{"id":"525a9a7b-4d38-4f0e-80c2-a5da037d1bce","arxiv_id":"1908.05708","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For confluently coupled rectangular random matrices, the large-n mean density of squared singular values is the second component of a three-measure vector equilibrium problem, with Meijer-G hard edge and sine/Airy universality.","lead":"This paper proves that the limiting distribution of the squared singular values of a product of two coupled random matrices is described by the solution to a three-measure equilibrium problem. The result resolves an open question posed by Akemann and Strahov and comes with an explicit spectral curve and local universality statements.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.5's strict inequality proof invokes Remark 3.8, which depends on Theorem 2.2, itself derived from Proposition 3.5 — a circular dependency.","rationale":"The reader's weakest assumption identified the one-cut/no-gap structure of the equilibrium minimizer as the main risk. I agree this is a relevant support condition, but the most concrete load-bearing flaw in the written proof is the circular dependency in Proposition 3.5: the strict inequality (3.42) is justified by Remark 3.8, whose proof comes from Theorem 2.2, which in turn relies on Proposition 3.5. This is a genuine structural gap in the manuscript as written, yet it is likely reorderable because Theorem 2.2 appears to use only the non-strict support and equality statements from Proposition 3.5, not (3.42). If the audit confirms this, the central claim stands but the presentation needs correction. Therefore the verdict remains conditional on fixing this dependency, matching the reader's CONDITIONAL determination, so no change to the verdict is recommended.","tokens_in":63874,"tokens_out":24542,"duration_ms":224619,"concrete_test":"Conduct a dependency audit of Section 3.6: list each place where the proof of Theorem 2.2 invokes (3.42), Proposition 4.3, or any strict inequality on (−q,0). If no such invocation occurs, prove (3.42) after Theorem 2.2 by evaluating the sign of the integral in (3.44) using the endpoint behaviors (2.9) and (2.11) and the absence of zeros of D1 on (−q,0), and then reorder the presentation accordingly. If Section 3.6 does use (3.42), the proof of Theorem 2.3 is genuinely circular and must be reworked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Proposition 3.5 establishes the strict inequality (3.42) by citing 'Remark 3.8 below', which asserts that ξ1−ξ2≠0 on (−q,0)∪(p,∞) because the discriminant of the spectral curve (3.61) has no roots there. This spectral curve is the content of Theorem 2.2, proved in Section 3.6 using the support structure supplied by Proposition 3.5 through Propositions 3.6–3.7. Thus, as written, Proposition 3.5 depends on Theorem 2.2, which depends on Proposition 3.5. The issue is not merely cosmetic: (3.42) is used through Proposition 4.3 to obtain Re φ1>0 near (−q,0) and φ1<0 on (−q,0), which is required for the exponential decay of jumps on the lens boundaries and on (−q,0) in the steepest descent analysis (Lemma 11.2). Without an independent proof of (3.42), the control of the Riemann-Hilbert problem, and hence the proof of the central claim Theorem 2.3, is incomplete. The footnote inside Proposition 3.5 claims the nonvanishing relies only on equalities (3.39), (3.41), (3.43), but the dependency chain still passes through Theorem 2.2; reordering the exposition would resolve this only if Theorem 2.2's proof never uses (3.42).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the squared singular values of the product X1 X2 of two coupled rectangular complex Gaussian matrices in the confluent case where the coupling matrix has a single singular value alpha < beta. The squared singular values form a determinantal point process whose correlation kernel Kn is expressed through mixed-type multiple orthogonal polynomials associated with modified Bessel functions. The authors introduce a vector equilibrium problem for three measures with an upper constraint on the first measure and an external field on the second, prove existence, uniqueness, and structural properties of the minimizer (Theorem 2.1), identify an explicit algebraic spectral curve of genus zero (Theorem 2.2), and then perform a Deift/Zhou steepest descent analysis of an associated 4x4 Riemann-Hilbert problem. This yields the main results: the limiting bulk density of the squared singular values is the second component of the equilibrium measure (Theorem 2.3), and the hard-edge scaling limit is the Meijer-G kernel (Theorem 2.4). The proof occupies most of the paper, with global, Airy, and Meijer-G parametrices, including a recent double-matching technique of Kuijlaars and Molag.","tokens_in":64113,"tokens_out":10992,"duration_ms":107534,"significance":"If correct, the paper resolves an open question raised by Akemann and Strahov by treating alpha and beta as independent parameters rather than coupled ones, and it extends the hard-edge results of Liu. The vector equilibrium formulation is natural and the explicit spectral curve with closed-form endpoints p and q is a concrete and valuable output. The four-sheeted Riemann surface and its rational uniformization are carefully developed, and the discriminant computations are explicit and checkable. The proof is technically substantial: the origin parametrix construction follows the Kuijlaars-Molag strategy with two matching conditions, and the auxiliary estimates in Lemmas 10.7, 10.12, and 12.1 are useful and clearly stated. The central results are falsifiable, and no free parameters are fitted to the limiting data. The main weakness is a circularity in the written order of the proof of the strict variational inequality (3.42), which feeds into the exponential decay estimates needed for the Riemann-Hilbert analysis.","major_comments":[{"comment":"The proof of the strict inequality (3.42) in Proposition 3.5 cites 'Remark 3.8 below', and Remark 3.8 is established after Theorem 2.1 using the discriminant of the spectral curve (2.16), i.e., after Theorem 2.2. Theorem 2.2 is proved in Section 3.6 using Proposition 3.7, which in turn relies on the support assertions of Proposition 3.5. On inspection, Theorem 2.2 does not appear to use the strict inequality (3.42) itself; the dependency chain passes through (3.38), (3.39), (3.41), and (3.43), as the footnote in Proposition 3.5 states. Nevertheless, the written order is circular, and since (3.42) is used in Proposition 4.3 and then in Lemma 11.2 to obtain exponential decay of the jumps, the proof of Theorem 2.3 is incomplete as written. The authors should either prove the nonvanishing of xi_1 - xi_2 on (-q,0) directly, or restructure the presentation so that the spectral curve and the discriminant are derived before the strict part of Proposition 3.5, without invoking (3.42) in that derivation.","section":"Sections 3.4-3.7, Proposition 3.5 and Remark 3.8"},{"comment":"The proof of Theorem 2.3 fully computes the limit only for x,y in a fixed compact subinterval (delta, p-delta) of the bulk. The theorem, however, claims uniform convergence on every compact subset of (0, infinity), which includes the soft edge x = p and the region x > p. These cases are dismissed with brief statements that they 'can be handled similarly' with an Airy parametrix or by the exponential decay of phi_2. Given that Theorem 2.3 is the central result, the endpoint and outside-support estimates should be written out, or the theorem statement should be narrowed to match the proof as presented.","section":"Section 12.1, proof of Theorem 2.3"}],"minor_comments":[{"comment":"The text says 'supp mu_3 = R_+' in the paragraph after (3.38), but the support of mu_3 should be R_- as stated in Theorem 2.1(c) and elsewhere; this is a typo.","section":"Section 3.4, proof of Proposition 3.5"},{"comment":"The display defining sigma_3 as the matrix [[0,1],[-1,0]] is not the standard third Pauli matrix and is inconsistent with the subsequent use of z^{a sigma_3} as a diagonal matrix. It should be diag(1,-1).","section":"Section 5, definition of sigma_3"},{"comment":"The abstract promises local Sine and Airy universality results, but these are not stated as formal theorems and the text explicitly says the details are omitted. The authors should either state these results as theorems with proofs or soften the abstract and the discussion in Section 2.4.","section":"Abstract and Section 2.4"},{"comment":"Lemma 3.1 establishes monotonicity, not strict monotonicity, yet the proof of Lemma 3.2 refers to a 'strictly increasing' quotient. The argument works with non-strict monotonicity, but the wording should be corrected.","section":"Lemma 3.2 and Proposition 3.4"},{"comment":"The caption mentions arcs gamma_k^+- for k = 1,2,3,4, but only k = 1,2,3 are defined.","section":"Figure 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically impressive and the central results are plausible, but the circular ordering around Proposition 3.5 and Remark 3.8 must be fixed before the proof of Theorem 2.3 can be considered complete. The authors should also align the abstract's universality claims with the formal theorem statements. I have no concerns about citation practice or novelty disclosure; the relation to Liu's work is appropriately acknowledged."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis is a serious piece of RMT, arguably the most complete treatment to date of the large-n limit for products of two coupled random matrices. What's actually new: a vector equilibrium problem with an upper constraint on the first measure and an external field on the second, an explicit spectral curve (genus-zero quartic with rational parametrization), and the hard-edge Meijer-G universality for arbitrary fixed alpha and beta, which previously required the parameters to be coupled. The main convergence result, Theorem 2.3, identifies the bulk density of squared singular values as the second component of the equilibrium measure, proved through a full 4x4 RH steepest descent analysis with local parametrices near 0, p, and -q. That is real work, and most of it looks solid.\n\nThe strongest section is the hard-edge analysis: the local parametrix near the origin uses the Bertola-Bothner Meijer-G construction with the Kuijlaars-Molag double matching, and the identification with the known Meijer-G kernel is legitimate because alpha,beta independence of the model kernel is proven first, before matching to Liu's result at a convenient parameter point.\n\nSoft spots, in order of severity. First, the proof of Proposition 3.5 uses Remark 3.8 to establish the strict inequality (3.42), and Remark 3.8 is stated after the discriminant computation in Section 3.7, which relies on the spectral curve Theorem 2.2. Theorem 2.2, in Section 3.6, uses the support structure from Proposition 3.5. That is a circular dependency as written. It is fixable: Theorem 2.2's proof does not appear to use (3.42), so reordering—prove the equalities and support structure, then the spectral curve, then (3.42)—would remove the circle. But a referee should force this to be explicit, because (3.42) is what controls the exponential decay on the lens boundaries in the RH estimates.\n\nSecond, sine and Airy universality are claimed in the abstract and in Section 2.4 but the details are not written. The authors say the ingredients are all there; for a paper already 79 pages that is acceptable, but the claims should be phrased as obtainable by the same methods rather than asserted results, or the reader is left trusting the authors' word. Third, the endpoint x = p in Theorem 2.3 is dismissed with a one-line 'handled similarly, with a worse error term'. Minor, but again a referee should ask for a sketch.\n\nNone of this is fatal. The central argument—VEP defined from the model parameters rather than fitted, and RH estimates independent of the hard-edge comparison—holds up. The citation pattern is fine; [42] is used only after the kernel's independence from alpha,beta is established, and the self-citation to [57] is appropriate since that is the source of the RH characterization.\n\nWho this is for: RMT researchers, particularly those working on RH steepest descent for biorthogonal ensembles and multi-matrix models. It deserves a serious peer review; with the circularity fixed and the universality claims properly qualified, it would be a valuable addition to the literature. Send it to review.","headline":"A substantial, well-executed RH analysis of coupled random matrix products; the main convergence and hard-edge results look right, but a circular dependency in Proposition 3.5 must be fixed and the sine/Airy claims qualified.","tokens_in":64664,"tokens_out":3332,"would_cite":true,"duration_ms":30255,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52","30E25","31A15","33C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The limiting density of squared singular values of a coupled random matrix product equals the second component of a three-measure equilibrium minimizer, with Meijer-G, sine, and Airy kernels at hard edge, bulk, and soft edge.","keywords":["products of random matrices","coupled two-matrix model","squared singular values","vector equilibrium problem","Riemann-Hilbert problem","multiple orthogonal polynomials","Meijer-G kernel","spectral curve"],"falsifier":"Run the model: generate $X_1,X_2$ from (1.1) with fixed $\\alpha<\\beta$ (for instance $\\alpha=1$, $\\beta=2$) at large $n$, form $X_1X_2$, and histogram the squared singular values scaled by $n^2$; then compare with $d\\mu_2/dx$ computed either numerically from the vector equilibrium problem or directly from the parametric formula $x=t/h(t)^2$, $d\\mu_2/dx=\\pi i\\,h(t)$ for $t\\in\\gamma^-_2$. A mismatch in the bulk, a wrong edge $p$, or a missing $x^{-2/3}$ divergence at the origin would refute Theorem 2.3. A cheaper check: solve the quartic discriminant (3.61) and verify that its two real roots equal the closed-form endpoints (2.10) and (2.8).","tokens_in":63629,"feed_emoji":"🎲","tokens_out":30727,"duration_ms":230246,"temperature":0.7,"pith_summary":"This paper resolves an open problem for correlated random matrices: what is the macroscopic spectrum of the product of two random matrices that are drawn together from a coupled model, rather than independently? Its main theorem proves that, when the coupling matrix has a single singular value, the limiting density of the squared singular values of the product is exactly the second component of the unique minimizer of a vector equilibrium problem — three measures on the line, the first constrained from above by an explicit singular density on the negative axis, the second carrying an external field on the positive axis. The minimizer is described in closed form: the positive measure is supported on $[0,p]$, diverging like $x^{-2/3}$ at the origin and vanishing like $(p-x)^{1/2}$ at $p$; the constraint is saturated on $(-\\infty,-q]$; and the endpoints $p,q$ are explicit algebraic functions of the two coupling parameters. The paper also shows that the same local universality seen for products of independent Gaussian matrices — the Meijer-G kernel at the hard edge, the sine kernel in the bulk, the Airy kernel at the soft edge — holds for the coupled product at every fixed admissible coupling, via a $4\\times 4$ Riemann-Hilbert steepest descent analysis of mixed-type multiple orthogonal polynomials for modified Bessel functions.","feed_headline":"Coupled random products obey one vector equilibrium law","feed_subtitle":"Settles the open density question for correlated products and shows the same universality as independent products.","key_machinery":"The argument is carried by three interlocking constructions. First, the vector equilibrium problem: three measures of total masses $1/2, 1, 1/2$ interacting through $I(\\nu_1)+I(\\nu_2)+I(\\nu_3)-I(\\nu_1,\\nu_2)-I(\\nu_2,\\nu_3)$, with the upper constraint $\\nu_1\\le\\sigma$ for the singular density $\\alpha/(\\pi\\sqrt{|x|})$ and the external field $2(\\beta-\\alpha)\\sqrt{x}$ on $\\nu_2$; the paper proves existence, uniqueness, and the one-cut structure of its minimizer using an iterative balayage argument together with a convexity criterion. Second, the four-sheeted Riemann surface glued across the three supports, which is shown to have genus $0$: its rational uniformization $z=t/h(t)^2$, $\\xi=h(t)$ rewrites the spectral curve as a quartic, and the origin is an index-3 branch point whose only integer-compatible ramification forces the $x^{-2/3}$ singularity of the density. Third, the $4\\times 4$ Riemann-Hilbert problem that encodes the correlation kernel from mixed-type multiple orthogonal polynomials with modified Bessel weights, analyzed by steepest descent along $Y\\to X\\to T\\to S\\to R$: the global parametrix is built from the uniformization, the parametrices at $p$ and $-q$ are Airy kernels, and the parametrix at the origin wraps the Meijer-G model problem for chains of coupled positive-definite matrices, matched with the two-circle technique of a fixed outer circle and a shrinking inner one.","core_discovery":"The central claim, Theorem 2.3, is that the correlation kernel $K_n$ of the determinantal point process formed by the squared singular values of $\\hat{Y}=X_1X_2$ satisfies, uniformly for $x$ in compact subsets of $(0,\\infty)$,\n$$\\lim_{n\\to\\infty} nK_n($n^{2}$x, $n^{2}$x) = \\frac{d\\mu_2}{dx}(x),$$\nwhere $(\\mu_1,\\mu_2,\\mu_3)$ is the unique minimizer of the energy $\\mathcal{E}(\\nu_1,\\nu_2,\\nu_3)=I(\\nu_1)+I(\\nu_2)+I(\\nu_3)-I(\\nu_1,\\nu_2)-I(\\nu_2,\\nu_3)+2(\\beta-\\alpha)\\int\\sqrt{x}\\,d\\nu_2(x)$ over triples with $|\\nu_1|=|\\nu_3|=1/2$, $|\\nu_2|=1$, constraint $\\nu_1\\le\\sigma$ on $\\mathbb{R}_-$ with $d\\sigma/dx=\\alpha/(\\pi\\sqrt{|x|})$, and $\\nu_2$ supported on $\\mathbb{R}_+$. The minimizer is structurally rigid: $\\operatorname{supp}(\\sigma-\\mu_1)=(-\\infty,-q]$, $\\operatorname{supp}\\mu_2=[0,p]$ with the explicit algebraic endpoints (2.10) and (2.8), densities with square-root vanishing at $p$ and $-q$ and an $x^{-2/3}$ blow-up at the origin, and $\\mu_3$ equal to half the balayage of $\\mu_2$ onto the negative axis. The Cauchy-transform combinations $\\xi_1,\\ldots,\\xi_4$ of these measures are the four branches of the spectral curve $\\xi^4-\\frac{\\alpha^2+\\beta^2}{z}\\xi^2+\\frac{\\alpha^2-\\beta^2}{z^2}\\xi+\\frac{\\alpha^2\\beta^2}{z^2}=0$, rationally parametrized by $(z,\\xi)=(t/h(t)^2,h(t))$ with $h(t)=\\frac{t^2-(\\alpha^2+\\beta^2)t+\\alpha^2\\beta^2}{\\beta^2-\\alpha^2}$, which yields an explicit parametric formula for the density $d\\mu_2/dx$. At the hard edge, Theorem 2.4 identifies the scaled kernel limit as the universal Meijer-G kernel $\\mathcal{K}_{\\nu,\\kappa}$, independent of $\\alpha,\\beta$; the sine and Airy limits in the bulk and at the soft edge are stated as further consequences of the same asymptotic analysis.","pith_inferences":["The equilibrium scheme (constraint on the left, external field on the right, free third measure) is presumably the template for longer chains of coupled products: any chain of measures with alternating interaction signs gives the same formal energy, and a genus-zero uniformization should again yield explicit densities. This extrapolation is ours, not the paper's.","The $x^{-2/3}$ blow-up at the hard edge is forced by an index-3 branch point, a mechanism that does not depend on the Bessel weights; if that reading is correct, the same exponent should appear at any hard edge where three sheets of a spectral curve meet, with a third-order Meijer-G kernel as the microscopic limit.","Because $\\mu_3$ is the half-balayage of $\\mu_2$, the whole minimizer is determined by $\\mu_2$ and the single endpoint $q$; a practical consequence is that the constrained equilibrium problem can be treated numerically as a one-function problem on $[0,p]$."],"forward_implications":["The zeros of the average characteristic polynomial of $\\hat{Y}^*\\hat{Y}$, scaled by $n^2$, accumulate according to $\\mu_2$, and the empirical law of the squared singular values converges weakly to the same measure.","The hard-edge scaling limit is the universal Meijer-G kernel at every fixed coupling $\\alpha<\\beta$, with the explicit microscopic scale $1/(n(\\beta^2-\\alpha^2))$; this extends the previously known hard-edge results from specially tuned couplings to the full admissible range.","In the bulk of the spectrum the sine kernel governs, and at the soft edge $x=p$ the Airy kernel governs; the paper states these as consequences of the steepest descent analysis with all ingredients present, although the endpoint details are not written out.","The spectral curve yields an explicit parametric formula for the limiting density — the graph of $d\\mu_2/dx$ is $(t/h(t)^2,\\pi i\\,h(t))$ along the contour $\\gamma^-_2$ — so the density can be evaluated numerically without solving any variational problem.","As the coupling parameter moves across its range, the model interpolates between the Laguerre-type regime and the product-of-two-Ginibre regime, and the theorem supplies the limiting density along the entire interpolation path rather than only at the endpoints."],"supporting_citations":[{"why":"Introduces the coupled two-matrix model, proves the squared singular values form a determinantal point process, and poses the open question about their limiting mean distribution that this paper resolves.","marker":"[7]"},{"why":"Gives the hard-edge phase transition and Meijer-G limits for the specially tuned coupling; Theorem 2.4 matches its parameters to identify the limiting kernel in general.","marker":"[42]"},{"why":"Shows the biorthogonal functions are mixed-type multiple orthogonal polynomials for modified Bessel weights, providing the 4x4 Riemann-Hilbert characterization from which the analysis starts.","marker":"[57]"},{"why":"Supplies the general Riemann-Hilbert theory of mixed-type multiple orthogonal polynomials, including the representation of the correlation kernel in terms of the RH solution.","marker":"[21]"},{"why":"Provides the potential-theoretic machinery, in particular the convexity criterion (Theorem IV.1.10(c)) that forces the one-cut support of the second equilibrium measure.","marker":"[53]"},{"why":"Constructs the bare Meijer-G parametrix for coupled positive-definite matrix chains that is used as the model problem for the local parametrix at the origin.","marker":"[11]"},{"why":"Introduces the two-circle matching technique that makes the local parametrix at the origin match simultaneously on a fixed outer circle and a shrinking inner circle.","marker":"[37]"},{"why":"Establishes existence and uniqueness of weakly admissible vector equilibrium problems, the framework guaranteeing the minimizer used in Theorem 2.3.","marker":"[34]"},{"why":"Identifies the Meijer G-kernels as the hard-edge universality class for products of Gaussian matrices, the class whose validity for the coupled product is established here.","marker":"[40]"},{"why":"Show the squared singular values of products of independent complex Gaussian matrices form a determinantal point process, the independent-product theory being relaxed and extended.","marker":"[3, 4]"}],"fun_headline_variants":["Coupled products obey vector equilibrium law","Universal spectral limits for coupled products","Squared singular values from vector equilibrium","Explicit spectral curve for coupled products","Coupled matrix products: one law, many kernels"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the equilibrium minimizer having a one-cut shape — one interval starting at the origin for the positive-axis measure and one unbounded interval of saturation for the negative-axis constraint — a structure the paper derives from a convexity computation together with a classical potential-theory criterion; if any admissible pair $\\alpha<\\beta$ produced a gap in the support or a second contact interval, the global parametrix and the main convergence statement would fail as written.","fun_headline_variants_meta":{"raw":{"variants":["Coupled products obey vector equilibrium law","Universal spectral limits for coupled products","Squared singular values from vector equilibrium","Explicit spectral curve for coupled products","Coupled matrix products: one law, many kernels"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000312,"raw_usage":{"total_tokens":1934,"prompt_tokens":1266,"completion_tokens":668,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":882,"completion_tokens_details":{"reasoning_tokens":605}},"tokens_in":882,"tokens_out":668,"duration_ms":6673,"temperature":1.0,"reasoning_tokens":605,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:05:56.600915+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the model: generate $X_1,X_2$ from (1.1) with fixed $\\alpha<\\beta$ (for instance $\\alpha=1$, $\\beta=2$) at large $n$, form $X_1X_2$, and histogram the squared singular values scaled by $n^2$; then compare with $d\\mu_2/dx$ computed either numerically from the vector equilibrium problem or directly from the parametric formula $x=t/h(t)^2$, $d\\mu_2/dx=\\pi i\\,h(t)$ for $t\\in\\gamma^-_2$. A mismatch in the bulk, a wrong edge $p$, or a missing $x^{-2/3}$ divergence at the origin would refute Theorem 2.3. A cheaper check: solve the quartic discriminant (3.61) and verify that its two real roots equal the closed-form endpoints (2.10) and (2.8).","supporting_citations":[{"cited_title":"Akemann and E","cited_arxiv_id":null,"evidence_quote":"Introduces the coupled two-matrix model, proves the squared singular values form a determinantal point process, and poses the open question about their limiting mean distribution that this paper resolves."},{"cited_title":"Liu, Singular values for products of two coupled random matrices: hard edge phase transition, Constr","cited_arxiv_id":null,"evidence_quote":"Gives the hard-edge phase transition and Meijer-G limits for the specially tuned coupling; Theorem 2.4 matches its parameters to identify the limiting kernel in general."},{"cited_title":"Zhang, Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices, J","cited_arxiv_id":null,"evidence_quote":"Shows the biorthogonal functions are mixed-type multiple orthogonal polynomials for modified Bessel weights, providing the 4x4 Riemann-Hilbert characterization from which the analysis starts."},{"cited_title":"Daems and A","cited_arxiv_id":null,"evidence_quote":"Supplies the general Riemann-Hilbert theory of mixed-type multiple orthogonal polynomials, including the representation of the correlation kernel in terms of the RH solution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the potential-theoretic machinery, in particular the convexity criterion (Theorem IV.1.10(c)) that forces the one-cut support of the second equilibrium measure."},{"cited_title":"Bertola and T","cited_arxiv_id":null,"evidence_quote":"Constructs the bare Meijer-G parametrix for coupled positive-definite matrix chains that is used as the model problem for the local parametrix at the origin."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the two-circle matching technique that makes the local parametrix at the origin match simultaneously on a fixed outer circle and a shrinking inner circle."},{"cited_title":"Hardy and A","cited_arxiv_id":null,"evidence_quote":"Establishes existence and uniqueness of weakly admissible vector equilibrium problems, the framework guaranteeing the minimizer used in Theorem 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the Meijer G-kernels as the hard-edge universality class for products of Gaussian matrices, the class whose validity for the coupled product is established here."}],"review_version":1}