma_{m-1}(a)^2+4-a})$ with $\\gamma_1(a)=\\sqrt{a}$, and the maximizing minors form a nested chain. For Wishart matrices, the scaled sets $\\{X_S^\\top X_S/p\\}$ converge in Hausdorff distance to $\\Gamma_{k,\\beta,\\mu}=\\{\\int xx^\\top d\\nu: \\mathcal{D}(\\nu_U\\|\\mu^{\\otimes |U|})\\le |U|/\\beta \\text{ for all } U\\subseteq[k]\\}$, so $T_{n,k}/p\\to\\sup_{Q\\in\\Gamma_{k,\\beta,\\mu}}\\lambda_{\\max}(Q)$; for standard Gaussian entries this supremum is $\\lambda_{k,\\beta}$, the unique root in $(1,\\infty)$ of $\\lambda-\\log\\lambda-1=2k/\\beta$.","pith_inferences":["Beyond the paper, the same Hausdorff-set route should apply to other ensembles, such as Wigner matrices with non-Gaussian entries or Wishart designs with correlated columns, whenever a Sanov-type large-deviation control of the scaled minors is available; only the limiting constraint set would change.","Beyond the paper, the nested-chain property for $a>2$ yields a directly testable prediction: for large $n$, the support of the maximizing $k\\times k$ minor should contain the support of the maximizing $(k-1)\\times(k-1)$ minor with high probability.","Beyond the paper, the explicit Wishart formula suggests a finite-sample heuristic that the transition in the restricted-isometry constant sharpens around $n\\approx e^{p/\\beta}$; simulations at moderate $n$ could measure the width of this transition and test the logarithmic scaling."],"forward_implications":["For deformed GOE with diagonal variance $a>2$, the extreme principal minor grows like $\\gamma_k(a)\\sqrt{2\\log n}$, and the optimizing minors form a nested chain as $k$ increases; this structural fact is tied to the failure of the usual Gumbel limit in that regime.","For Wishart matrices with $p/\\log n\\to\\beta\\in(0,\\infty)$, the largest normalized principal-minor eigenvalue converges to the largest eigenvalue over the entropy-constrained set $\\Gamma_{k,\\beta,\\mu}$, and for Gaussian entries this limit is the explicit $\\lambda_{k,\\beta}>1$ solving $\\lambda-\\log\\lambda-1=2k/\\beta$.","In the Wishart endpoint regimes, $p/\\log n\\to\\infty$ gives the universal limit $T_{n,k}/p\\to1$, while $p/\\log n\\to0$ gives a tail-driven limit $T_{n,k}/(k\\log n)\\to1/\\kappa_\\mu$ where $\\kappa_\\mu$ is the exponential tail rate of the squared entries.","For Wigner matrices with growing $k$, $M_{n,k}/(2\\sqrt{k\\log(n/k)})\\to1$ when $k/n\\to0$, and $M_{n,k}/\\sqrt{n}\\to E(c)$ when $k/n\\to c\\in(0,1]$, with $E$ increasing, concave, $E(1)=2$, and explicit rates at both endpoints.","In compressed sensing with Gaussian sensing matrices, the restricted-isometry constant of order $k$ converges to $\\lambda_{k,\\beta}-1$, so the condition $\\beta>2k/(\\delta-\\log(1+\\delta))$ makes the restricted isometry property hold with probability tending to one."],"supporting_citations":[{"why":"Established first-order limits for deformed GOE with a≤2 and the Wishart p/log n→∞ regime, leaving the a>2 case open that this paper resolves.","marker":"Cai et al. [2021]"},{"why":"Extended Wishart principal-minor extremes to non-Gaussian entries under moment conditions; this paper fills the p/log n∈(0,∞) regime.","marker":"Hu et al. [2023]"},{"why":"Studied the k=2 deformed GOE and Wishart cases; the k=2 constant here matches and the nested-chain structure explains the distributional change.","marker":"Jiang and Qi [2024]"},{"why":"Provided Gumbel fluctuations and eigenvector structure for fixed-size GOE principal minors, the distributional context for the new first-order limits.","marker":"Feng et al. [2026]"},{"why":"Supplied the Sanov large-deviation theorem used to derive the KL-divergence constraints in the Wishart Hausdorff limit.","marker":"Dembo and Zeitouni [2010]"},{"why":"Supplied the uniform-convergence-in-probability result used to upgrade pointwise Hausdorff convergence to uniform convergence.","marker":"Newey [1991]"},{"why":"Provided the GOE edge spectral theorem and Haar eigenvector facts used in the growing-k Wigner analysis (Theorem 5).","marker":"Anderson et al. [2010]"}],"fun_headline_variants":["Hausdorff approach yields exact extreme principal-minor limits","Wigner and Wishart extreme minors: explicit limits via convex sets","Phase transition at a=2 in extreme principal minors solved","Extreme principal-minor laws now explicit via Hausdorff distance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the Wishart results, the load-bearing premise is that the entry distribution has a finite exponential moment and that $p/\\log n$ converges to a positive constant; if either fails, the proof's truncation step and entropy constraints no longer guarantee that the limiting set captures the extreme principal-minor eigenvalues.","fun_headline_variants_meta":{"raw":{"variants":["Hausdorff approach yields exact extreme principal-minor limits","Wigner and Wishart extreme minors: explicit limits via convex sets","Phase transition at a=2 in extreme principal minors solved","Extreme principal-minor laws now explicit via Hausdorff distance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000522,"raw_usage":{"total_tokens":2641,"prompt_tokens":1180,"completion_tokens":1461,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":796,"completion_tokens_details":{"reasoning_tokens":1389}},"tokens_in":796,"tokens_out":1461,"duration_ms":10701,"temperature":1.0,"reasoning_tokens":1389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:35:23.345109+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a Gaussian Wishart matrix with $p=\\beta\\log n$, $\\beta=2$, $k=2$, and $n$ large; Corollary 1 predicts $T_{n,2}/p$ concentrates near the root $\\lambda\\approx 4.505$ of $\\lambda-\\log\\lambda-1=2$, so a persistent deviation in the sample mean of $T/p$ across many independent replicates would refute the claim. Alternatively, for deformed GOE with $a=4$ and $k=3$, $M_{n,3}/\\sqrt{2\\log n}$ should concentrate near $\\gamma_3(4)=2.9$.","supporting_citations":[],"review_version":1}