{"id":"a68568bd-e565-49e8-9ba1-f2c5e47621fd","arxiv_id":"2507.13922","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Driver-Hall-Kemp family of multiplicative Brownian motions on GL_N(C) strongly converges, almost surely and jointly with deterministic matrices, to the free multiplicative Brownian motion as N tends to infinity.","lead":"Random matrices evolving by multiplicative Brownian motion on the general linear group are shown to converge almost surely, in spectrum and trace, to a free counterpart. This gives the first strong convergence result for the full Driver-Hall-Kemp family, extending a unitary-group result to non-normal matrices.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.9 is false for general (λ,τ): left- and right-invariant multiplicative Brownian motions do not have the same one-time law, so the variance estimate collapses.","rationale":"The reader's weakest assumption, Proposition 6.9, is indeed the load-bearing point: the covariance decomposition in Lemma 4.1 and the subsequent O(1/N^4) variance estimate both require that the right-invariant multiplicative Brownian motions eG and Ĝ introduced in (27) have the same one-time distribution as the left-invariant process G. The reader asked for verification of Proposition 6.9. Direct computation shows that the claim is not merely unproven but false: the left- and right-invariant generators differ on GL_N(C) for N≥2 and λ>0. The second-order coefficients of the two generators are, respectively, (λ/N)(GG^*)_{ik}δ_{jl} and (λ/N)δ_{ik}(G^*G)_{lj}, which are not equal as differential operators. A concrete evaluation at a non-unitary point exhibits the difference, so the heat kernels and therefore the one-time laws differ. The unitary case (λ,τ)=(1,0) is special because the diffusion stays in U(N), where the metric is bi-invariant; the general parameter range treated by Theorem 1.1 has no such symmetry. Since the proof of Theorem 1.1 depends essentially on the false equality, the submitted argument is not correct as written. The main theorem might still be true and repairable, but the current proof must be rejected or substantially revised.","tokens_in":43455,"tokens_out":41174,"duration_ms":438018,"concrete_test":"Evaluate the two generators at g0 = [[1,1],[0,1]] on f(g) = g_{11}\\overline{g_{22}}. The left generator's second-order term is (1/2)(λ/N)(g0 g0^*)_{12} δ_{11} = λ/(2N), while the right generator's corresponding term is (1/2)(λ/N)δ_{12}(g0^* g0)_{21} = 0. Since the generators differ at a point, the semigroups e^{tΔ_L} and e^{tΔ_R} differ for t>0; equivalently, a Monte Carlo simulation for N=2, λ=1, τ=1, small t, comparing E[(G_t)_{11}\\overline{(G_t)_{22}}] with E[(eG_t)_{11}\\overline{(eG_t)_{22}}], should show a discrepancy of order t^2.","verdict_should_be":"REJECT","load_bearing_attack":"The proof's central variance estimate (24) relies on Lemma 4.1, which uses Proposition 6.9 to assert that the right-invariant processes eG and Ĝ are independent copies of the left-invariant process G. Proposition 6.9 claims that G_t and eG_t have the same distribution for every admissible (λ,τ). This is not true for N≥2 and λ>0. The left- and right-invariant generators are different operators on GL_N(C). For the left-invariant G, d⟨G_{ij}, \\overline{G_{kl}}⟩/dt = (λ/N)(G G^*)_{ik} δ_{jl}; for the right-invariant eG, d⟨eG_{ij}, \\overline{eG_{kl}}⟩/dt = (λ/N) δ_{ik} (eG^* eG)_{lj}. At the point g0 = [[1,1],[0,1]], applied to f(g) = g_{11}\\overline{g_{22}}, the left generator contributes (1/2)(λ/N)(g0 g0^*)_{12} = λ/(2N), while the right generator contributes 0. Thus the two generators differ at a point. Since both diffusions are elliptic on GL_N(C), the heat kernels e^{tΔ_L}δ_e and e^{tΔ_R}δ_e differ for t>0, so G_t and eG_t have different laws. The equality does hold for the unitary case (λ,τ)=(1,0) because the process stays in U(N), where the Hilbert–Schmidt metric is Ad(U(N))-invariant, but it fails for the general parameters treated in the paper. Lemma 4.1's identification of eG_t and Ĝ_t as independent copies of G_t is therefore invalid, and the O(1/N^4) variance estimate, which is essential for the Borel–Cantelli step and Theorem 1.1, is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the family of multiplicative (λ,τ)-Brownian motions on GL_N(C) introduced by Driver–Hall–Kemp, and claims almost sure strong convergence of their finite-dimensional marginals, jointly with strongly converging deterministic matrices, to the corresponding free multiplicative (λ,τ)-Brownian motion. The proof strategy is a multiplicative interpolation between the matrix process and the free process, combined with an estimate of the covariance of smooth functions of polynomials in the matrix process, Helffer–Sjöstrand representation, and a Borel–Cantelli step. The paper also contains an appendix proving weak convergence with an O(1/N^2) rate.","tokens_in":43825,"tokens_out":14785,"duration_ms":168788,"significance":"If the main theorem were established, it would be a substantial extension of strong convergence for multiplicative Brownian motions from the unitary case (λ,τ)=(1,0) of Collins–Dahlqvist–Kemp to the full two-parameter family, and the proposed multiplicative interpolation would be a novel technical contribution. The appendix's self-contained proof of weak convergence at rate O(1/N^2) is also a useful contribution. However, the central variance estimate that drives the main theorem depends on a claimed equality of laws (Proposition 6.9) that is false for general admissible (λ,τ); accordingly, the main results are not supported by the present proof.","major_comments":[{"comment":"Proposition 6.9 is false for general admissible (λ,τ). The reasoning that the left-invariant and right-invariant Laplacians are 'relative to one same and single metric' is not valid: a left-invariant metric and a right-invariant metric determined by the same inner product at the identity coincide only when that inner product is Ad(GL_N(C))-invariant, and the Hilbert–Schmidt inner product Tr(A*B) is not Ad(GL_N(C))-invariant. Concretely, let N≥2, λ>0, and compare the two generators at g0=[[1,1],[0,1]] acting on f(g)=g_{11}\\overline{g_{21}}. For the left-invariant process dG_t = G_t dZ_t, the Itô correction gives (λ/N)(g0 g0^*)_{12}=λ/N, while for the right-invariant process d\\tilde G_t = dZ_t \\tilde G_t it gives (λ/N)δ_{12}(...)=0, using d⟨Z_{ij},\\bar Z_{kl}⟩=(λ/N)δ_{ik}δ_{jl}dt. The two generators are therefore different operators. The cited [26, Theorem 2.7] does not justify the claimed equality because it concerns heat kernels for a common metric. Thus the equality of the one-time laws of G_t and \\tilde G_t is not established and is, in fact, false for general parameters.","section":"Section 6.2, Proposition 6.9"},{"comment":"Lemma 4.1 explicitly relies on Proposition 6.9 to assert that \\tilde Q_{t,0} and \\hat Q_{t,0} are independent copies of Q(G_t). Since Proposition 6.9 fails, the endpoint identification h(0)=E[tr R_1(\\tilde G_t)]E[tr R_2(\\hat G_t)] = (E tr R(G_t))^2 is false, and the derivative computation in (28)–(34) no longer computes the covariance of the two resolvent traces. Consequently the variance estimate (24), which is the basis for the Borel–Cantelli step and for Theorem 1.1, is not established.","section":"Section 4, Lemma 4.1 and Eq. (24)"},{"comment":"The proof that P P*(gu \\hat g_{t-u}, A_N) shares the same spectrum as P P*(g_t, A_N) relies on Proposition 6.10, which is itself derived from Proposition 6.9 and inherits its failure. The same flaw therefore invalidates the claimed O(1/N^4) variance bound for the functions f_{N,δ}, and with it the spectral inclusion argument leading to Theorem 1.1.","section":"Section 4, paragraph after Eq. (38)"}],"minor_comments":[{"comment":"In the second SDE of (27), the initial condition is written G^{(ℓ)}_{λ,τ}(0)=I_N, but the process being defined is \\hat G^{(ℓ)}_{λ,τ}; this is a typo.","section":"Eq. (27)"},{"comment":"The text refers to 'the multi-time statement in Theorem 1.2', but no Theorem 1.2 is stated; the intended reference is likely Theorem 1.1 or Corollary 1.2.","section":"Section 3, first paragraph"},{"comment":"The sentence beginning 'for any s1 and s2' is incomplete; it appears to be a leftover from an earlier draft.","section":"Proof of Proposition 5.4"},{"comment":"The phrase 'almost sure strong convergence' is slightly nonstandard, since strong convergence in this context already includes an almost sure statement; this is a wording issue.","section":"Abstract"}],"recommendation":"reject","confidential_remarks":"The main theorem may be true and the multiplicative interpolation idea is promising, but the proof rests on a false equality of laws. A revision would need to replace the interpolation pair (\\tilde G, \\hat G) with processes whose endpoint laws match G_t, or else develop a different covariance estimate. Because the false lemma is load-bearing and central to the claimed variance estimate, I cannot recommend acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: the claimed theorem is new and nontrivial, and the multiplicative interpolation machinery is a real contribution. The weak convergence part in the appendix (pure trace polynomial calculus, generator computation, 1/N^2 control) is careful and correct as far as I can tell. The variance estimate of Theorem 1.3 would be a useful result if the proof held up. So the paper deserves to be taken seriously.\n\nThe soft spot is not soft. Proposition 6.9 states that the left-invariant process G_t and the right-invariant process tG_t have the same one-time distribution for every admissible (lambda,tau). The proof in Section 6.2 is one paragraph: it asserts both Laplacians are relative to the same metric and cites [26, Theorem 2.7]. That assertion is wrong for the Hilbert-Schmidt inner product on gl_N(C), which is not Ad(GL_N(C))-invariant. The explicit check matches the stress-test note: at g0 = [[1,1],[0,1]], for f(g)=g_11 \\bar{g_22}, the left generator contributes lambda/(2N) and the right generator contributes 0. So for N>=2 and lambda>0 the one-time laws differ. The equality does hold for (lambda,tau)=(1,0), where the process lives on U(N), but not in the generality claimed.\n\nThis is load-bearing. Lemma 4.1 uses Proposition 6.9 to treat tG_t and hG_t as independent copies of G_t, and the variance estimate O(1/N^4) in Section 4 collapses without that identification. The same false identification enters through Proposition 6.10 in the free reduction. I do not see a way around it with the present arguments. This is not a gap in presentation; it is a false statement.\n\nRecommendation: do not desk-reject, because the problem is important and the method may be repairable, but do not accept. Send to a referee with the request to focus on Proposition 6.9 first. If the authors can prove a corrected version, or rework the proof to avoid needing the equality, the strong convergence result would be a major contribution.","headline":"New interpolation method and a substantial result if true, but a false equality of left- and right-invariant laws sinks the main proof as written.","tokens_in":44340,"tokens_out":5083,"would_cite":false,"duration_ms":62775,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B10","60B20","46L54","22E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that multiplicative Brownian motions on GL_N(C) converge strongly to the free multiplicative Brownian motion for every admissible variance-covariance pair.","keywords":["multiplicative Brownian motion","general linear group","strong convergence","free probability","free multiplicative Brownian motion","random matrices","operator norm","spectral inclusion"],"falsifier":"Compute the generator of the right-invariant multiplicative Brownian motion on pure trace polynomials and compare it with the operator $\\Delta$ from Lemma 6.1; a mismatch for any admissible $(\\lambda,\\tau)$ (e.g. a non-real $\\tau$) would falsify Proposition 6.9 and break the covariance estimates.","tokens_in":43258,"feed_emoji":"🎲","tokens_out":10539,"duration_ms":109015,"temperature":0.7,"pith_summary":"The paper proves that the multiplicative $(\\lambda,\\tau)$-Brownian motion $G_{\\lambda,\\tau}(t)$ on $GL_N(\\mathbb{C})$ strongly converges, as $N\\to\\infty$, to the free multiplicative $(\\lambda,\\tau)$-Brownian motion $g_{\\lambda,\\tau}(t)$, for every admissible variance $\\lambda\\ge0$ and complex covariance $\\tau$. Almost surely, for any fixed times and any noncommutative polynomial $P$, both the normalized trace $\\mathrm{tr}_N\\,P(G_t,G_t^*,G_t^{-1},(G_t^{-1})^*,A_N)$ and the operator norm $\\|P(\\cdot)\\|$ converge to their free counterparts, with deterministic matrices $A_N$ allowed to converge strongly as well. This strengthens the previously known convergence in $*$-distribution, which does not control spectra because $G_t$ is not normal. It extends the unitary case $(\\lambda,\\tau)=(1,0)$ to the full family of Brownian motions on $GL_N(\\mathbb{C})$ introduced in [32].","feed_headline":"Strong convergence proved for all GL(N) Brownian motions","feed_subtitle":"Traces and norms of polynomials match the free multiplicative Brownian motion almost surely.","key_machinery":"The proof is carried by a multiplicative interpolation scheme together with a sharp estimate on the mean trace of products of resolvents and monomials (Theorem 5.1). One interpolates between the left-invariant Brownian motion $G$ and a right-invariant independent copy $\\widetilde{G}$ (or a large-dimension copy $K$), using the equality of their one-time laws (Proposition 6.9) to cancel the first-order drift terms and leaving only trace-type quadratic-covariation terms. The surviving terms are bounded by rewriting them as covariances of normalized traces and applying a variance estimate (Theorem 1.3) of order $O(1/N^4)$ for $C^3$ functions with compact support, obtained via the Helffer–Sjöstrand representation formula and a Borel–Cantelli eigenvalue-counting argument.","core_discovery":"On the paper's own terms, the central assertion is Corollary 1.2: for any admissible $(\\lambda,\\tau)$, the finite-dimensional marginals of $G_{\\lambda,\\tau}$ converge almost surely in the strong sense to the free multiplicative $(\\lambda,\\tau)$-Brownian motion $g$ that is free from the strong limit $a$ of the deterministic matrices. The norm convergence is obtained from a spectral-inclusion theorem (Theorem 1.1): almost surely, for large $N$, the spectrum of any self-adjoint polynomial in the Brownian motions at several times, their inverses and adjoints, and the deterministic matrices is contained in a $\\delta$-neighborhood of the spectrum of the same polynomial in the free variables. Because the free tuple is shown to be strongly convergent, Proposition 2.2 converts this inclusion into the equality of norms.","pith_inferences":["The equality of left- and right-invariant laws (Proposition 6.9) is the load-bearing hinge; a direct computation of the pure-trace generator of the right-invariant process would either confirm or refute it for all admissible $(\\lambda,\\tau)$, and the proof would likely adapt if only a weaker comparison (up to $O(1/N^2)$) held.","The interpolation technique appears to be insensitive to the specific values of $\\lambda$ and $\\tau$ beyond the trace form of the quadratic covariations, so the same scheme may prove strong convergence for other multiplicative diffusions with additional drifts or different noise laws.","A concrete test is to compute the limiting variance in Theorem 1.3 for the complex case $(\\lambda,\\tau)=(1,1)$ and compare it with known fluctuation results for unitary Brownian motion, probing whether the $O(1/N^4)$ rate is sharp in the non-normal regime.","The spectral-inclusion approach could yield quantitative bounds on the norm of polynomials as $N$ grows, once the dependence of the constants on $\\delta$ and the polynomial degree is made explicit."],"forward_implications":["For every admissible $(\\lambda,\\tau)$, the empirical spectral measure of any fixed polynomial in $G_{\\lambda,\\tau}(t)$ and its inverse and adjoint converges almost surely, and the spectrum converges in Hausdorff distance to the free limit.","The result holds jointly with any strongly convergent family of deterministic matrices $A_N$, so the Brownian motion can be combined with other strongly convergent variables in a free probability model.","The variance estimate $\\mathrm{Var}[\\mathrm{tr}_N f(P P^*(G_t,\\dots))] = O(1/N^4)$ for compactly supported $C^3$ functions is quantitative and may be used as input for fluctuation results at the scale $1/N$.","Weak convergence of finite-dimensional marginals (Theorem 3.2) is established with an explicit $O(1/N^2)$ error using the pure trace polynomial generator.","The spectral-inclusion theorem applies to multi-time marginals, not just a fixed time, because of the stationary independent multiplicative increments of the process."],"supporting_citations":[{"why":"Establishes strong convergence in the unitary case (λ,τ)=(1,0); the result this paper generalizes.","marker":"[22]"},{"why":"Introduces the family of multiplicative (λ,τ)-Brownian motions and provides the model studied here.","marker":"[32]"},{"why":"Defines the free multiplicative (λ,τ)-Brownian motion and studies its Brown measure; the target limit object.","marker":"[40]"},{"why":"Proves *-distribution convergence of the two-parameter family on GL_N(C); supplies the weak-convergence baseline.","marker":"[49]"},{"why":"Develops the interpolation technique used to compare random matrices and free variables.","marker":"[23]"},{"why":"Provides the asymptotic expansion method for smooth functions of GUE and deterministic matrices, adapted for the resolvent estimate.","marker":"[62]"},{"why":"Invoked in the proof of Proposition 6.9 to identify the heat kernels of left- and right-invariant Laplacians.","marker":"[26]"},{"why":"Proves convergence of the Brownian motion on GL_N to the free multiplicative Brownian motion, used for the weak-convergence step.","marker":"[14]"},{"why":"Introduced strong convergence for GUE matrices and the linearization/norm-control framework that motivates this work.","marker":"[37]"}],"fun_headline_variants":["Almost sure strong convergence for all GL(N) Brownian motions","Norm convergence proven for all elliptic Brownian motions on GL(N)","GL(N) Brownian motions converge to free limit almost surely","Spectral inclusion secures strong convergence for GL(N)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the left-invariant Brownian motion and its right-invariant counterpart driven by the same noise have the same one-time distribution for all allowed parameters; this one-paragraph claim is not automatic because the inner product on the Lie algebra is not Ad-invariant.","fun_headline_variants_meta":{"raw":{"variants":["Almost sure strong convergence for all GL(N) Brownian motions","Norm convergence proven for all elliptic Brownian motions on GL(N)","GL(N) Brownian motions converge to free limit almost surely","Spectral inclusion secures strong convergence for GL(N)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000555,"raw_usage":{"total_tokens":2614,"prompt_tokens":885,"completion_tokens":1729,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":1660}},"tokens_in":501,"tokens_out":1729,"duration_ms":14234,"temperature":1.0,"reasoning_tokens":1660,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:18:03.770268+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the generator of the right-invariant multiplicative Brownian motion on pure trace polynomials and compare it with the operator $\\Delta$ from Lemma 6.1; a mismatch for any admissible $(\\lambda,\\tau)$ (e.g. a non-real $\\tau$) would falsify Proposition 6.9 and break the covariance estimates.","supporting_citations":[{"cited_title":"The spectral edge of unitary Brownian motion","cited_arxiv_id":null,"evidence_quote":"Establishes strong convergence in the unitary case (λ,τ)=(1,0); the result this paper generalizes."},{"cited_title":"The complex-time Segal–Bargmann transform","cited_arxiv_id":null,"evidence_quote":"Introduces the family of multiplicative (λ,τ)-Brownian motions and provides the model studied here."},{"cited_title":"Hall and Ching-Wei Ho","cited_arxiv_id":null,"evidence_quote":"Defines the free multiplicative (λ,τ)-Brownian motion and studies its Brown measure; the target limit object."},{"cited_title":"The large-N limits of Brownian motions onGLN (C)","cited_arxiv_id":null,"evidence_quote":"Proves *-distribution convergence of the two-parameter family on GL_N(C); supplies the weak-convergence baseline."},{"cited_title":"On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices","cited_arxiv_id":null,"evidence_quote":"Develops the interpolation technique used to compare random matrices and free variables."},{"cited_title":"Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices.Comm","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic expansion method for smooth functions of GUE and deterministic matrices, adapted for the resolvent estimate."},{"cited_title":"On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall isomorphisms.Jour- nal of Functional Analysis, 133(1):69–128, 1995","cited_arxiv_id":null,"evidence_quote":"Invoked in the proof of Proposition 6.9 to identify the heat kernels of left- and right-invariant Laplacians."},{"cited_title":"Free convolution operators and free Hall transform.Journal of Functional Analysis, 265(11):2645–2708, 2013","cited_arxiv_id":null,"evidence_quote":"Proves convergence of the Brownian motion on GL_N to the free multiplicative Brownian motion, used for the weak-convergence step."},{"cited_title":"A new application of random matrices: Ext(C∗ red(F2)) is not a group.Annals of Mathematics, pages 711–775, 2005","cited_arxiv_id":null,"evidence_quote":"Introduced strong convergence for GUE matrices and the linearization/norm-control framework that motivates this work."}],"review_version":1}