{"id":"8c03405e-cf45-457b-95f0-f266660e7a0c","arxiv_id":"1908.08150","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit Brown measures are computed for free circular Brownian motion with self-adjoint initial condition and for free multiplicative Brownian motion with unitary initial condition.","lead":"This paper derives explicit formulas for the Brown measure, the non-self-adjoint analogue of an eigenvalue distribution, for two natural random-matrix processes with arbitrary self-adjoint or unitary starting points. The formulas connect those eigenvalue clouds to classical free convolution laws and reduce to a simple annulus law when the starting point is Haar-distributed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Multiplicative result depends on an only-outlined real-analytic extension of S̃(t,λ,z)=S(t,λ,z²) at z=0 in §4.4; without it the Laplacian formula and absence of boundary mass are not justified.","rationale":"The reader's weakest_assumption identifies exactly the real-analytic extension of the regularized trace in the multiplicative case. My independent reading of §4.4 confirms that all subsequent Laplacian formulas and the push-forward identification rely on this regularity, and the paper itself flags the gap by deferring to [17, Section 7.4]. The additive case is fully self-contained and appears sound. The Haar-unitary case has an independent R-diagonal computation in the appendix, but that does not cover a general unitary initial condition. I therefore would keep the reader's CONDITIONAL verdict: the main result is plausible and well supported in the additive case, but the multiplicative proof needs a complete analytic-extension argument or a fuller citation before the density formula is fully established.","tokens_in":42579,"tokens_out":8335,"duration_ms":83942,"concrete_test":"Independently complete the §4.4 extension argument: for a two-atom unitary law such as μ=(1/3)δ_{e^{2πi/5}}+(2/3)δ_{e^{3πi/4}}, t=0.8, compute the Jacobian determinant of V(t,λ0,ε0)=(t,λ(t;λ0,ε0),z(t;λ0,ε0)) in (t,θ,ρ,δ) coordinates using (4.36)–(4.40) and verify it is nonzero on the whole of Δ_{t,μ} together with its boundary. If the determinant vanishes anywhere, the inverse-function-theorem step fails and the density formula is not justified.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim of Theorem 1.2 is that the Brown measure of u b_t is absolutely continuous with density r^{-2}w_t(θ) on Δ_{t,μ}. The proof in §4.4 requires that the regularized trace S̃(t,λ,z)=S(t,λ,z²) extend real-analytically to a neighborhood of (t,λ,0) for λ in Δ_{t,μ}; this is what lets the authors pass from values along Hamiltonian characteristics to the limit ε→0 and to differentiate twice under the limit (e.g., ∂²s_t/∂θ² = (d/dθ)m_t(θ) just before (4.60)). The paper states \"we will give the main lines below why it holds\" and then refers to [17, Section 7.4] for details, but the supplied inverse-function-theorem sketch does not contain a complete verification for the arbitrary unitary initial condition (4.17). In particular, Lemma 4.22 proves that the proposed density has total mass 1, but it does not by itself prove that the distributional Laplacian of s_t equals that density; if the analytic extension failed, additional boundary or atomic mass could coexist with the absolutely continuous part. Since Theorem 4.23 and hence Theorem 1.2 rest on this extension, the multiplicative side is conditional on a gap that is acknowledged in the text itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the Brown measures of two free non-normal processes: the free circular Brownian motion with a self-adjoint initial condition, x0+c_t, and the free multiplicative Brownian motion with an arbitrary unitary initial condition, u b_t. The main results are stated as Theorems 1.1 and 1.2. For x0+c_t, the support is the closure of Λ_t={a+ib: |b|<v_t(a)}, the density is w_t(a) db da with w_t(a)=(1/(πt))ψ_t'(a), and the push-forward under the natural map Ψ_t is exactly the law of x0+s_t. For u b_t, the support is the closure of Δ_{t,μ}={r e^{iθ}: r_t(θ)<r<1/r_t(θ)}, the density has the polar form (1/r²)w_t(θ), and the push-forward under Γ_t is the law of u u_t. In the Haar-unitary case the Brown measure is the annulus law with density 1/(2πt r²). The additive part is derived self-containedly: the Hamilton-Jacobi PDE is obtained by free Itô calculus, the Hamiltonian ODEs are solved explicitly, the Laplacian of S(t,λ,0) is computed both outside and inside Λ_t, and the normalization argument shows that this gives a probability measure. The multiplicative part follows the same strategy but is less self-contained: the real-analytic extension of the regularized trace at ε=0 is taken from Driver-Hall-Kemp [17] and is only sketched in Section 4.4. The push-forward target laws are external and independently known from Biane and from the second author's earlier work, so the Brown measures are computed rather than fitted.","tokens_in":42870,"tokens_out":8147,"duration_ms":81128,"significance":"If the results are correct, the paper provides explicit density formulas for Brown measures in two natural non-normal free processes and reveals a clean relation between these Brown measures and the subordination functions for free additive and multiplicative convolution. The additive part is a clear strength: the PDE derivation, the ODE solution, the Laplacian computation, and the normalization are all present and verifiable. The multiplicative part is a substantial extension of Driver-Hall-Kemp and includes a new Haar-unitary result that is corroborated independently by the R-diagonal/Haagerup-Larsen calculation in the appendix. The push-forward statements are not fitted: the target laws of x0+s_t and u u_t are independent external comparison objects. The main weakness is that the multiplicative proof depends on a real-analytic extension result that the text itself says is only outlined; this is a correctness risk rather than a circularity. The central claims are plausible and the evidence in the paper is strong, but the multiplicative theorems should not be stated unconditionally until the missing analytic regularity step is supplied or explicitly reduced to [17, Theorem 7.4].","major_comments":[{"comment":"The proof of Theorem 4.23 requires that S̃(t,λ,z)=S(t,λ,z²) extends real-analytically to a neighborhood of (σ,λ,0) for λ∈Δ_{t,μ}. The text describes this as 'the key' and gives an inverse-function-theorem sketch, but the verification is not complete for the arbitrary unitary initial condition (4.17). In particular, the passage from values along Hamiltonian characteristics to the limit ε→0, and the interchange leading to equation (4.60), are not fully justified. Since Theorem 4.23 and hence Theorem 1.2 rest on this extension, this is a load-bearing gap. Please provide a complete proof, or give a precise reduction to [17, Theorem 7.4] that covers the arbitrary unitary initial condition.","section":"§4.4, paragraph before Theorem 4.23"},{"comment":"The statement that 'in the case λ∉Δ_{t,μ}, ΔλS(t,λ,0)=0 and hence the Brown measure is supported in Δ_{t,μ}' is asserted at the start of Section 4 without a proof analogous to the additive Theorem 3.8. The normalization in Lemma 4.22 shows only that the proposed absolutely continuous density has total mass one on Δ_{t,μ}; it does not rule out extra boundary or atomic mass unless one already knows that the Brown measure is supported inside Δ_{t,μ}. This is not a presentation issue: Theorem 4.23's conclusion that the Brown measure of g_t is supported in Δ_{t,μ} depends on this assertion. Please add a proof of harmonicity of S(t,λ,0) outside the candidate support, or state explicitly that Theorem 4.23 is conditional on the analytic extension and on the support-containment argument.","section":"§4, opening discussion, and §4.4, Lemma 4.22"},{"comment":"The continuous extension of ε0^t from Δ_{t,μ} to Δ_{t,μ}\\(∂Δ_{t,μ}∩T), with value zero on (∂Δ_{t,μ}\\T), is stated without a proof of the boundary limit. The formula (4.50) is derived for interior points, and the boundary behavior is used in Corollary 4.20 and in Theorem 4.23 to assign initial data on the boundary. Please include the limiting argument, or fold it into the analytic-extension proof requested above, so that the boundary assignment is fully documented.","section":"§4.3, Proposition 4.18 and Corollary 4.20"}],"minor_comments":[{"comment":"In the statement of Theorem 1.2, 'suppρt' should read 'suppμt' to avoid confusion with the additive case notation.","section":"Theorem 1.2, point 1"},{"comment":"The text writes 'where λ0=a0+b0'; this should be 'λ0=a0+ib0'.","section":"§4.2, below Eq. (4.24)"},{"comment":"The sentence 'the upper bound (4.59) follows from (4.8) and Proposition 4.21' appears to cite the wrong equation: (4.8) is a characterization of U_{t,μ}, while the needed bound dφ/dθ≤2 is Lemma 4.8 (or equation (4.12)). Please correct the reference.","section":"§4.4, last paragraph of proof of Theorem 4.23"},{"comment":"The phrase 'using using Haagerup-Larsen's formula' contains a duplicated 'using' and should be corrected.","section":"§1.2, paragraph on the Haar-unitary case"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern raised by the skeptic matches my reading: the multiplicative theorems are conditional on the real-analytic extension of S̃ at ε=0, and the manuscript itself acknowledges that this is only outlined. I agree with the reader's conditional verdict. I would ask the authors to add a complete proof of that extension, or a full reduction to [17, Section 7.4], and to prove the harmonicity of S outside Δ_{t,μ} before stating Theorems 4.23 and 1.2 unconditionally. The additive part is sound and could stand on its own; the multiplicative part is the advertised extension and needs the missing support before publication. I found no concerns about novelty or citation patterns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. The additive result is genuinely new and self-contained: the paper converts Biane–Lehner's implicit formula for x0+ct into an explicit density wt(a)=ψt'(a)/(πt), proves support Λt={|Im z|<vt(Re z)}, and shows the push-forward is the law of x0+st. I checked the Hamilton–Jacobi derivation, the ODE solution, and the Laplacian computation; they are coherent, and the normalization argument (Lemma 3.11) is clean. The multiplicative result is also new for arbitrary unitary initial conditions and gives the nice density r^{-2}wt(θ) on the delimited region Δt,μ; the Haar unitary case gives the annulus law, which is also checked independently in Appendix A by Haagerup–Larsen, a good cross-check.\n\nWhere are the soft spots? The multiplicative proof has a load-bearing gap, acknowledged in the text. Section 4.4 needs S̃(t,λ,z)=S(t,λ,z²) to extend real-analytically to z=0; otherwise the passage from regularized trace to the distributional Laplacian and the absence of boundary mass isn't justified. The paper gives only a sketch (\"we will give the main lines below why it holds\") and refers to Driver–Hall–Kemp Section 7.4. The sketch includes an inverse-function argument, but it is not a complete verification for the arbitrary unitary initial condition; the Jacobian check is asserted. Lemma 4.22 shows total mass 1 of the candidate density, but that alone doesn't rule out an atomic or singular extra component if the analytic continuation failed. So Theorem 1.2 is conditional on a missing detail. I don't think it's wrong, and I'd be surprised if the extension fails for the same structural reasons as in DHK, but as written it is a gap in a proof, not a mere exposition choice. The additive theorem is not affected by this.\n\nAnother honest limitation the authors themselves flag: random matrix convergence for U_N G_N(t) to u b_t is open even for u=I; the eigenvalue simulations are just simulations. That's stated clearly, so no points deducted.\n\nCitation pattern looks appropriate: the paper leans on [17] and on the second author's earlier work [38,39] for subordination functions, and it says so. No hidden fitting; the Brown measures are computed, not defined to match targets.\n\nWho is this for? Free probabilists and random matrix theorists working on Brown measures. The additive theorem is a clear improvement over Biane–Lehner for self-adjoint initial data; the multiplicative theorem, once the regularity detail is supplied, answers a natural question. I'd send it to a serious referee — the main results are likely correct and significant — but with a request to substantially expand Section 4.4 or to state the regularity as a lemma with proof. My own verdict would be conditional acceptance.","headline":"Strong, mostly self-contained computation of Brown measures for circular and multiplicative Brownian motions with non-trivial initial conditions; the multiplicative half rests on a lightly documented regularity step.","tokens_in":43370,"tokens_out":2539,"would_cite":true,"duration_ms":24870,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L54","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper computes explicit densities for the Brown measures of $x_0+c_t$ and $u b_t$.","keywords":["Brown measure","free circular Brownian motion","free multiplicative Brownian motion","subordination function","Hamilton-Jacobi equation","free probability","random matrices","annulus law"],"falsifier":"Simulate eigenvalues of $U_N G_N(t)$ with $N=2000$ and $U_N$ a unitary matrix whose empirical spectral measure approximates $\\frac{1}{3}\\delta_{e^{i\\pi/3}}+\\frac{2}{3}\\delta_{e^{4\\pi i/5}}$; after binning by argument, compare the radial density to $\\frac{1}{r^2}w_t(\\theta)$ with $w_t(\\theta)=\\frac{1}{2\\pi t}\\frac{d\\varphi}{d\\theta}$. A systematic departure from the $1/r^2$ scaling inside the predicted support would refute the formula.","tokens_in":42388,"feed_emoji":"🌀","tokens_out":10289,"duration_ms":86758,"temperature":0.7,"pith_summary":"The paper computes the eigenvalue density, in the large-matrix limit, of a non-Hermitian random matrix with i.i.d. complex Gaussian entries deformed by adding a self-adjoint matrix or by multiplying a unitary matrix. The limit objects are $x_0+c_t$, a free circular Brownian motion with self-adjoint initial condition, and $u b_t$, a free multiplicative Brownian motion with unitary initial condition. For both, the Brown measure is shown to be absolutely continuous, with a density written explicitly in terms of the subordination function of the corresponding free additive or multiplicative convolution. The densities are constant along vertical lines (additive case) or inversely proportional to $r^2$ along radial lines (multiplicative case), and a natural push-forward map sends each Brown measure to the law of $x_0+s_t$ or $u u_t$.","feed_headline":"Explicit density found for Brown measure of x0+ct","feed_subtitle":"The density is constant on vertical lines and maps to the law of x0+st.","key_machinery":"The central object is the regularized logarithmic determinant $S(t,\\lambda,\\varepsilon)=\\tau[\\log(x_{t,\\lambda}^*x_{t,\\lambda}+\\varepsilon)]$, with $x_{t,\\lambda}=\\lambda-(x_0+c_t)$ or $\\lambda-u b_t$. For each $\\lambda$ the paper solves a first-order Hamilton–Jacobi equation, $\\partial_t S=\\varepsilon(\\partial_\\varepsilon S)^2$ in the additive case and a higher-dimensional analogue in the multiplicative case, by integrating Hamilton's equations. The lifetime of the Hamiltonian trajectory is tuned by the initial condition $\\varepsilon_0$ so that $\\varepsilon(t)\\to 0$ exactly at the chosen time $t$; the Laplacian of $S(t,\\lambda,0)$ then gives the Brown-measure density. The boundary curves $v_t(a)$ and $r_t(\\theta)$ are determined by the subordination functions $F_t$ (with left inverse $H_t$) and $\\eta_t$ (with left inverse $\\Phi_{t,\\bar\\mu}$), which connect $x_0+s_t$ to $x_0$ and $u^*u_t$ to $u^*$.","core_discovery":"For a self-adjoint $x_0$ with spectral law $\\mu$, the Brown measure $\\rho_t$ of $x_0+c_t$ is supported on $\\overline{\\Lambda_t}=\\{a+ib:|b|\\le v_t(a)\\}$, where $v_t(a)$ is the unique positive solution of $\\int \\frac{d\\mu(x)}{(a-x)^2+v^2}=\\frac{1}{t}$. Inside the support the density is $w_t(a)\\,da\\,db$ with $w_t(a)=\\frac{1}{\\pi t}\\psi_t'(a)$, where $\\psi_t(a)=H_t(a+iv_t(a))$ and $H_t(z)=z+tG_{x_0}(z)$; the density does not depend on $b$. The map $\\Psi_t(a+ib)=H_t(a+iv_t(a))$ pushes $\\rho_t$ forward to the law of $x_0+s_t$, and these two properties characterize $\\rho_t$ uniquely. For a unitary $u$ with law $\\mu$, the Brown measure of $u b_t$ is supported on $\\overline{\\Delta_{t,\\mu}}=\\{re^{i\\theta}:r_t(\\theta)\\le r\\le 1/r_t(\\theta)\\}$, where $r_t(\\theta)$ is defined through the subordination function for $u^*u_t$. Its density in polar coordinates is $\\frac{1}{r^2}w_t(\\theta)$, with $w_t(\\theta)=\\frac{1}{2\\pi t}\\frac{d\\varphi}{d\\theta}$, and the push-forward under $\\Gamma_t(re^{i\\theta})=\\Phi_{t,\\bar\\mu}(r_t(\\theta)e^{i\\theta})$ is the law of $u u_t$. When $u$ is Haar unitary, the support becomes the annulus $e^{-t/2}\\le|z|\\le e^{t/2}$ and $w_t(\\theta)=\\frac{1}{2\\pi t}$, giving the annulus law.","pith_inferences":["Inference: the Hamilton–Jacobi route here suggests that the same PDE, with suitable initial conditions, could compute Brown measures of other deformed processes as long as a subordination function is available, since the PDE itself does not depend on the initial law.","Inference: the explicit annulus law for Haar-initialized $u b_t$ gives a sharp test case for the open convergence problem of empirical eigenvalue distributions of $GL(N)$ Brownian motions, because any discrepancy from the $1/r^2$ profile would show up in simulations.","Inference: the density formulas reduce Brown-measure computation to evaluating the boundary radius functions $v_t$ or $r_t$; numerical evaluation of those functions could provide a fast proxy for eigenvalue histograms without diagonalizing large matrices."],"forward_implications":["The Brown measure of $x_0+c_t$ is uniquely pinned down by its support, vertical constancy of the density, and the push-forward to $x_0+s_t$.","The empirical eigenvalue density of $X_N+\\sqrt{t}Z_N$ for a self-adjoint $X_N$ converging to $x_0$ is approximated by $w_t(a)$ in the large-$N$ limit, with the vertical spread governed by $v_t(a)$.","For unitary initial conditions, the Brown measure of $u b_t$ always satisfies the radial scaling $\\rho(re^{i\\theta})\\propto 1/r^2$ inside each angular sector.","If $u$ is Haar distributed, the Brown measure is exactly the rotationally invariant annulus law with density $\\frac{1}{2\\pi t r^2}$.","The support of the Brown measure of $u b_t$ is symmetric under $z\\mapsto 1/\\bar z$, and the number of its connected components decreases with $t$."],"supporting_citations":[{"why":"Supplies the subordination function $F_t/H_t$ and the density of $x_0+s_t$, which the additive density and push-forward formulas are built on.","marker":"[10]"},{"why":"Provides the free Itô product rule used to derive the PDE for $S(t,\\lambda,\\varepsilon)$.","marker":"[14]"},{"why":"Supplies the Hamilton–Jacobi method and the real-analytic continuation argument for the regularized trace that the multiplicative computation relies on.","marker":"[17]"},{"why":"Supplies the subordination function $\\eta_t$, the boundary radius $r_t(\\theta)$, and the density of $u u_t$ used in the multiplicative push-forward.","marker":"[39]"},{"why":"Gives the almost-sure convergence of empirical eigenvalue distributions to the Brown measure, connecting the computed density to random-matrix eigenvalues.","marker":"[32]"}],"fun_headline_variants":["Explicit Brown measure densities for free circular and multiplicative Brownian motions","Explicit Brown measures for free convolution with circular and multiplicative noise","Brown measure of x0+ct: density constant on vertical lines","Explicit Brown measure for u bt: annulus when u is Haar unitary","Explicit densities for Brown measures of x0+ct and u bt"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The multiplicative computation assumes that the regularized trace $S(t,\\lambda,z^2)$ extends real-analytically to a neighborhood of $(t,\\lambda,0)$ for every $\\lambda$ in the predicted support; if that extension fails, the Laplacian limit might carry extra boundary mass that the explicit density does not count.","fun_headline_variants_meta":{"raw":{"variants":["Explicit Brown measure densities for free circular and multiplicative Brownian motions","Explicit Brown measures for free convolution with circular and multiplicative noise","Brown measure of x0+ct: density constant on vertical lines","Explicit Brown measure for u bt: annulus when u is Haar unitary","Explicit densities for Brown measures of x0+ct and u bt"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001638,"raw_usage":{"total_tokens":6760,"prompt_tokens":1443,"completion_tokens":5317,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1059,"completion_tokens_details":{"reasoning_tokens":5224}},"tokens_in":1059,"tokens_out":5317,"duration_ms":33403,"temperature":1.0,"reasoning_tokens":5224,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:48:12.059326+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate eigenvalues of $U_N G_N(t)$ with $N=2000$ and $U_N$ a unitary matrix whose empirical spectral measure approximates $\\frac{1}{3}\\delta_{e^{i\\pi/3}}+\\frac{2}{3}\\delta_{e^{4\\pi i/5}}$; after binning by argument, compare the radial density to $\\frac{1}{r^2}w_t(\\theta)$ with $w_t(\\theta)=\\frac{1}{2\\pi t}\\frac{d\\varphi}{d\\theta}$. A systematic departure from the $1/r^2$ scaling inside the predicted support would refute the formula.","supporting_citations":[{"cited_title":"On the free convolution with a semi-circular distribution","cited_arxiv_id":null,"evidence_quote":"Supplies the subordination function $F_t/H_t$ and the density of $x_0+s_t$, which the additive density and push-forward formulas are built on."},{"cited_title":"Stochastic calculus with respect to free Brownian motion and analysis on Wigner space","cited_arxiv_id":null,"evidence_quote":"Provides the free Itô product rule used to derive the PDE for $S(t,\\lambda,\\varepsilon)$."},{"cited_title":"The Brown measure of the free multiplicative Brownian motion","cited_arxiv_id":"1903.11015","evidence_quote":"Supplies the Hamilton–Jacobi method and the real-analytic continuation argument for the regularized trace that the multiplicative computation relies on."},{"cited_title":"On the free convolution with a free multiplicative analogue of the normal distribution","cited_arxiv_id":null,"evidence_quote":"Supplies the subordination function $\\eta_t$, the boundary radius $r_t(\\theta)$, and the density of $u u_t$ used in the multiplicative push-forward."},{"cited_title":"Random regularization of Brown spectral measure","cited_arxiv_id":null,"evidence_quote":"Gives the almost-sure convergence of empirical eigenvalue distributions to the Brown measure, connecting the computed density to random-matrix eigenvalues."}],"review_version":1}