{"id":"db07e5b1-d6d1-48b9-aed8-23e4b4689531","arxiv_id":"1908.03304","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The N-scaled fluctuations of empirical eigenvalue measures of generalized Wishart processes and related particle systems converge to explicit Gaussian processes, yielding CLTs for Wishart, Dyson Brownian motion, and Ornstein-Uhlenbeck matrix eigenvalues.","lead":"This paper proves central limit theorems for the random fluctuations of eigenvalue distributions of generalized Wishart matrix processes, as the matrix dimension grows. It recovers known CLTs for Dyson Brownian motion and Wishart processes under more general initial conditions and adds a CLT for symmetric Ornstein-Uhlenbeck matrix processes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1 is conditional on full-sequence convergence of L^N to a deterministic μ_t, a hypothesis imported from a companion paper and for which uniqueness is left open; the covariance (2.3) depends on μ_t, so the central CLT is only a subsequential statement unless that convergence is verified.","rationale":"The reader's weakest_assumption identifies exactly the condition on which the central theorem hinges: the convergence of the empirical measures to a deterministic limit μ_t. My reading of the proof confirms that this hypothesis is used to obtain the deterministic covariance and to justify the centering; without it, the CLT is not asserted. The paper is transparent about this, noting the uniqueness of μ_t is open. The reader's CONDITIONAL verdict is therefore appropriate: the theorem is a conditional statement that the paper does not unconditionally establish for the general class. I do not see a fatal internal inconsistency in the proof of Theorem 2.1, and the applications that rely on the companion paper for convergence are plausible. The remaining issues cited by the reader (the likely sign error in the OU L_t(x) formula, the unproved Theorem 2.2, and wording flaws in Lemma 3.2/Corollary 3.3) are secondary to the conditional-convergence concern, but they do not change the verdict. The most load-bearing concern is the convergence/uniqueness assumption, and the concrete test above would settle whether this concern actually limits the paper's central claim.","tokens_in":30987,"tokens_out":60910,"duration_ms":561132,"concrete_test":"For a coefficient pair (b_N, G_N) satisfying (2.2) but outside the Wishart/Dyson/OU examples, solve the limit equation (1.5) numerically from μ_0 = δ_0 using two independent time-stepping schemes; if two distinct limit measures emerge, the covariance (2.3) is not well-defined for the full sequence and Theorem 2.1 cannot hold without an additional uniqueness assumption. Separately, check the companion paper Song et al. (2019): if it proves only tightness and subsequential convergence for the three applications, then the CLTs in Section 3 are subsequential, and the covariance in (2.3) should be verified to be identical along every converging subsequence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central theorem, Theorem 2.1, is not an unconditional CLT: it assumes that the random empirical measures {L^N(t)} converge weakly to a deterministic {μ_t}. The proof of Theorem 2.1 uses this convergence in two essential places: to pass from N⟨f,L^N−μ⟩ increments to the centered process Q^N (via (2.10) and (2.11)), and to identify the limit of the martingale quadratic variation (2.6) as 2∫⟨f_1'f_2'G(x,x), μ_u⟩du. If the hypothesis fails, the conclusion is not asserted. Section 1 explicitly states, after equation (1.5), that conditions for uniqueness of such limits are still unknown for the general system, and the paper says 'without loss of generality' one may assume full-sequence convergence. This is not WLOG: passing to a subsequence can change the limit μ_t, and hence the covariance in (2.3) depends on which subsequential limit is chosen. The paper therefore delivers a conditional theorem for the general setting, with the hypothesis verified only in the three applications, where the companion paper Song et al. (2019) supplies convergence. The most load-bearing assumption is this convergence/uniqueness; if it fails or is only subsequential, the advertised Gaussian limit is not a well-defined statement for the original sequence. This is a limitation of scope rather than a proof error, but it is the load-bearing point on which the central claim rests.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fluctuations of the empirical eigenvalue measure of generalized Wishart processes and of related interacting particle systems. The main theoretical result, Theorem 2.1, states that under rate conditions on the coefficients and under the assumption that the full sequence of empirical measures converges weakly to a deterministic limit μ_t, the N-scaled centered fluctuation process Q_t^N(f) converges in distribution to a centered Gaussian process whose covariance is 2∫_0^{t∧s}⟨f_i'(x)f_j'(x)G(x,x), μ_u⟩du. The proof uses Itô's formula, a martingale decomposition, and Rebolledo's martingale CLT. Section 2.2 states an analogous CLT for a more general particle system, with proofs omitted. The applications in Section 3 treat the Wishart process, Dyson's Brownian motion, and a symmetric Ornstein-Uhlenbeck matrix process; for these cases the authors prove moment bounds via comparison principles and obtain recursive equations for fluctuation functionals of polynomial test functions.","tokens_in":31253,"tokens_out":6046,"duration_ms":66618,"significance":"If the results are valid, the paper gives a useful and fairly general CLT for eigenvalue fluctuations of matrix diffusions, extending earlier work of Cabanal-Duvillard and Anderson-Guionnet-Zeitouni to more general initial conditions and to a new Ornstein-Uhlenbeck example. The stochastic-calculus proof of Theorem 2.1 is clean and gives an explicit covariance formula in terms of the limit measure and the coefficient limit G. The paper also provides a comparison principle and moment bounds that are of independent interest for non-colliding particle systems. The main qualification is that the general theorem is conditional on full-sequence convergence of the empirical measures, and the uniqueness of the limit equation is left open; in addition, the particle-system CLT of Section 2.2 is stated without proof.","major_comments":[{"comment":"The manuscript says that 'up to considering a subsequence, the theory is here developed, without loss of generality, by assuming the convergence of the whole sequence'. This reduction is not a genuine WLOG because uniqueness of solutions to the limit equation (1.5) is explicitly left open. A subsequential limit μ may depend on the subsequence, and since the covariance (2.3) is a functional of μ, different subsequences could lead to different Gaussian limits. Theorem 2.1 should therefore be formulated either as a conditional CLT under full-sequence convergence or as a subsequential CLT, with the dependence of the covariance on the chosen limit made explicit.","section":"Section 1, Eq. (1.5); Theorem 2.1"},{"comment":"Theorem 2.2 is a central advertised result, but it is stated with the sentence that its proof and the proofs of the associated corollaries are 'similar to those of Theorem 2.1 ... and thus omitted'. The particle system (2.21) has a different drift normalization, a different noise coefficient, and a different test-function class than the eigenvalue system (1.2), so Theorem 2.2 is not a formal corollary of Theorem 2.1. A rigorous treatment requires either a full proof or a detailed indication of how the covariance ∫⟨f_i'f_j'σ̃^2, μ_u⟩du emerges from the quadratic variation of the relevant martingale.","section":"Section 2.2, Theorem 2.2 and Corollaries 2.3-2.4, Proposition 2.2"},{"comment":"The induction argument for convergence of (L_t^N(x^k),...,L_t^N(x)) has a joint-convergence gap. The proof shows separately that (̃Q_t^N(x^k),...,̃Q_t^N(x)) converges to a Gaussian vector and, by induction, that (L_t^N(x^{k-1}),...,L_t^N(x)) converges. But separate convergence does not imply joint convergence of ̃Q_t^N(x^k) with the lower-order L^N vector. Since L_t^N(x^k) is defined as ̃Q_t^N(x^k) plus a function of lower-order L^N terms, the argument needs an explicit joint-tightness or Cramér-Wold step. The same issue appears in the inductive parts of Theorems 3.3 and 3.4.","section":"Section 3.2, proof of Theorem 3.2, around Eq. (3.18)"}],"minor_comments":[{"comment":"There are numerous typographical errors, including 'repsectively', 'flunctuations', 'Scetion 2.2', and 'eignenvalues'; these should be corrected in revision.","section":"Throughout"},{"comment":"The author name 'Ma/suppress lecki' appears to be a LaTeX corruption of 'Maślęcki' or 'Malecki' and should be typeset consistently and correctly in the text and references.","section":"Throughout"},{"comment":"The phrase 'Choosing p = ln2 N' is ambiguous: it should be written as (ln N)^2 or as a differently named exponent variable, especially because p is also used as the moment order in condition (2.15).","section":"Corollary 2.2, proof"},{"comment":"The condition lim N‖N G_N(x,y)-G(x,y)‖_{L∞(R²)}=0 should be stated with care when G is unbounded, as in the Wishart example where G(x,y)=x+y; the norm is understood in the extended sense, and the text would benefit from a remark clarifying this.","section":"Theorem 2.1, condition (2.2)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful part of this paper is the general CLT for eigenvalue fluctuations of generalized Wishart processes, and especially the application to the symmetric OU matrix process, which I don't think was in the literature. The Dyson and Wishart applications mostly recover known results with slightly more general initial conditions, which is honest and useful. The proof of Theorem 2.1 is the real meat: Itô's formula, a martingale decomposition, and Rebolledo's martingale CLT, with the covariance falling out as 2∫⟨f'g'G, μ⟩. That argument is coherent and, as far as I can tell, correct.\n\nThe comparison principle in Section 3.1 is a nice addition. It gives moment bounds that let the authors push the test functions from the bounded class F to polynomials. That is what makes the recursive formulas in Theorems 3.2–3.4 possible. Credit where due: the paper is also upfront that the central theorem is conditional on the empirical measures converging to a deterministic limit μ, and that uniqueness of such limits is open. That is not circular; the covariance is computed from the assumed limit, not fitted.\n\nThe soft spots are real but not fatal. First, the central theorem is genuinely conditional, and the phrase \"without loss of generality\" before assuming full-sequence convergence is a bit loose: passing to a subsequence could change the covariance. The applications get around this via the companion paper Song–Yao–Yuan, but the general statement is weaker than it looks. Second, the entire particle-system section, Theorem 2.2 and its corollaries, is stated without proof. The paper says the proofs are similar, and they probably are, but for a standalone contribution that is a notable omission. Third, the formula for Lt(x) in the OU theorem looks off to me — there seems to be a sign error in the deterministic integral term. I would not bet heavily on that without checking carefully, but it deserves a look. There are also minor wording issues in Lemma 3.2 and Corollary 3.3, but they do not affect the main line.\n\nWho is this for? People working on random matrix diffusions and eigenvalue fluctuations. The paper is worth a serious referee: the central theorem is plausible and mostly proved, the OU application is new, and the comparison principle is independently useful. I would send it to review, with the expectation that the authors fill in the Section 2.2 proofs and fix the OU formula before publication.","headline":"A sound but conditional CLT for generalized Wishart eigenvalue fluctuations, with a genuinely new OU application; the main proof holds up, but the paper leans on an unproved convergence hypothesis and omits proofs for the particle-system half.","tokens_in":31816,"tokens_out":1150,"would_cite":true,"duration_ms":15768,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60F05","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a broad class of matrix diffusions, the eigenvalue histogram's N-scaled centered fluctuations converge to a Gaussian process, with covariance fixed by the limit measure and the local diffusion coefficient G(x,x).","keywords":["Dyson's Brownian motion","Wishart process","generalized Wishart process","squared Bessel particle system","central limit theorem","empirical measure fluctuations","Ornstein-Uhlenbeck matrix process","random matrix"],"falsifier":"Simulate Dyson's Brownian motion (3.25) for large $N$, take $f(x)=x$, and estimate the variance of the centered fluctuation $L_t(x)-L_0(x)$ at $t=1$; the theorem predicts this variance is exactly $2$, so a statistically significant deviation from $2$ would refute the covariance formula. Alternatively, find a coefficient pair satisfying (2.2) for which the limit equation (1.5) has two distinct solutions with the same initial measure; subsequences selecting different limits would then show no single Gaussian limit describing the full sequence.","tokens_in":30740,"feed_emoji":"📈","tokens_out":12607,"duration_ms":122022,"temperature":0.7,"pith_summary":"The paper establishes central limit theorems for eigenvalue fluctuations of generalized Wishart processes and a wider class of particle systems. It shows that, after subtracting an initial value and a deterministic drift, the N-scaled centered fluctuation process $Q_t^N(f)=N\\langle f,L^N(t)-\\mu_t\\rangle$ converges to a Gaussian process. The limiting covariance is $2\\int_0^{t\\wedge s}\\langle f_i'(x)f_j'(x)G(x,x),\\mu_u\\rangle\\,du$, so the random spread of eigenvalues around their high-dimensional limit has size $1/\\sqrt{N}$ and is described entirely by the limit measure $\\mu_t$ and the coefficient limit $G(x,x)$. The same argument recovers earlier fluctuation results for Dyson's Brownian motion and the Wishart process under more general initial conditions, and produces a new fluctuation theorem for the eigenvalues of a symmetric Ornstein-Uhlenbeck matrix process.","feed_headline":"Eigenvalue fluctuations are Gaussian for generalized Wishart processes","feed_subtitle":"A covariance formula tells exactly how far spectral empirical measures wander from their large-N limit over time.","key_machinery":"The mechanism is Itô's formula applied to linear statistics of the eigenvalues, which splits $\\langle f,L^N(t)\\rangle$ into an initial value, a martingale $M_f^N(t)$ with quadratic variation $(4/N)\\int_0^t\\langle (f'g_Nh_N)^2,L^N(s)\\rangle\\,ds$, and drift terms. The proof shows $Q_t^N(f)-N M_f^N(t)$ tends to zero uniformly in $t$, then reads the Gaussian limit off the limit of the quadratic variation via a martingale central limit theorem (Lemma 4.1). A secondary mechanism is the comparison principle of Section 3.1, which sandwiches a target process between two comparison processes with known moment bounds and thereby extends the CLT to polynomial test functions and to drifts that merely converge to a constant.","core_discovery":"The central discovery is structural: for eigenvalues of the matrix stochastic differential equation (1.1), the fluctuation field $Q_t^N(f)$ has a Gaussian limit whose covariance depends only on the product $f_i'(x)f_j'(x)$, the coefficient limit $G(x,x)$, and the limit measure $\\mu_t$, not on the finer dynamics. The proof identifies $Q_t^N(f)$ with a martingale term $N M_f^N(t)$ up to a remainder that vanishes in $L^p$ uniformly in time; the quadratic variation of that martingale converges to $2\\int_0^t\\langle (f')^2G(x,x),\\mu_s\\rangle\\,ds$, so a martingale central limit theorem forces the Gaussian limit. In the three applications, restricting to polynomial test functions gives more: the limiting fluctuation processes $L_t(x^n)$ satisfy explicit linear recursions driven by Gaussian processes, yielding distributional descriptions such as $L_t(x)=L_0(x)+G_t(x)$ for the Wishart and Dyson cases.","pith_inferences":["If the empirical measures converge only along a subsequence, the same proof gives a subsequential Gaussian CLT; uniqueness of solutions to the limit equation (1.5) would automatically upgrade this to a full statement, so resolving that uniqueness is the natural next step.","The covariance formula suggests a broader universality principle: for interacting particle systems with a deterministic mean-field limit, centered fluctuations of smooth statistics should be Gaussian with covariance driven by the local diffusion coefficient and the limit measure, independent of the detailed interaction kernel.","The recursive moment equations yield an algorithm to simulate the limiting fluctuation processes for Wishart, Dyson, and OU ensembles without simulating $N$ particles; finite-$N$ Monte Carlo checks of variance predictions such as $2t$ for Dyson's Brownian motion would test the theory directly.","For the Ornstein-Uhlenbeck matrix process, the explicit time-change relation with Brownian motion identifies its eigenvalue fluctuations as a deterministic time change of Dyson fluctuations, which predicts the covariance (3.41) and may extend to other matrix flows with explicit solutions."],"forward_implications":["For the Wishart, Dyson, and symmetric Ornstein-Uhlenbeck flows, the limiting fluctuation process is Gaussian on the polynomial basis and satisfies explicit recursions; for example $L_t(1)=0$ and $L_t(x)$ is the initial fluctuation plus a Gaussian martingale.","The covariance formula gives a direct computation rule for asymptotic covariances of linear spectral statistics: integrate $f_i'f_j'G(x,x)$ against the limit measure over the time overlap.","The comparison principle extends the CLT to generalized particle systems without a matrix representation or explicit stationary density, as long as the drift converges uniformly at rate $o(1/N)$.","Initial data need only be convergent and polynomially moment-bounded; the null-initial-condition restriction of earlier CLTs is removed."],"supporting_citations":[{"why":"Supplies the limit equation for the empirical measures and the Itô-formula expansion that the CLT takes as its input.","marker":"Song et al. (2019)"},{"why":"Introduces the generalized Wishart matrix SDE and proves the eigenvalue SDE (1.2) has a non-colliding strong solution.","marker":"Graczyk and Ma/suppress lecki (2013)"},{"why":"Introduces the particle system (2.21) and the monotonicity condition (3.3) on which the comparison principle rests.","marker":"Graczyk and Ma/suppress lecki (2014)"},{"why":"Established the earlier CLTs for Dyson's Brownian motion and the Wishart process under null initial condition, which this paper recovers and extends.","marker":"Cabanal-Duvillard (2001)"},{"why":"Provided the stationarity and moment-bound techniques and the relaxed-initial-condition CLT for Dyson's Brownian motion used in Section 3.","marker":"Anderson et al. (2010)"},{"why":"Comparison theorem for multidimensional SDEs that underlies Lemma 4.2 and the comparison principle in Theorem 3.1.","marker":"Geiß and Manthey (1994)"},{"why":"Identifies the symmetric Ornstein-Uhlenbeck matrix process and its Wigner-law limit, extended here to a fluctuation CLT.","marker":"Chan (1992)"},{"why":"Introduced the Wishart process, the eigenvalue process treated in Theorem 3.2.","marker":"Bru (1991)"}],"fun_headline_variants":["Wishart eigenvalue fluctuations obey Gaussian limits","Central limit theorem for spectral measures of Wishart processes","Gaussian fluctuations confirmed for generalized Wishart eigenvalues","Eigenvalue jitter is Gaussian in Wishart process limits","How spectral measures fluctuate: a CLT for Wishart processes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the eigenvalue histogram converges, as the dimension grows, to one deterministic limit curve $\\mu_t$; the paper assumes this convergence rather than proving it, and for general coefficients uniqueness of such a curve is left open, so if convergence fails or only occurs along a subsequence the stated Gaussian limit is not asserted.","fun_headline_variants_meta":{"raw":{"variants":["Wishart eigenvalue fluctuations obey Gaussian limits","Central limit theorem for spectral measures of Wishart processes","Gaussian fluctuations confirmed for generalized Wishart eigenvalues","Eigenvalue jitter is Gaussian in Wishart process limits","How spectral measures fluctuate: a CLT for Wishart processes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2811,"prompt_tokens":833,"completion_tokens":1978,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":449,"completion_tokens_details":{"reasoning_tokens":1901}},"tokens_in":449,"tokens_out":1978,"duration_ms":14786,"temperature":1.0,"reasoning_tokens":1901,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:18:52.904239+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate Dyson's Brownian motion (3.25) for large $N$, take $f(x)=x$, and estimate the variance of the centered fluctuation $L_t(x)-L_0(x)$ at $t=1$; the theorem predicts this variance is exactly $2$, so a statistically significant deviation from $2$ would refute the covariance formula. Alternatively, find a coefficient pair satisfying (2.2) for which the limit equation (1.5) has two distinct solutions with the same initial measure; subsequences selecting different limits would then show no single Gaussian limit describing the full sequence.","supporting_citations":[{"cited_title":"High-dimensional limits of eigenvalue distributions for general Wishart process","cited_arxiv_id":"1901.02190","evidence_quote":"Supplies the limit equation for the empirical measures and the Itô-formula expansion that the CLT takes as its input."},{"cited_title":"and Ma ecki, J","cited_arxiv_id":null,"evidence_quote":"Introduces the generalized Wishart matrix SDE and proves the eigenvalue SDE (1.2) has a non-colliding strong solution."},{"cited_title":"and Ma ecki, J","cited_arxiv_id":null,"evidence_quote":"Introduces the particle system (2.21) and the monotonicity condition (3.3) on which the comparison principle rests."},{"cited_title":"W., Guionnet, A., and Zeitouni, O","cited_arxiv_id":null,"evidence_quote":"Provided the stationarity and moment-bound techniques and the relaxed-initial-condition CLT for Dyson's Brownian motion used in Section 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the symmetric Ornstein-Uhlenbeck matrix process and its Wigner-law limit, extended here to a fluctuation CLT."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the Wishart process, the eigenvalue process treated in Theorem 3.2."}],"review_version":1}