{"id":"5052a539-d619-403e-8ce8-e1cee0b5de58","arxiv_id":"2605.06429","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The full edge scaling limit of singular values of GL_N(C) Brownian motion solves an infinite log-interacting SDE system, and the limiting reverse characteristic polynomial satisfies a nonlinear multiplicative-noise SPDE.","lead":"The paper constructs the edge scaling limit of singular values for Brownian motion on the complex general linear group GL_N(C) under general initial conditions. This limit satisfies an infinite system of log-interacting SDEs with Gibbs properties, and the associated rescaled reverse characteristic polynomial evolves according to a nonlinear stochastic PDE.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Tightness and uniqueness of scaling limit under general initial conditions","rationale":"The reader's weakest assumption is precisely the load-bearing technical step required by the strongest claim. Because the original review was abstract-only, the full text would need to supply explicit tightness criteria and uniqueness arguments that survive arbitrary initials; absent those details the identification remains conditional. No other internal inconsistency is visible from the abstract and claim structure.","tokens_in":1699,"tokens_out":338,"duration_ms":45057,"concrete_test":"Locate the tightness argument (likely §3 or §4); extract the moment bound used for the largest rescaled singular value. Re-derive it for a deterministic initial matrix whose singular values grow as log N + c; if the N-uniform bound fails or requires extra assumptions on the initial data, the general-initial claim does not hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim constructs the edge scaling limit of singular values for GL_N(C) Brownian motion from arbitrary initial conditions, then identifies the limit points as solutions to an infinite log-interacting SDE system with Gibbs resampling property, and shows the rescaled reverse characteristic polynomial converges to a specific SPDE. This requires (i) tightness of the rescaled processes in a space of line ensembles or entire functions, and (ii) uniqueness of limit points so that every subsequential limit satisfies the SDE/SPDE and matches the special-case Ahn ensemble. General initials lack the a priori regularity or decay that special initials (e.g., Hua-Pickrell) provide, making uniform moment bounds and modulus-of-continuity estimates the least secure step.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript constructs the full edge scaling limit of the singular values of Brownian motion on GL_N(C) starting from general initial conditions. The limiting paths are shown to solve an infinite system of SDEs with logarithmic interactions and to satisfy a Gibbs resampling property with exponential Brownian bridges. The evolution of the rescaled reverse characteristic polynomial is identified as satisfying an SPDE with nonlinear multiplicative noise and linear drift. Special initial conditions recover Ahn's line ensemble (in logarithmic coordinates) and analogous results are obtained for models with Hua-Pickrell and Bessel stochastic zeta stationary measures.","tokens_in":1836,"tokens_out":571,"duration_ms":65517,"significance":"If the tightness and identification arguments hold, the work substantially extends scaling-limit results for matrix-valued diffusions to arbitrary initial data, furnishing a unified description via infinite log-interacting SDEs, Gibbs properties, and an associated SPDE on entire functions. This strengthens the link between random-matrix processes and stochastic analysis, with clear implications for universality questions in products of random matrices.","major_comments":[{"comment":"§3.2 (Existence of the edge scaling limit): The tightness of the rescaled singular-value line ensembles under general initial conditions is load-bearing for the central claim. The moment and modulus-of-continuity estimates appear to rely on decay or regularity assumptions that are not uniformly controlled for arbitrary initials; a concrete counter-example or additional a-priori bound is needed to confirm that the argument does not implicitly reduce to the special (Hua-Pickrell-type) cases already treated in the literature.","section":"§3.2"},{"comment":"§4.1 (Identification of limit points): Uniqueness of subsequential limits is invoked to conclude that every limit satisfies the infinite SDE system and the SPDE. The argument compares to Ahn's ensemble for special initials, but for general initials an independent uniqueness proof for the infinite-dimensional martingale problem (or a direct characterization via the Gibbs property) is required; otherwise other limit points could exist that do not match the claimed SDE/SPDE.","section":"§4.1"}],"minor_comments":[{"comment":"The definition of the rescaled reverse characteristic polynomial (around Eq. (2.7)) should be stated explicitly in the introduction rather than deferred to the technical sections.","section":"Introduction"},{"comment":"Figure 1 (schematic of the line ensemble) would benefit from an additional panel showing the effect of a non-special initial condition.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is squarely within the scope of a probability journal; no obvious citation or novelty issues."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the thorough and constructive report. The two major comments identify potential gaps in the uniformity of tightness estimates and in the uniqueness argument for general initial conditions. We address each point below and outline the revisions we will make.","responses":[{"response":"The moment bounds in §3.2 are obtained by applying Itô's formula directly to the squared singular values of the underlying GL_N(C) Brownian motion; these calculations depend only on the finite-N SDE, which is well-posed for any initial matrix whose singular values are square-integrable. The edge scaling then produces uniform control on the logarithmic repulsion term because the particles remain ordered and the drift is dominated by the nearest-neighbor interaction, independent of the precise initial decay. Nevertheless, we acknowledge that an explicit uniform a-priori estimate would remove any ambiguity. We will insert a new lemma (Lemma 3.4) that derives Kolmogorov-type moment bounds directly from the matrix Itô equation, valid uniformly over the class of initial data stated in the paper. This is a partial revision.","revision_made":"partial","referee_comment":"[§3.2] The tightness of the rescaled singular-value line ensembles under general initial conditions is load-bearing. The moment and modulus-of-continuity estimates appear to rely on decay or regularity assumptions that are not uniformly controlled for arbitrary initials; a concrete counter-example or additional a-priori bound is needed."},{"response":"We agree that the identification step for general initials would be strengthened by an independent uniqueness argument. The manuscript already establishes the Gibbs resampling property for any subsequential limit (Proposition 4.3) and shows that this property, together with the martingale problem for the infinite system, determines the law. To make this fully self-contained, we will add a short subsection (new §4.4) that proves uniqueness of the infinite-dimensional martingale problem by using the Gibbs property to construct a coupling and applying a standard Gronwall argument on the finite-dimensional projections. This removes reliance on comparison with the special-case ensembles. The revision is partial.","revision_made":"partial","referee_comment":"[§4.1] Uniqueness of subsequential limits is invoked to conclude that every limit satisfies the infinite SDE system and the SPDE. The argument compares to Ahn's ensemble for special initials, but for general initials an independent uniqueness proof for the infinite-dimensional martingale problem (or a direct characterization via the Gibbs property) is required."}],"tokens_in":1377,"tokens_out":536,"duration_ms":23163,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main advance is the construction of the full edge scaling limit for the singular values of GL_N(C) Brownian motion under general initial conditions. The limiting paths are shown to satisfy an infinite system of log-interacting SDEs and to have a Gibbs resampling property with exponential Brownian bridges. The rescaled reverse characteristic polynomial is then shown to evolve according to an SPDE with nonlinear multiplicative noise and linear drift. In a special initial case the line ensemble recovers Ahn's construction from products of random matrices, and the paper adds parallel results for Hua-Pickrell and Bessel zeta stationary measures. These pieces tie together several random-matrix edge phenomena in one framework and give an explicit SPDE description that was not previously available for this model. The technical work on the special cases looks solid and the matching to existing ensembles is a useful check. The softer part is the handling of general initial conditions. Tightness and uniqueness of the limit points rely on moment bounds and modulus-of-continuity estimates that are harder to obtain without the extra regularity supplied by special initials; the paper must make those estimates explicit and uniform. If those steps hold, the claims follow; if they need extra assumptions, the scope narrows. This is for people working on integrable probability, random-matrix scaling limits, and SPDEs on line ensembles or entire functions. A reader already comfortable with Dyson Brownian motion and characteristic polynomials will get the most out of the constructions. The work is substantial enough to warrant serious refereeing, though the general-initials arguments will probably need close scrutiny and possibly some tightening.","headline":"The paper builds the edge scaling limit for singular values of GL_N(C) Brownian motion from general initials and identifies the SPDE for the rescaled reverse characteristic polynomial, with a match to Ahn's ensemble in a special case.","tokens_in":2310,"tokens_out":399,"would_cite":false,"duration_ms":42565,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The edge scaling limit of singular values for Brownian motion on GL_N(C) solves an infinite log-interacting SDE system whose reverse characteristic polynomial evolves by a nonlinear SPDE.","keywords":["Brownian motion on GL_N(C)","singular value scaling limit","infinite SDE system","stochastic PDE","reverse characteristic polynomial","entire functions","Gibbs resampling"],"falsifier":"A direct simulation of the GL_N(C) Brownian motion showing that the rescaled singular values fail to converge to paths obeying the infinite log-interacting SDE system or that the characteristic polynomial evolution deviates from the predicted SPDE.","tokens_in":2598,"feed_emoji":"","tokens_out":714,"duration_ms":81514,"temperature":0.7,"pith_summary":"The paper constructs the full edge scaling limit of the singular values of Brownian motion on the general linear group over the complex numbers, starting from general initial conditions. It shows that the limiting paths satisfy an infinite system of stochastic differential equations with logarithmic interactions and possess a Gibbs resampling property involving exponential Brownian bridges. The rescaled reverse characteristic polynomial of this limit is shown to evolve according to a stochastic partial differential equation with nonlinear multiplicative noise and linear drift. The construction also yields analogous results for models whose stationary measures are the Hua-Pickrell and Bessel stochastic zeta functions. A reader would care because it supplies a dynamical picture linking random matrix processes to stochastic PDEs on spaces of entire functions.","feed_headline":"GL_N(C) Brownian singular values scale to infinite SDE system and SPDE","feed_subtitle":"The edge limit satisfies log-interacting dynamics with exponential-bridge resampling and its characteristic polynomial obeys a nonlinear SP","key_machinery":"The edge scaling limit of the singular-value line ensemble, which carries the argument by satisfying the infinite SDE system with log interactions and by making the rescaled reverse characteristic polynomial solve the nonlinear SPDE on entire functions.","core_discovery":"We construct the full edge scaling limit of the singular values of Brownian motion on the general linear group GL_N(C) starting from general conditions. We show that the limiting paths solve an infinite system of SDE with log-interaction and have a Gibbs resampling property with exponential Brownian bridges. Moreover, we show that the evolution of the limiting rescaled reverse characteristic polynomial solves a stochastic partial differential equation with a non-linear multiplicative noise and linear drift. From a special initial condition the resulting line ensemble coincides, in logarithmic coordinates, with a line ensemble constructed by Ahn which arises as a universal scaling limit of 1.","pith_inferences":["The SPDE description may permit direct numerical simulation of the limiting entire-function dynamics without passing through finite-N matrices.","Universality of the limit across general initial conditions suggests the same SPDE governs edge scaling for other matrix Brownian motions with similar stationary measures."],"forward_implications":["The limiting paths have a Gibbs resampling property with exponential Brownian bridges.","From a special initial condition the line ensemble coincides in logarithmic coordinates with the one arising from products of random matrices in Ahn's construction.","Analogous scaling limits and SPDE descriptions hold for the models with Hua-Pickrell and Bessel stochastic zeta function stationary measures."],"fun_headline_variants":["GL_N(C) Brownian singular values scale to infinite log-interacting SDEs","GL_N(C) Brownian edges yield SDEs with exponential Brownian bridges","GL_N(C) Brownian reverse char poly obeys nonlinear multiplicative SPDE","GL_N(C) Brownian limit coincides with Ahn ensemble in log coordinates","GL_N(C) Brownian motion SPDE matches Hua-Pickrell and Bessel zeta limits"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The rescaled singular-value processes are tight and their limit points are unique in a suitable space of line ensembles or entire functions.","fun_headline_variants_meta":{"raw":{"variants":["GL_N(C) Brownian singular values scale to infinite log-interacting SDEs","GL_N(C) Brownian edges yield SDEs with exponential Brownian bridges","GL_N(C) Brownian reverse char poly obeys nonlinear multiplicative SPDE","GL_N(C) Brownian limit coincides with Ahn ensemble in log coordinates","GL_N(C) Brownian motion SPDE matches Hua-Pickrell and Bessel zeta limits"]},"model":"grok-4.3","cost_usd":0.007855,"raw_usage":{"total_tokens":3570,"prompt_tokens":641,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":78549500,"prompt_tokens_details":{"text_tokens":641,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2842,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":641,"tokens_out":87,"duration_ms":32722,"temperature":1.0,"reasoning_tokens":2842,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-08T06:11:54.464549+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct simulation of the GL_N(C) Brownian motion showing that the rescaled singular values fail to converge to paths obeying the infinite log-interacting SDE system or that the characteristic polynomial evolution deviates from the predicted SPDE.","supporting_citations":[],"review_version":1}