{"id":"893828d1-e7aa-42b0-8889-390bdafc24fb","arxiv_id":"2412.06086","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The conditional zero-count of a Gaussian power series is given by a new q-series, and the hole probability has a t-independent leading term that differs from the unconditional case.","lead":"This paper computes the exact distribution of the number of zeros in a disk for a random power series whose first coefficient is fixed. It corrects a prior conjecture about the probability of having no zeros and reveals new extreme-value statistics.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central formulas rely on the unproved process-level identification with [8]; if that identification is incomplete, the results describe the eigenvalue process rather than the conditional GAF zeros.","rationale":"The paper is a high-quality technical contribution: the q-series asymptotics, saddle-point arguments, and exact formulas are largely self-contained and impressive. The weakest point in the logical chain is the identification of the scaling limit of the eigenvalue point process of A_{n,τ} with the zero set of the conditional Gaussian power series. This identification is essential: without it, the central p.g.fl. (1.12) and MGF (1.13) are statements about the random matrix ensemble, not about the GAF zeros featured in the title and abstract. The reader correctly flagged that this identification is quoted from [8] and not reproved. A secondary technical gap is the interchange of limit and Mellin inversion in the proof of Theorem 2.1, which would also need rigorous justification, but the [8] dependence is more load-bearing because it connects the entire paper to its stated subject. The suggested concrete test—a direct derivation of the p.g.fl. from the GAF—would settle the concern. Since the concern is real but does not, on the present evidence, invalidate the central claims, the conditional verdict is appropriate and no change is needed.","tokens_in":41304,"tokens_out":9678,"duration_ms":94654,"concrete_test":"Independently derive the p.g.fl. of the conditional zero set directly from the GAF: take φ_a^{(N)}(z)=a+∑_{k=1}^N c_k z^k, condition on |a|²=t, compute the finite-N p.g.fl. or correlation functions explicitly, and pass N→∞. If the limit does not equal (1.12), the reliance on [8] is invalid. As a weaker check, prove tightness for the sequence of eigenvalue counting functions of A_{n,τ} uniformly in n and show that any subsequential limit has the p.g.fl. (2.3), so that the limiting point process is uniquely identified and matches the conditional zero set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central object is the zero set of the conditional Gaussian power series φ_a(z)=a+∑_{k≥1} c_k z^k. The key formulas (1.12), (1.13), and (2.20) are derived from Theorem 2.1, which computes the scaling limit of the p.g.fl. of absolute squares of eigenvalues of A_{n,τ} but does not itself identify that limit with the conditional zero set. The identification is imported from Forrester and Ipsen [8]. The paper's own description of [8] says it 'obtained the correlation functions of the zero set' after passing to the limit in the eigenvalue correlation functions. Correlation functions alone determine a point process only under additional tightness or boundedness conditions, and those conditions are neither stated nor proved here. If [8] did not establish process-level convergence, or if the limiting process is not uniquely determined by its correlation functions in this non-determinantal setting, then Theorems 1.1, 1.2, and all subsequent asymptotics describe the eigenvalue process rather than the zeros of φ_a. This is not an internal inconsistency, but a concrete correctness risk in the chain connecting the random matrix computation to the stated GAF results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the radial statistics of the zero set of the conditional Gaussian power series φ_a(z)=a+∑_{k≥1} c_k z^k with |a|²=t, focusing on the point process Q_a of absolute squares of zeros. The main results are: (i) an explicit q-series formula for the probability generating functional of Q_a, hence the full conditional distribution of the counting function N_q; (ii) asymptotic normality and large-deviation estimates for N_q in the limit q→1⁻; (iii) extreme-value limits of Fréchet and Gumbel type in the regimes a→0 and a→∞, respectively; and (iv) precise asymptotic forms for the probabilities P_k(t;q), including a hole probability whose leading term is independent of t>0 but differs from the t=0 case and matches the hyperbolic GAF. The proofs are carried out via the random matrix ensemble A_{n,τ}=U_n diag(√τ,1,…,1) in the scaling limit n→∞, nτ=t, using Mellin transforms, q-binomial identities, and saddle-point asymptotics for a new q-series.","tokens_in":41548,"tokens_out":15531,"duration_ms":142073,"significance":"The paper is potentially significant: it offers exact solvability for a non-determinantal point process, introduces a new q-series with a general asymptotic method, and establishes surprising hole-probability behavior that corrects a conjecture in [8]. If the central identification between the limiting eigenvalue process and the conditional GAF zero process is made rigorous, the results would substantially extend the Peres–Virág theory and the Forrester–Ipsen analysis. The asymptotic machinery for the q-series is creative and likely to be useful beyond this paper. However, the current version has a load-bearing gap in the process-level identification and several asymptotic proofs are only sketched; these issues must be resolved before the results can be accepted as proved theorems.","major_comments":[{"comment":"The proof asserts that the point process with p.g.fl. (2.3) is the conditional zero set of φ_a, with the identification imported from Forrester and Ipsen [8]. As the manuscript itself describes, [8] obtains correlation functions of the zero set by passing to the limit in the eigenvalue correlation functions. Convergence of correlation functions does not by itself imply weak convergence of point processes; one needs tightness and a uniqueness argument (e.g., moment determinacy with suitable growth bounds). The paper does not supply these, nor does it prove that the factorial moments of the process defined by (2.3) match those of the zero set. Since every subsequent statement about the conditional GAF zeros—Theorems 1.1, 1.2, 1.7, 1.9, 1.11, 1.13, and the marginals in Section 2.3—rests on this identification, this is a load-bearing gap. Please either prove directly that the p.g.fl. (2.3) is the p.g.fl. of the conditional zero process (for instance by comparing factorial moments computed from the correlation functions of [8] and checking moment determinacy), or establish weak convergence of the eigenvalue processes by a tightness argument and invoke a standard point-process convergence theorem.","section":"Section 2.1, Proof of Theorems 1.1 and 1.2"},{"comment":"The proof of Theorem 1.13 is presented as a standard saddle-point calculation but leaves several non-standard steps unjustified. The Euler–Maclaurin expansion of f(k,e^{-s/k};z) is stated with a uniform O(1/k) error only for complex z avoiding the cuts (-∞,0] and [e^s,∞); however, the steepest-descent contour necessarily approaches z=0 and the accumulating poles at z=e^s, and uniform control on the contour is not demonstrated. The resulting asymptotic (1.28) is claimed with relative error O(1/k), and this precision is later used to match with Theorem 1.10. A rigorous error analysis of the saddle-point contribution and of the contour deformation is needed.","section":"Section 5.3, Proof of Theorem 1.13"},{"comment":"The evaluation of the hole probability uses a saddle-point integral whose quadratic term vanishes and whose saddle point escapes to -∞. The rescaling around (4.48) and the subsequent estimates are described informally ('Evidently', 'it is plain to see'), and the control of the cubic and higher-order terms in the exponent—which contribute at the claimed order—is not written out. Since the leading-order independence of t is a central and surprising claim, the asymptotic argument should be made fully rigorous.","section":"Section 4.4, Proof of Theorem 1.11"}],"minor_comments":[{"comment":"The conclusion 'convergence in distribution' for N_n,τ(q) is drawn from convergence of the generating functions; please add the required tightness argument (for example, using the finiteness of E[(1+x)^{N_{n,τ}(q)}] for some x>0) to justify the step.","section":"Section 2.1, Corollary 2.2"},{"comment":"The interchange of the n→∞ limit with the Mellin inversion integral is not justified: the convergence in (2.11) is stated only uniformly on compact sets in ℜs>0, while the inversion is over an unbounded vertical contour. Please provide decay estimates for the integrand or deform the contour so that the limit can be taken under the integral.","section":"Section 2.1, Proof of Theorem 2.1"},{"comment":"The convergence in distribution of the m-th smallest absolute square of eigenvalues to that of the zeros is again attributed to [8]; this is part of the gap described in the first major comment and should be re-derived or stated as a precise theorem from [8] with a proof or reference.","section":"Section 2.3, Remark 2.5"},{"comment":"There are several typographical and formatting issues, such as 'F or' appearing at the beginnings of many paragraphs, which appear to be conversion artifacts; the authors should proofread the final version carefully.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The central doubt is the process-level identification with the conditional GAF zeros. If the authors can provide the missing tightness/moment-determinacy argument, the paper would be a strong contribution to mathematical probability and random matrix theory. The asymptotic computations are impressive but need full rigor in the saddle-point estimates, especially in Theorems 1.11 and 1.13. The paper fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The reader's h(1)=0 objection to Corollary 2.2 is a red herring: for q<1 the test function x·1_{[0,q]} does vanish at 1, so Theorem 2.1 applies directly. The real soft spot is the process-level identification.\n\nWhat is genuinely new: the conditional moment generating function (1.13), its q-series structure, and the saddle-point analysis that powers Theorems 1.9–1.13. The hole probability result (1.26) refuting the Forrester–Ipsen conjecture is surprising and convincing, and the consistency checks—averaging over φ(0) recovers the unconditional product—support the correctness of the algebra. The paper is honest about importing the identification with [8], and the random-matrix-side computations appear rigorous.\n\nThe main weakness is exactly that identification. The paper quotes [8] for the statement that the eigenvalue process of A_{n,τ} converges to the conditional GAF zero set, but does not reprove it or even state clearly what notion of convergence [8] established. If [8] only gives correlation functions, then in this non-determinantal setting one would need an extra argument to identify the limiting p.g.fl. with that of the zeros. That is a genuine gap. It does not invalidate the random matrix results, but it leaves the interpretation of the main theorems as statements about GAF zeros relying on an external citation.\n\nThe proof of Theorem 1.13 is also more sketched than the rest: the saddle-point computation is plausible, but the uniformity claims are asserted rather than proved. That is worth fixing but is not fatal.\n\nWho is this for? People working on Gaussian analytic functions, random matrix outliers, and q-series asymptotics. It is a substantial contribution and deserves a serious referee. I would send it to a probability or mathematical physics journal, and ask the authors to either prove or carefully state the convergence assumption from [8], and to expand the proof of Theorem 1.13.","headline":"Genuinely new exact results for conditioned GAF zeros, with a real but manageable gap in the process-level identification from [8]; deserves serious peer review.","tokens_in":42070,"tokens_out":4917,"would_cite":true,"duration_ms":49270,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","60G55","60F05","33D15","30C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The zeros inside a disk of a Gaussian power series conditioned on its value at the origin are controlled exactly by a q-series moment generating function, which yields asymptotic normality and a hole probability that is independent of the…","keywords":["Gaussian analytic functions","zeros of random power series","radial zeros statistics","hole probability","q-series","random sub-unitary matrices","Lambert-W function","large deviations"],"falsifier":"Fix $t>0$ and compute $P_0(t;e^{-\\varepsilon})$ for very small $\\varepsilon=-\\log q$, either by high-precision summation of the series (1.14) or by direct simulation of the conditioned Gaussian power series. Theorem 1.11 predicts $$\\log P_0(t;$e^{{-\\varepsilon}}$)=-\\frac{1}{\\varepsilon}\\left(\\frac{$w^{2}$}{2}+w+\\frac{\\$pi^{2}$}{3}\\right)-\\frac12\\log(\\varepsilon t)+t-\\frac{t}{w}+O(\\varepsilon),\\quad w=W_{-1}(-\\varepsilon t).$$ A stable departure from the $-\\frac12\\log\\varepsilon$ term, or any leading-order dependence on $t$, would refute the central asymptotics.","tokens_in":41118,"feed_emoji":"🧮","tokens_out":10623,"duration_ms":93459,"temperature":0.7,"pith_summary":"This paper studies the zeros inside a disk of the Gaussian power series $\\varphi(z)=\\sum_{k\\ge 0} c_k z^k$ with independent standard complex normal coefficients, conditioned on $|\\varphi(0)|^2=t$. The central result is an exact $q$-series formula for the probability generating function of $N_q$, the number of zeros in the disk of radius $\\sqrt{q}$, and a full asymptotic description of this distribution as $q\\to 1^{-}$, the limit that sees the entire zero set. The authors prove that $N_q$ becomes asymptotically normal with mean $\\sim(1-q)^{-1}$ and variance $\\sim(1-q)^{-1}/2$, and they compute moderate and large deviation rates in that limit. They also determine the conditional hole probability $P_0(t;q)=\\Pr\\{N_q=0\\mid|\\varphi(0)|^2=t\\}$: to leading order it is independent of $t>0$, yet different from the $t=0$ case, and it coincides with the hole probability of the hyperbolic Gaussian analytic function with coefficients $\\sqrt{k+1}\\,c_k$. A careful reader would care because the conditional zero process is not determinantal and not a product of independent variables, and the paper develops an asymptotic method for a new class of $q$-series to handle it.","feed_headline":"A single q-series sets the law for conditioned Gaussian zeros","feed_subtitle":"Zeros counts are asymptotically normal, and the hole probability is independent of t > 0 yet different from the unconditioned case.","key_machinery":"The load-bearing object is the conditional moment generating function $F(x,t;q)=\\sum_{k=0}^{\\infty}(-x)^k q^{k(k-1)/2}(q;q)_k^{-1}e^{t(1-q^{-k})}$, a $q$-series that does not factor into a product. All later results flow from its contour-integral representation (4.9),\n$$F(x,t;q)=\\frac{(q;q)_\\infty}{2\\pi i}\\oint_C \\frac{$e^{{-\\log x\\,\\log z/\\log q}}$}{(z;q)_\\infty}\\,$e^{{t(1-z)}}$\\,dz,$$\nwith $C$ a Hankel contour around the positive real axis, which reduces the $q\\to 1^{-}$ asymptotics to a saddle-point problem. The saddle-point equation is solved in terms of the two real branches $W_0$ and $W_{-1}$ of the Lambert-$W$ function. On the random-matrix side, the joint eigenvalue density of the sub-unitary matrices $A_{n,\\tau}=U_n\\,\\mathrm{diag}(\\sqrt{\\tau},1,\\dots,1)$ carries a delta function enforcing the product conservation law $|\\det A_{n,\\tau}|^2=\\tau$, and the Mellin transform of that density, combined with the $q$-binomial theorem, produces the $q$-series.","core_discovery":"On the paper's own terms, the discovery is that conditioning the Gaussian power series $\\varphi(z)=\\sum_{k\\ge 0} c_k z^k$ on $|\\varphi(0)|^2=t$ leaves the radial zero statistics exactly encoded by the $q$-series\n$$\\mathbb{E}\\{(1+x)^{N_q}\\mid |\\varphi(0)|^2=t\\}=\\sum_{m=0}^{\\infty} x^m\\frac{$q^{{m(m-1)/2}}$}{(q;q)_m}\\,$e^{{t(1-q^{-m}}$)},$$\nwhere $(q;q)_m=(1-q)(1-q^2)\\cdots(1-q^m)$. From this identity the paper derives the asymptotic normality of $N_q$ as $q\\to 1^{-}$, precise moderate and large deviation tail estimates, and the hole-probability asymptotics (Theorem 1.11), whose leading term is independent of $t>0$ but differs from the $t=0$ value and matches the hyperbolic GAF with coefficients $\\sqrt{k+1}\\,c_k$. It further shows that for every fixed $k\\ge 1$, $P_k(t;q)=e^t P_k(0;q)$ asymptotically as $q\\to 1^{-}$, with $P_k(0;q)=\\Pr\\{N_q=k-1\\}$ in the unconditional process. The same machinery yields extreme-value limits: as the conditioning value tends to $0$, the smallest modulus follows a Fr\\'echet law after scaling, while for large conditioning values the rescaled point process of absolute squares converges to a Poisson process.","pith_inferences":["The $q$-series (1.13) is arguably a new special function; the same Hankel-contour and Lambert-$W$ saddle-point method should extend to other rank-one multiplicative or additive deformations of unitary or Hermitian ensembles, where Mellin transforms of determinants produce similar $e^{t(1-q^{-k})}$ factors.","The discontinuity at $t=0$—hole probability independent of $t>0$ but different at exactly $t=0$—suggests a phase transition in the zero process under conditioning; one could look for a non-commuting double limit $t\\to 0$, $q\\to 1$ in (1.26).","The relation $P_k(t;q)\\sim e^t P_{k-1}(q)$ hints at an asymptotic stochastic picture: conditioning on $|\\varphi(0)|^2=t$ effectively inserts one zero near the origin and reweights the whole configuration by $e^t$; making this precise might yield a Gibbsian or Palm-type description of the conditional zero set.","Because the same sub-unitary ensemble models resonance widths in open chaotic maps, the Fr\\'echet and Poisson extremal laws derived here give concrete predictions for the distribution of the smallest and largest resonance widths in the one-channel scaling limit."],"forward_implications":["As $q\\to 1^-$, the conditional count $N_q$ is asymptotically normal: after centring by $(1-q)^{-1}$ and scaling by $\\sqrt{(1-q)/2}$, its law tends to a standard Gaussian, independently of the conditioning value $t$.","The hole probability for fixed $t>0$ satisfies Theorem 1.11, so the leading exponential rate is $-\\varepsilon^{-1}(w^2/2+w+\\pi^2/3)$ with $w=W_{-1}(-\\varepsilon t)$; this makes the leading order independent of $t>0$ but distinct from the $t=0$ case and equal to the hyperbolic GAF hole probability.","For fixed $k\\ge 1$, $P_k(t;q)=e^t P_k(0;q)(1+o(1))$ as $q\\to 1^-$, so conditioning multiplies each fixed-count probability by $e^t$ and shifts the index by one.","In the tail regime $k\\to\\infty$, $q=e^{-s/k}$, the probabilities (1.28)--(1.29) give a rate function $\\Psi(s)$ and pre-factor $A(s)$, refining the large-deviation results and matching the moderate-deviation scale.","The extreme smallest moduli obey Fr\\'echet and Poisson limits in the two opposite conditioning regimes ($a\\to 0$ and $a\\to\\infty$), as stated in Theorem 1.7 and Theorem 3.5."],"supporting_citations":[{"why":"Establishes the determinantal structure of the unconditional zero set and the product formula (1.8) for its moment generating function, which the paper generalizes.","marker":"[5]"},{"why":"Supplies the joint eigenvalue density for truncated unitary matrices, a starting point for the Mellin-transform computation of the eigenvalue process.","marker":"[7]"},{"why":"Provides the process-level convergence of eigenvalues of $A_{n,\\tau}$ in the scaling limit $n\\to\\infty$, $n\\tau=t$ to the conditioned zero set; this identification is the bridge from random matrices to zeros.","marker":"[8]"},{"why":"Introduces the ensemble $A_{n,\\tau}$ and gives its closed-form joint eigenvalue density and correlation functions, which are used to derive the limiting p.g.fl.","marker":"[9]"},{"why":"Gives the Hankel-contour integral representation for q-series and the uniform estimate for $\\log(x;q)_\\infty$ that underpin the saddle-point analysis.","marker":"[48]"},{"why":"Determines the hole-probability asymptotics for the family of hyperbolic Gaussian analytic functions, the comparison result that Theorem 1.11 matches.","marker":"[29]"},{"why":"Provides the large-deviation rate function for the unconditional zero count, which the conditional-rate results reproduce and refine.","marker":"[26]"}],"fun_headline_variants":["Conditioned Gaussian zeros obey a single q-series law","Zero counts in GAFs: one q-series rules them all","Hole probability for conditioned GAF zeros: t-independent","A q-series unlocks conditioned Gaussian zero statistics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the paper's reliance on the already-established process-level convergence of the eigenvalues of the random sub-unitary matrix $A_{n,\\tau}$ to the zero set of the conditioned Gaussian power series in the scaling limit $n\\to\\infty$, $n\\tau=t$; all the q-series formulas describe the zeros only if this convergence holds.","fun_headline_variants_meta":{"raw":{"variants":["Conditioned Gaussian zeros obey a single q-series law","Zero counts in GAFs: one q-series rules them all","Hole probability for conditioned GAF zeros: t-independent","A q-series unlocks conditioned Gaussian zero statistics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000549,"raw_usage":{"total_tokens":2753,"prompt_tokens":1210,"completion_tokens":1543,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":826,"completion_tokens_details":{"reasoning_tokens":1478}},"tokens_in":826,"tokens_out":1543,"duration_ms":10400,"temperature":1.0,"reasoning_tokens":1478,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:00:57.911288+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix $t>0$ and compute $P_0(t;e^{-\\varepsilon})$ for very small $\\varepsilon=-\\log q$, either by high-precision summation of the series (1.14) or by direct simulation of the conditioned Gaussian power series. Theorem 1.11 predicts $$\\log P_0(t;$e^{{-\\varepsilon}}$)=-\\frac{1}{\\varepsilon}\\left(\\frac{$w^{2}$}{2}+w+\\frac{\\$pi^{2}$}{3}\\right)-\\frac12\\log(\\varepsilon t)+t-\\frac{t}{w}+O(\\varepsilon),\\quad w=W_{-1}(-\\varepsilon t).$$ A stable departure from the $-\\frac12\\log\\varepsilon$ term, or any leading-order dependence on $t$, would refute the central asymptotics.","supporting_citations":[{"cited_title":"Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process,","cited_arxiv_id":null,"evidence_quote":"Establishes the determinantal structure of the unconditional zero set and the product formula (1.8) for its moment generating function, which the paper generalizes."},{"cited_title":"Truncations of random unita ry matrices,","cited_arxiv_id":null,"evidence_quote":"Supplies the joint eigenvalue density for truncated unitary matrices, a starting point for the Mellin-transform computation of the eigenvalue process."},{"cited_title":"A generalisation of the r elation between zeros of the complex Kac polynomial and eigenvalues of truncated unitary matrices,","cited_arxiv_id":null,"evidence_quote":"Provides the process-level convergence of eigenvalues of $A_{n,\\tau}$ in the scaling limit $n\\to\\infty$, $n\\tau=t$ to the conditioned zero set; this identification is the bridge from random matrices to zeros."},{"cited_title":"Spectra of random matrices close to unit ary and scattering theory for discrete-time systems,","cited_arxiv_id":null,"evidence_quote":"Introduces the ensemble $A_{n,\\tau}$ and gives its closed-form joint eigenvalue density and correlation functions, which are used to derive the limiting p.g.fl."},{"cited_title":"Uniform q-series asymptotics for staircase polygons,","cited_arxiv_id":null,"evidence_quote":"Gives the Hankel-contour integral representation for q-series and the uniform estimate for $\\log(x;q)_\\infty$ that underpin the saddle-point analysis."},{"cited_title":"Hole prob ability for zeroes of Gaussian Taylor series with finite radii of convergence,","cited_arxiv_id":null,"evidence_quote":"Determines the hole-probability asymptotics for the family of hyperbolic Gaussian analytic functions, the comparison result that Theorem 1.11 matches."},{"cited_title":"Overcrowding for zeros of Hyperbolic Gaussian analytic functions","cited_arxiv_id":"2209.05854","evidence_quote":"Provides the large-deviation rate function for the unconditional zero count, which the conditional-rate results reproduce and refine."}],"review_version":1}