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This paper proves that for Gaussian analytic functions with power-exponential weights e^{-|z|^β}, conditioning on the absence of zeros in a disk forces the scaled zero set to converge to a limiting measure with a forbidden annulus 1<|z|<e^{

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For every β>0, conditioning a power-exponential Gaussian analytic function to have no zeros in D(0,r) makes the scaled zero measure converge to a limit supported on {|z|=1}∪{|z|≥e^{1/β}}, avoiding {1<|z|<e^{1/β}}.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection A credible extension of Ghosh–Nishry with a checkable variational core, but the lower-bound proof uses a tail estimate outside its valid range and the main theorem is not established as written. the 2 major comments →

arxiv 2602.24193 v3 pith:PDIXOULN submitted 2026-02-27 math.CV math.PR

Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights

Yun-Heng Du This is my paper
classification math.CV math.PR MSC 30D2030C1560F10
keywords Gaussian analytic functionshole phenomenonpower-exponential weightsforbidden regionzero counting measurelogarithmic energylarge deviationshole probability
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes the hole phenomenon for the family of Gaussian analytic functions F_β(z)=Σ ξ_n z^n / sqrt(Γ((2/β)(n+1))), the random power series associated with the weight e^{-|z|^β} for every β>0. The main theorem says that if F_β has no zeros in the disk D(0,r), then as r→∞ the rescaled zero counting measure (1/r^β)[Z_r^β] converges vaguely in distribution to an explicit limit supported on the unit circle and on the exterior of the disk of radius e^{1/β}. The limit assigns zero mass to the annulus 1<|z|

Core claim

The central discovery is that the conditional zero distribution is asymptotically deterministic and contains a hole. Let [Z_r^β] be the zero counting measure of F_β(z/r) conditioned on the event {F_β has no zeros in D(0,r)}. The paper proves that (1/r^β)[Z_r^β] converges vaguely in distribution to μ_0^β = (βe/2)m_{S^1} + (β/2) bm_β|_{|z|≥e^{1/β}}, where m_{S^1} is the normalized uniform measure on the unit circle and bm_β is the measure with polar density (β/(2π))r^{β-1} dr dθ. Consequently the annulus {1<|z|<e^{1/β}} carries no mass in the limit, which is the forbidden region; its width varies with β. The proof proceeds through sharp exponential asymptotics for hole probabilities and a cond

What carries the argument

The engine of the proof is a logarithmic-energy variational principle with external field |z|^β. For each α,β one minimizes the functional I_{α,β}(μ)=2 sup_w (U_μ(w)-|w|^β/(βα))-Σ(μ), where U_μ is the logarithmic potential and Σ the logarithmic energy, over probability measures with a constraint on the mass inside the unit disk. Proposition 1.3 gives explicit minimizers μ^β_{α,p} — built from a uniform circle component plus radial shells of bm_β — obtained by the method of undetermined coefficients; rescaling by β/2 and sending α→∞ yields the limiting measure μ_0^β. In parallel, the infinite series defining F_β is truncated to a dominant random polynomial plus a negligible tail, and Rouché's

Load-bearing premise

The whole result depends on the Section 4 control of the high-degree tail T_N(z): the asserted bound |T_N(z)|≤exp(-2^{2β-1} e r^β) on |z|≤r is invoked to prove the lower bound on hole probabilities, yet the cited Lemma 2.1 requires α≥(4B)^β and the application uses α=4βe, which fails this condition for β>2; if that bound does not hold, Proposition 1.4, Lemma 1.5, and Theorem 1.1 all lose support.

What would settle it

Verify the Section 4 tail estimate: for β=3, B=1, α=12e, check whether Lemma 2.1's argument actually yields |T_N(z)|≤exp(-2^{5} e r^3) on |z|≤r. Since α=4βe<4^β for β>2, the lemma's hypothesis is not met; either supply a direct proof of the asserted bound or exhibit a configuration of Gaussian coefficients for which the tail exceeds it with probability larger than exp(-C r^{3β}). If no such bound holds, the lower-bound half of Proposition 1.4 collapses, and with it the main theorem as proved.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For every β>0, the conditional zero measure has the same two-piece support: a circle component at |z|=1 and an exterior component beyond |z|=e^{1/β}, with no mass in between.
  • The hole probability P[n_{F_β}(r)=0] decays as exp(-(βe^2/8)r^{2β}+O(r^β log^2 r)); setting β=2 recovers the known Gaussian entire function asymptotics.
  • The forbidden region shrinks to the empty set as β→∞ and expands to the exterior of the closed unit disk as β→0+, so the phenomenon persists across the whole family.
  • The conditional large-deviation estimate of Lemma 1.5 controls fluctuations of linear statistics under the zero-count constraint with Gaussian-type tail decay in λ^2, and is the step that upgrades convergence of counts to convergence of the full scaled measure.
  • Corollary 1.2 gives a distributional limit for the conditioned scaled zero sets, so smooth compactly supported statistics converge in distribution to integrals against μ_0^β.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unproved high-degree tail bound in Section 4 deserves scrutiny before the lower-bound half of Proposition 1.4 is accepted for β>2: the text sets α=4βe and asserts |T_N(z)|≤exp(-2^{2β-1}e r^β), but Lemma 2.1's stated hypothesis α≥4^β is violated for β>2 and no separate proof is given.
  • If the tail estimate can be repaired, the same variational mechanism should extend to any radially symmetric weight Q for which the associated energy functional has the two-shell minimizer structure; this is the natural reading of the paper's Question 1.6 and could be tested on weights such as (1+|z|^2)^s.
  • The β→∞ limit, where the annulus collapses, invites a separate high-β analysis: conditioning on an empty disk may force zeros onto the unit circle with a thin exterior cloud, a transition the present paper does not address and which numerical experiments could probe.
  • For simply connected domains other than disks, the picture from neighbouring results suggests the forbidden region should become the quadrature domain of the limiting measure; verifying that here would connect the annulus to domain geometry.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies Gaussian analytic functions F_β(z)=Σ ξ_n z^n / sqrt(Γ(2/β(n+1))) associated with the radial weight e^{-|z|^β}, β>0. The main theorem (Theorem 1.1) asserts that, conditionally on having no zeros in D(0,r), the expected linear statistics of the scaled zeros admit the asymptotic r^β ∫ φ dμ_0^β + O(r^{β/2} log^2 r), where μ_0^β = (βe/2)m_{S^1} + (β/2) bm_β restricted to |z|≥e^{1/β}; equivalently, the scaled zero measure converges vaguely in distribution to μ_0^β (Corollary 1.2), which has a forbidden annulus 1<|z|<e^{1/β}. The proof strategy follows Ghosh–Nishry: truncate the series at degree N, relate zero counts of F_β to those of the polynomial P_{N,L}, prove large-deviation estimates for hole probabilities by minimizing an energy functional I_{α,β}, and then transfer the estimates to the conditional distribution. Proposition 1.3 gives explicit variational minimizers; Proposition 1.4 gives sharp hole probabilities; Lemma 1.5 gives a conditional large-deviation bound for linear statistics; these are assembled to prove Theorem 1.1 and Corollary 1.2.

Significance. If the proof were complete, the result would be a substantial and natural generalization of the Ghosh–Nishry hole phenomenon from the Gaussian entire function (β=2) to the entire family of power-exponential weights. The explicit limiting measure, the variational characterization, and the geometric description of the forbidden annulus as a function of β are attractive and likely to be influential. The paper is also commendable for its transparent structure and for avoiding fitted constants: the limiting measure comes from a genuine variational problem. However, two load-bearing points are not established as written: the high-degree tail estimate in the lower-bound proof of Proposition 1.4 (Section 4) and the energy-gap inequality Claim 5.1 in the proof of Lemma 1.5. Without these, the main theorem is not proven in the current text, although the gaps appear repairable.

major comments (2)
  1. [Section 4, 'Control of the high-degree tail (k>N)'] The proof applies Lemma 2.1 with B=1 and α=4βe, but Lemma 2.1 requires α≥(4B)^β=4^β. This fails for β>2 (e.g., for β=3, 4βe≈32.6<64). Thus the claimed bound |T_N(z)|≤exp(−2^{2β−1}er^β) is not justified. Moreover, even if the lemma's hypothesis were met, its conclusion with α=4βe would give |T_N|≤exp(−2βe(1+logβ)r^β), not the displayed exponent; and for any admissible α the exponent would be −(α/2)log(α/4)r^β, not −2^{2β−1}er^β. This tail estimate is explicitly used to ensure |T_N|<(1/4)|ξ_{k0}|b_{k0} in the Rouché step. Without it, the lower bound in Proposition 1.4 is unproved, and the chain Proposition 1.4 → Lemma 1.5 → Theorem 1.1 collapses. The gap is plausibly repairable by choosing α=4^β e or by proving a direct tail estimate, but the present text contains no such repair.
  2. [Section 5, Claim 5.1] Equation (5.1) states the quantitative energy-gap inequality I_{α,β}(ν)−I_{α,β}(μ_{α,p}^β) ≥ λ^2 for all ν∈L_{φ,τ,λ} with ν(D)≤p/α. This is asserted to be 'a restatement of [12, Claim 7.4] adapted to our setting', but no proof or detailed verification is given. The functional I_{α,β}, the constraint set, and the minimizer μ_{α,p}^β all depend on β, and the quadratic gap in λ is not a formal consequence of the β=2 argument without additional work. This inequality is load-bearing: it produces the factor exp(−(2π/D(φ))κ′^2) in the proof of Lemma 1.5, which is the main tool for Theorem 1.1. The author should either supply a proof of Claim 5.1 or provide a precise reference to a result that covers the β-family.
minor comments (5)
  1. [Section 3, near 'The upper bounds for parts (2) and (3)'] The text refers to 'parts (2) and (3) of Proposition 1.3' where the context (hole probabilities) indicates Proposition 1.4. Please correct the cross-reference.
  2. [Section 1, introduction] The name 'Zeltoni-Zelditch' should be 'Zeitouni-Zelditch'.
  3. [Section 3, proof of Proposition 1.3] The minimizer is introduced with 'we conjecture that the minimizer is...' even though the subsequent verification proves the claim. The wording 'we claim' or 'we set' would be more accurate.
  4. [Section 2, Lemma 2.8 and Section 3] In Section 3, Lemma 2.8 is used with α∈[log r,2log r]. This is compatible with Lemma 2.1's condition α≥(4B)^β only for sufficiently large r; the text should state this explicitly, since the condition is a hypothesis of the lemma being invoked.
  5. [Throughout] The notation for linear statistics appears both as n_{F_β}(φ;r) and n_F(φ;r); unify the notation for readability.

Circularity Check

0 steps flagged

No significant circularity: the limiting measure is obtained from an independently solved variational problem, not from tuning or self-citation.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 1.1 is built on Proposition 1.4 (sharp hole probabilities) and Lemma 1.5 (conditional large deviations for linear statistics). Proposition 1.4's upper bound works by truncating the Gaussian series to a random polynomial, writing its joint zero density from the i.i.d. Gaussian coefficients and the explicit Jacobian, then bounding the resulting integral by minimizing the logarithmic energy functional I_{α,β}. The minimizer in Proposition 1.3 is introduced as an ansatz guided by the β=2 case, but the paper then verifies it directly: it computes U_{μ^β_{α,p}} explicitly, shows the required equality on the support, and invokes the classical comparison lemma (Zeitouni–Zelditch Lemma 29) to conclude minimality. It is not assumed to be the minimizer in order to obtain the limiting measure; the computation stands independently. The limiting measure μ_0^β is then obtained by taking β/2 μ^β_{α,0} as α→∞, again independent of the conditional expectation it later predicts. The lower bound in Section 4 uses Rouché's theorem with a dominant monomial term and direct Gaussian small-ball estimates; no parameter is fitted to the target expectation. Lemma 1.5 and the final proof of Theorem 1.1 use these estimates plus standard large-deviation machinery. Citations to Ghosh–Nishry and others supply techniques and known lemmas, not the target result, and there is no self-citation load-bearing chain: the present author is not among the authors of [12]. The one notable issue in Section 4—applying Lemma 2.1 with α=4βe when Lemma 2.1 requires α≥(4B)^β=4^β, which fails for β>2—is a correctness gap in the tail control as written, not a circularity: it does not make any derived quantity equivalent to its input by construction. Accordingly the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

No fitted parameters or invented entities; the central claim rests on a chain of standard theorems in potential theory and random analytic functions, plus several quoted lemmas from Ghosh–Nishry and Krishnapur whose β-adaptation is not always demonstrated.

axioms (9)
  • standard math Stirling's formula bounds (2.1) for Γ(2/β(k+1))
    Used throughout to bound coefficients and the A_{N,L} normalization in Lemma 3.1.
  • domain assumption Lemma 2.5: P(−∫ log|F_β(re^{iθ})| dθ/2π ≥ u) ≤ exp(−exp(Cε u)) + P(log M_{F_β}(ε) ≤ −Bε u + √u), quoted from the proof of [17, Lemma 7]; no proof given.
    Used in Lemma 2.4 to bound overcrowding probability n_{F_β}(r)>r^{2β}; the adaptation to β is asserted, not proved.
  • standard math Rosenbloom's theorem (Theorem 2.7) lower bound for minimum of |f| on a punctured disk
    Used in Lemma 2.9 to compare F_β with its polynomial truncation; cited [18], accepted theorem.
  • standard math Lemma 2.10 ([12, Lemma 3.8]) comparing zero counts of f and f+g when |g|<min|f| on a domain
    Used in Lemma 2.9 to relate zeros of F_β to zeros of P_N; cited from prior work.
  • standard math Saff–Totik Theorem I.4.1 (Theorem 3.4) identifying the external-field sup over |z|≤α^{1/β}
    Used in Claim 3.3 to localize the energy B_{α,β}; cited [16].
  • standard math Lemma 3.5 ([9, Lemma 29]) sufficient condition for minimizers of I_{α,β} in terms of g_μ
    Key comparison principle used to verify the claimed minimizers of Proposition 1.3.
  • domain assumption Existence/uniqueness and radial symmetrization inequality for I_{α,β}: strict convexity and lower semicontinuity (cited to [19]), plus I(μ_rad)≤I(μ) asserted in text
    Used to reduce the variational problem to radial measures and to guarantee a unique minimizer; the symmetrization inequality is not proved in the paper.
  • ad hoc to paper Claim 5.1: energy gap inequality I_{α,β}(ν)−I_{α,β}(μ_{α,p}^β) ≥ λ^2 for ν∈L_{φ,τ,λ} with ν(D)≤p/α, stated as a restatement of [12, Claim 7.4]
    Central large-deviation input for Lemma 1.5; no proof of the β-adaptation is provided.
  • standard math Fact ([12], Claim 4.7): (1/N^2)∑_{j≠k} log|z_j−z_k| ≤ Σ(μ_t^z) − C log t / N
    Used to compare I*_β with I_{α,β}; quoted from [12].

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights." pith.science (2026). https://pith.science/paper/PDIXOULN

@misc{pith2026260224193,
  author       = {Pith},
  title        = {Pith review of: Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDIXOULN}},
  note         = {Machine review of arXiv:2602.24193}
}
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abstract

We establish the \emph{hole phenomenon} for the Gaussian analytic function \[ F_{\beta}(z)=\sum_{n=0}^{\infty}\frac{\xi_{n}}{\sqrt{\Gamma\bigl(\frac{2}{\beta}(n+1)\bigr)}}\,z^{n}, \] associated with the power-exponential weight $e^{-|z|^{\beta}}$ on $\mathbb{C}$, where $\beta>0$. Under the condition that $F_{\beta}(z)$ has no zeros in $D(0,r)$, the scaled zero counting measure converges to a limiting measure $\mu_{0}^{\beta}$ vaguely in distribution. This limit exhibits a \emph{forbidden region} \[ \bigl\{1<|z|<e^{1/\beta}\bigr\}, \] which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case $\beta=2$), who first revealed this striking conditional convergence and the emergence of a hole. Our analysis extends their phenomenon to the entire family of power-exponential weights.

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Reference graph

Works this paper leans on

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This paper was first reviewed by deepseek-v4-flash on August 2, 2026.