{"id":"0c028ca0-7d27-41c6-97a6-4103bbfeb84e","arxiv_id":"2508.11505","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Space-time Fisher-Hartwig asymptotics for the Hermitian Ornstein-Uhlenbeck process imply that fractional powers of its characteristic polynomial converge to a two-dimensional Gaussian multiplicative chaos measure on an infinite strip.","lead":"This mathematics preprint derives exact space-time asymptotics for characteristic polynomials of a random matrix model that evolves in time, and shows that fractional powers of the polynomial converge to a two-dimensional Gaussian multiplicative chaos measure. The result upgrades previously single-time random matrix theory to a dynamical setting and builds a new bridge to Liouville quantum gravity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract-only review: the key unverified link is uniform-in-time Fisher-Hartwig error bounds integrable over t, which the abstract does not state; without seeing them the infinite-strip GMC claim is unsupported but not contradicted.","rationale":"The reader's weakest assumption is exactly the uniform-in-time controllability and integrability of the space-time Fisher-Hartwig error terms, which is necessary to pass from pointwise asymptotics to a genuinely two-dimensional GMC measure on an infinite strip. I agree with that identification. Since only the abstract was provided, no internal inconsistency can be established; the concern is a missing verification rather than a demonstrated error. Therefore the appropriate verdict remains UNVERDICTED, and my pass does not change the reader's assessment. A concrete test would be to inspect the remainder bounds in the full text and confirm they are strong enough to integrate over time. If those bounds are present and correct, the concern would be resolved.","tokens_in":878,"tokens_out":1794,"duration_ms":23806,"concrete_test":"In the full text, locate the theorem stating the space-time Fisher-Hartwig asymptotics and check the stated remainder bound. Verify that it is uniform in the time parameter and integrable over t (e.g., sup over compact spectral sets of the error is O(e^{-c|t|}) or O((1+|t|)^{-p}) with p>1). Then trace the use of this bound in the GMC convergence proof: confirm that the same integrability is used to control the total mass or Laplace transform over the infinite strip. If the stated bound is weaker, test the proof by attempting to relax it to a non-integrable decay and see whether the GMC limit still exists; if it does not, the infinite-strip assertion is unproven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the convergence of fractional powers of the characteristic polynomial to a 2D Gaussian multiplicative chaos (GMC) measure on an infinite strip. This requires more than pointwise-in-time Fisher-Hartwig asymptotics: the remainder terms in the asymptotics must be controlled uniformly in the time parameter and must be integrable (or sufficiently decaying) over t ∈ R so that single-time estimates can be integrated into a bona fide space-time measure. The abstract states 'space-time Fisher-Hartwig asymptotics' but gives no statement of uniformity or remainder decay. If the error terms are only o(1) for each fixed t, or decay like t^{-α} with α ≤ 1, the cumulative contribution over the infinite time axis could diverge and the claimed GMC convergence would fail or require a different normalization. The 'subcritical phase' condition must also align with the integrability of the limiting field; the abstract does not specify the precise γ-regime or the nature of the infinite-strip measure (e.g., reflected or periodic). Because the full text is unavailable, this is not a detected flaw but the load-bearing assumption most in need of verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims space-time Fisher-Hartwig asymptotics for characteristic polynomials of the stationary Hermitian Ornstein-Uhlenbeck process, with root and jump type singularities, generalizing prior static results of Krasovsky, Its-Krasovsky, and Charlier, and deriving as consequences: (i) convergence of fractional powers of the absolute characteristic polynomial (and of the associated exponential eigenvalue counting process) to a two-dimensional Gaussian multiplicative chaos measure on an infinite strip in the subcritical phase; (ii) leading-order Gaussian fluctuations of the log-characteristic polynomial; and (iii) optimal bulk rigidity for non-intersecting Brownian motions. The paper is presented as an abstract only; no full text was available to the referee.","tokens_in":1092,"tokens_out":2142,"duration_ms":26717,"significance":"If the claims hold, the paper offers a substantial dynamical extension of the interface between random matrix theory and Gaussian multiplicative chaos, adding a second connection after Bourgade-Falconet and generalizing the single-time GMC convergence of Berestycki-Webb-Wong, as well as the maximum and rigidity results of Lambert-Paquette and Claeys-Fahs-Lambert-Webb. The stated program is coherent: the subcritical regime is the natural place for such convergence, the exponent is not fitted but an input parameter, and the work rests on established external results rather than circular reasoning. The chief value lies in the space-time uniformity required to pass from single-time asymptotics to an infinite-strip GMC, and in the stated consequences for non-intersecting Brownian motions. The abstract alone, however, does not provide enough detail to verify the load-bearing technical estimates.","major_comments":[{"comment":"The central inference from 'space-time Fisher-Hartwig asymptotics' to a 2D Gaussian multiplicative chaos measure on an infinite strip requires more than pointwise-in-time asymptotics. One needs remainder bounds that are uniform in the time parameter and sufficiently integrable/decaying in t so that the single-time estimates can be integrated over the whole real line. The abstract does not state any uniformity or decay condition on the error terms. If the errors are only o(1) for each fixed t, or decay too slowly, the infinite-strip measure may fail to exist or require a different normalization. This is the key load-bearing point that must be visible in the full text.","section":"Abstract, GMC convergence claim"},{"comment":"The 'subcritical phase' is named but not specified quantitatively. The claims require that the stated condition on the fractional exponent matches the standard GMC phase (e.g., the analogue of gamma < sqrt(2), depending on normalization) and that the limiting measure is neither zero, nor infinite, nor trivial on the infinite strip. The abstract does not indicate whether the subcritical condition is proved to be equivalent to the integrability condition of the limiting field, or merely assumed. This needs to be addressed explicitly.","section":"Abstract, subcritical phase condition"},{"comment":"The word 'optimal' is a strong quantitative claim. The abstract gives no precise scale or exponent for the bulk rigidity, nor the matching lower and upper bounds that would establish optimality. This is a secondary but still advertised consequence; the details need to be stated in the full text so the claim can be checked.","section":"Abstract, 'optimal bulk rigidity'"}],"minor_comments":[{"comment":"The phrase 'exponential eigenvalues counting process' is not standard and is not defined in the abstract. Please define or rename this object.","section":"Abstract, terminology"},{"comment":"The exponent in 'fractional powers of the absolute value' is not named. Introducing a symbol (e.g., gamma) and its range in the abstract would improve precision.","section":"Abstract, notation"},{"comment":"The geometry of the 'infinite strip' is unclear: what are its coordinates, width, and boundary conditions? A sentence specifying the strip would help a reader see how the space-time process is parameterized.","section":"Abstract, 'infinite strip'"},{"comment":"The abstract cites multiple prior works by author-year. In a full paper, please ensure that the distinctions among the new results and each cited work are explicitly stated in the introduction (e.g., which prior result is the single-time special case).","section":"References"}],"recommendation":"uncertain","confidential_remarks":"The manuscript is provided only as an abstract. The abstract's claims are internally plausible and no circularity is evident, but the decisive technical question—whether the space-time Fisher-Hartwig remainders are uniform in time and integrable over the infinite time axis—cannot be assessed without the full text. I would recommend a full-text review before any editorial decision; if the uniformity claims are proven, the paper could be a strong contribution, but I cannot certify soundness from the abstract alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You'll want to know this: it's an abstract-only review, but the abstract describes a genuine program. The authors claim space-time Fisher-Hartwig asymptotics for the stationary Hermitian Ornstein-OU process, extending the static results of Krasovsky, Its-Krasovsky, and Charlier. If correct, the payoff is a two-dimensional GMC measure on an infinite strip, plus leading-order log-characteristic polynomial fluctuations and optimal bulk rigidity for non-intersecting Brownian motions. It would also be the second RMT-LQG connection after Bourgade-Falconet, which is a big deal in integrable probability.\n\nThe paper does something right immediately: it anchors itself to established results rather than trying to reinvent the wheel. The abstract reads coherently, with no obvious circularity and no fitted parameters. It names the subcritical phase and cites the relevant single-time GMC convergence from Berestycki-Webb-Wong. That is the natural next step after those works, so the novelty is real, not manufactured.\n\nNow the soft spot. The entire edifice rests on an unstated uniform-in-time estimate. Single-time Fisher-Hartwig asymptotics are not enough to build a measure on an infinite strip. You need the remainder to be controlled uniformly in t and to decay enough that integrating over t doesn't blow up. The abstract says 'space-time Fisher-Hartwig asymptotics' but gives no statement of the error term's behavior. If it's only o(1) for each fixed t, or decays like t^{-α} with α ≤ 1, the cumulative contribution over the infinite time axis could diverge and the GMC limit would either fail or require a different normalization. Also, the subcritical threshold has to line up with the integrability of the limiting field. I can't check any of that from the abstract, and neither can anyone else. That's not a detected flaw—it's the load-bearing unknown. A referee should be sent to probe exactly this.\n\nI should also note the citation pattern looks healthy. It cites the static FH literature and the GMC literature, and the claimed connection to Bourgade-Falconet is plausible. No red flags there.\n\nWho is this for? People working in integrable probability, random matrix theory, and anyone tracking LQG connections. It deserves a serious referee because the program is well-posed and the consequences are significant if the technical estimates hold. My honest take: send it to review, but the referee should be instructed to explicitly verify the uniform-in-time remainder control and the construction of the infinite-strip GMC measure. Without that, the paper would be incomplete.\n\nI'd bring it to the reading group and cite it if the proofs check out.","headline":"If the proofs deliver what the abstract promises, this is a significant space-time generalization of Fisher-Hartwig asymptotics and a real RMT-LQG bridge; the one thing a referee must check is the uniform-in-time control of the remainders.","tokens_in":1639,"tokens_out":1541,"would_cite":true,"duration_ms":20830,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","60G57","60J65"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes space-time Fisher-Hartwig asymptotics for the characteristic polynomial of the stationary Hermitian Ornstein-Uhlenbeck process, and derives from them the convergence to a two-dimensional Gaussian multiplicative chaos","keywords":["Fisher-Hartwig asymptotics","Gaussian multiplicative chaos","Ornstein-Uhlenbeck process","characteristic polynomial","non-intersecting Brownian motions","random matrix theory","Liouville quantum gravity","bulk rigidity"],"falsifier":"Compute the ratio of the $k$-th moment of the absolute value of the characteristic polynomial with a root singularity to the leading Fisher-Hartwig term, taking the supremum over time in a fixed interval, while $N \\to \\infty$. If this ratio diverges for any fixed $k$ below the subcritical threshold, the asserted uniformity in time fails, and the infinite-strip GMC convergence would be false.","tokens_in":716,"feed_emoji":"📈","tokens_out":3661,"duration_ms":41742,"temperature":0.7,"pith_summary":"The paper seeks to prove that adding time to a standard random matrix model preserves universal statistics. It claims Fisher-Hartwig asymptotics with root and jump singularities, uniformly in time, for the characteristic polynomial of the stationary Hermitian Ornstein-Uhlenbeck process, whose eigenvalues evolve as non-intersecting Brownian paths. If correct, fractional powers of the modulus of this polynomial converge to a random fractal measure on an infinite strip, the logarithm of the polynomial has Gaussian leading-order fluctuations, and the Brownian paths enjoy optimal bulk rigidity. These results would be the dynamical forms of known static statements and would offer a second bridge from random matrix theory to Liouville quantum gravity.","feed_headline":"Eigenvalue Brownian paths converge to a fractal strip measure","feed_subtitle":"New space-time asymptotics give leading log fluctuations and optimal rigidity for non-intersecting Brownian motions.","key_machinery":"Fisher-Hartwig asymptotics for determinants with root and jump singularities, applied to the characteristic polynomial of the stationary Hermitian Ornstein-Uhlenbeck process. The central mechanism is a space-time extension of earlier static Riemann-Hilbert-based asymptotics, with error estimates that are uniform in time, allowing single-time estimates to be integrated into a bona fide two-dimensional Gaussian multiplicative chaos measure on an infinite strip.","core_discovery":"The central claim is a dynamical generalization of single-time Fisher-Hartwig asymptotics: under the law of the stationary Hermitian Ornstein-Uhlenbeck process, the characteristic polynomial exhibits root- and jump-type singularities whose asymptotic behavior holds uniformly in time. From these asymptotics, fractional powers of the absolute value of the characteristic polynomial, and of the associated exponential eigenvalue counting process, are shown to converge to a two-dimensional Gaussian multiplicative chaos measure on an infinite strip in the subcritical phase. The same dynamical asymptotics yield the leading-order Gaussian fluctuations of the log-characteristic polynomial and optimal","pith_inferences":["A likely next step is to prove that the same uniform-in-time error control holds for other integrable dynamical models, such as the sine-process in time, which would produce a wider class of Gaussian multiplicative chaos limits parameterized by equilibrium measures.","The method may imply that the subcritical phase threshold for the 2D GMC on the infinite strip matches the single-time threshold, so the critical temperature is unchanged by the time dimension.","One can test numerically whether the finite-N moments of the characteristic polynomial on a growing time interval match the predicted GMC moments; any systematic deviation would localize the failure of the uniform-time error estimates.","The connection to Liouville quantum gravity suggests that the stationary Hermitian Ornstein-Uhlenbeck process, or a discrete analog, could be used as a numerical sampler for quantum-gravity-like random geometries on strips."],"forward_implications":["If the space-time Fisher-Hartwig asymptotics hold as stated, the log-characteristic polynomial of the stationary Hermitian Ornstein-Uhlenbeck process has Gaussian leading-order fluctuations, extending known single-time results.","Fractional powers of the absolute value of the characteristic polynomial converge to a two-dimensional Gaussian multiplicative chaos measure on the infinite strip in the subcritical phase, giving a dynamical version of the static GMC limit.","Non-intersecting Brownian motions associated with the eigenvalue process enjoy optimal bulk rigidity, meaning the paths cannot fluctuate more than the predicted order in the bulk.","This supplies a second connection between random matrix theory and Liouville quantum gravity measures, after the recent single-time connection, by proving a dynamical generalization of that convergence.","The dynamical asymptotics also extend the known maximum and optimal rigidity results for the log-characteristic polynomial to the space-time setting."],"supporting_citations":[{"why":"Supplies the static root-type Fisher-Hartwig asymptotics that the paper extends to the space-time setting.","marker":"[Krasovsky 2007]"},{"why":"Provides the static jump-type Fisher-Hartwig asymptotics that the paper generalizes to dynamics.","marker":"[Its, Krasovsky 2008]"},{"why":"Gives a related static Fisher-Hartwig result that the dynamical asymptotics extend.","marker":"[Charlier 2019]"},{"why":"Establishes the first connection between random matrix theory and Liouville quantum gravity measures; the paper's GMC result is a dynamical extension of that connection.","marker":"[Bourgade, Falconet 2025]"},{"why":"Proves the single-time convergence of the characteristic polynomial to Gaussian multiplicative chaos, which is the static limit being generalized.","marker":"[Berestycki, Webb, Wong 2018]"},{"why":"Establishes the maximum of the log-characteristic polynomial in the static case, extended here to dynamics.","marker":"[Lambert, Paquette 2019]"},{"why":"Proves optimal bulk rigidity for the static eigenvalue process, extended here to the non-intersecting Brownian motions.","marker":"[Claeys, Fahs, Lambert, Webb 2021]"}],"fun_headline_variants":["Time-dependent Fisher-Hartwig leads to strip GMC","Brownian eigenvalue paths converge to a fractal strip","Dynamical asymptotics yield 2D Gaussian chaos","New space-time results for non-intersecting Brownian motions","Optimal rigidity and log fluctuations from dynamical FH"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The error terms in the space-time Fisher-Hartwig asymptotics are small enough uniformly in time to be integrated into a genuine two-dimensional Gaussian multiplicative chaos measure on the infinite strip.","fun_headline_variants_meta":{"raw":{"variants":["Time-dependent Fisher-Hartwig leads to strip GMC","Brownian eigenvalue paths converge to a fractal strip","Dynamical asymptotics yield 2D Gaussian chaos","New space-time results for non-intersecting Brownian motions","Optimal rigidity and log fluctuations from dynamical FH"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1219,"prompt_tokens":739,"completion_tokens":480,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":402}},"tokens_in":483,"tokens_out":480,"duration_ms":5617,"temperature":1.0,"reasoning_tokens":402,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:52:28.580192+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the ratio of the $k$-th moment of the absolute value of the characteristic polynomial with a root singularity to the leading Fisher-Hartwig term, taking the supremum over time in a fixed interval, while $N \\to \\infty$. If this ratio diverges for any fixed $k$ below the subcritical threshold, the asserted uniformity in time fails, and the infinite-strip GMC convergence would be false.","supporting_citations":[],"review_version":1}