{"id":"40ccff5e-5df5-4c69-86fe-d741861c1989","arxiv_id":"2504.15083","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A survey of quantitative equidistribution of zeros of random holomorphic sections toward equilibrium currents, emphasizing pluripotential theory and Bergman kernels, with two theorem variants proved.","lead":"This paper is a survey of recent results, mostly by the authors themselves, on where the zeros of random polynomials and random holomorphic sections tend to cluster. It explains how pluripotential theory and Bergman kernels are used to prove that zeros equidistribute toward an equilibrium current, with explicit rates in higher dimensions.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.25 is stated only for smooth volume forms but is applied to μ = ρλ_K with ρ^{-λ} ∈ L^1 (assumption H2); the needed Bergman-kernel rate is neither proved nor cited, so the quantitative rates in Theorems 4.2, 4.4, 4.5 and 4.7 rest on an unstated extension.","rationale":"The survey is an exposition of the authors' published line of work [67], with the survey-specific contribution being the proofs of Theorems 4.2 and 4.6 as 'variants' of results in [67]. I read it in good faith: the underlying theorems are published in a top journal, so the realistic defect is an internal incompleteness, not incorrect mathematics. Tracing every quantitative statement in Section 4, all roads lead to the uniform rate (1/2k)log~B_k → φ_K with error O(log k/k) for the L^2 Bergman kernel function of the measure in force. The survey states this rate only for smooth volume forms (Theorem 3.25) and then, inside the proofs of Theorems 4.2 and 4.5, applies it (or an unproven equivalent) to the (H2) class μ = ρλ_K with ρ^{-λ} ∈ L^1. The citation to Theorem 3.24 in the proof of Theorem 4.2 cannot carry the step, since Theorem 3.24 concerns the sup-norm envelope φ_{K,k}, not the L^2 kernel; the quantitative bridge between the two is exactly Theorem 3.25. Proposition 3.14 gives only O(1/k) in one direction and o(1) in the other. This is the single most load-bearing point: the rates in Theorems 4.2, 4.4, 4.5 and 4.7 — the advertised content of the survey — collapse to unquantified convergence if the rate is unavailable. The concern is internal (the proofs do not follow from the stated theorems), not a disagreement with any external consensus. I concur with the reader's weakest_assumption and with the conditional verdict: the fix is to extend Theorem 3.25 to (H2) (the ingredients — Theorem 3.22's polynomial growth and Theorem 3.24's envelope rate — are present in the survey, making the extension plausible) or to cite an authoritative version; the concrete test settles which deficiency needs repair. The duplicated references ([21]/[22], [54]/[55], [82]/[83], [57]/[58], [87]/[88]) are real but cosmetic.","tokens_in":23859,"tokens_out":23595,"duration_ms":190592,"concrete_test":"Compare the survey's Theorem 3.25 with its cited source [67, Theorem 2.6]: if the published theorem already covers μ = ρ Leb_K with ρ^{-λ} ∈ L^1, then the applications in Sections 4.2 and 4.4 are backed by the literature and the gap is a mis-restatement to fix in revision; if [67] also requires a smooth volume form, request the authors to derive the (H2) extension in the text from Theorem 3.22 (polynomial growth) and Theorem 3.24 (envelope rate), or to relax the statements of Theorems 4.2, 4.4, 4.5 and 4.7. This single comparison is decisive because the survey's proofs cite no other result connecting ~B_k to φ_K with a rate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"All quantitative results in Section 4 pass through the uniform rate ||(1/2k)log~B_k − φ_K||_{C^0(X)} = O(log k/k) for the L^2 Bergman kernel function ~B_k of μ. The only stated theorem giving this rate is Theorem 3.25, whose hypothesis is: 'Let μ be a smooth volume form on K.' Theorems 4.2, 4.4, 4.5 and 4.7 instead assume (H2): μ = ρλ_K with ρ^{-λ} ∈ L^1(λ_K), a strictly larger class (ρ may vanish on subsets or be unbounded). Two concrete failures: (i) In the proof of Theorem 4.2 the line 'By (H2) and Theorem 3.24, we see that ||k^{-1} log ||p_k|| − V_{K,Q}||_{L∞(Cn)} = O(log k/k)' does not follow from the cited statement — Theorem 3.24 is a rate for the sup-norm envelope φ_{K,k}, not for the L^2 quantity k^{-1}log||p_k|| = (1/2k)log~B_k; bridging the two at a rate is precisely the content of Theorem 3.25. Proposition 3.14 bridges them only up to o(1) (plus O(1/k) in one direction), too weak for O(log k/k). (ii) In Section 4.4, Theorem 3.25 is invoked directly with μ satisfying (H2). Theorem 3.22 does supply polynomial growth sup_K B_k ≤ Ck^N under exactly the (H2) class, and this is probably the extra ingredient needed to extend Theorem 3.25; but the survey neither states nor proves that extension. Hence the rates of Theorems 4.2, 4.4, 4.5, 4.7 are not established by the results as stated in this manuscript. The claims themselves are published results from [67], so the gap is likely patchable rather than fatal; still, as written, the proofs are incomplete at a load-bearing junction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is a survey of recent results, largely due to the authors, on the asymptotic equidistribution of zeros of random holomorphic sections and random polynomials associated with Bernstein-Markov triples (K, Q, μ). It reviews Bergman kernel functions, extremal plurisubharmonic envelopes, Bernstein-Markov measures, and Hölder regularity results, then states quantitative expectation and large-deviation estimates for random zeros. The main advertised results are Theorem 4.2 (expectation of the random zero current with error O((C_k + log k)/k)), Theorem 4.4, Theorem 4.5 (polynomial-tail large deviation), and Theorem 4.7 (equidistribution along algebraic curves), all under a hypothesis (H2) that the measure μ is ρλ_K with ρ^{-λ} ∈ L^1(λ_K). The survey also proves several preparatory results and lists open problems.","tokens_in":24355,"tokens_out":7718,"duration_ms":61701,"significance":"If the quantitative results are correct, the survey is a useful and timely synthesis: it connects pluripotential theory, Bergman kernel asymptotics, and random polynomial theory, and it collects the recent progress in [67] with explicit statements and some proofs. The detailed examples (unit ball, interval, polydisk), the discussion of the relationship to the Shiffman–Zelditch results, and the set of open problems (scaling limits, large deviations with rate k^2, hole events) will be helpful for researchers entering the area. A notable strength is that several auxiliary results, such as Lemma 4.3 and the Bernstein–Markov inequality in Theorem 3.20, are proved in the text. The central quantitative theorems are, however, taken from the authors' own paper [67], and the survey's proofs of those theorems contain a gap: the Bergman kernel rate is applied to a measure class wider than the class covered by the stated theorem. The gap is local and likely repairable by invoking the polynomial-growth result (Theorem 3.22) or by citing the corresponding result in [67], but as it stands the proof of the advertised rates is incomplete.","major_comments":[{"comment":"The uniform estimate ||(1/2k) log ~B_k - φ_K||_{C^0(X)} = O(log k/k) is stated in Theorem 3.25 only for μ a smooth volume form on K, yet it is applied throughout Section 4 to measures satisfying (H2), i.e. μ = ρλ_K with ρ^{-λ} ∈ L^1(λ_K). This class is strictly larger than the class of smooth volume forms. The proof of Theorems 4.5 and 4.7 in §4.4 invokes Theorem 3.25 directly ('Theorem 3.25 implies ∫_L |ψ_k − V_{K,Q}| ω^m_{FS,n} ≲ log k/k'), and the proof of Theorem 4.2 requires the same rate for ~B_k under (H2). Theorem 3.22 supplies the needed polynomial upper bound sup_K B_k ≤ C k^{2n_K(λ+1)/(αλ)} under exactly the (H2) hypotheses, and the proof note after Theorem 3.25 indicates that this bound is the basis of the rate, so the extension is very plausible; nevertheless, as the manuscript stands, the quantitative rates in Theorems 4.2, 4.4, 4.5, 4.6 and 4.7 rest on an unstated extension of Theorem 3.25. The authors should either state and prove the extended Bergman-kernel rate for the (H2) class or restrict the hypotheses of the four theorems accordingly.","section":"§4.4 and §4.2; Theorem 3.25 vs. (H2)"},{"comment":"The line 'By (H2) and Theorem 3.24, we see that ||k^{-1} log ||p_k|| − V_{K,Q}||_{L∞(C^n)} = O(log k/k)' is not a consequence of the cited result. Theorem 3.24 (Proposition 3.24) gives the rate O(log k/k) for the sup-norm envelope φ_{K,k}, whereas k^{-1} log ||p_k|| equals (1/2k) log ~B_k, the L^2 Bergman kernel function. The bridge between these two quantities supplied by Proposition 3.14 is only pointwise convergence plus one-sided O(1/k) and subexponential bounds, which does not imply a two-sided O(log k/k) estimate. The proof therefore has a load-bearing gap at this step; it would be repaired by invoking the (appropriately extended) Theorem 3.25, but not by Theorem 3.24 as stated.","section":"§4.2, proof of Theorem 4.2"}],"minor_comments":[{"comment":"The reference list contains duplicate entries: [21] and [22] are both Bloom–Levenberg, 'Random polynomials and pluripotential-theoretic extremal functions'; [82] and [83] are both Shiffman–Zelditch, 'Equilibrium distribution of zeros of random polynomials'; and [54] and [55] are both Hammersley's paper. These duplicates should be removed and the in-text citations renumbered.","section":"References"},{"comment":"The sentence 'The crucial ingredient in the proof of Theorems 4.5 and 4.6 is a polynomial growth of Bergman kernel functions, see Theorem 3.25' is imprecise: the polynomial growth is Theorem 3.22, while Theorem 3.25 is the quantitative convergence of the normalized log Bergman kernel. The sentence should be corrected to point to the right theorem.","section":"§4.4, paragraph before Lemma 4.11"},{"comment":"In the proof of Theorem 4.5, the inequality (4.8) is applied to u := p(dk), but p(dk) is a vector of sections, not a fixed unit vector in C^{d_k}. The argument should specify that for each z one applies (4.8) to the normalized vector (s_j(z))/||(s_j(z))|| and then integrates.","section":"§4.4, proof of Theorem 4.5"},{"comment":"There are several typos and minor notational inconsistencies: 'quantiative' in Section 1; 'Berstein-Markov' in Theorem 3.10; 'Cauchy-Riema nn' in the statement of Theorem 3.22; and the notation 'LebC^m' versus 'LebC' is used without comment. These should be cleaned up.","section":"Throughout"},{"comment":"The statement of Theorem 4.7 is not self-contained: the algebraic curve L is introduced only in the sentences preceding the statement, and the definition of μ_L depends on that discussion. The statement should be reformulated so that L and μ_L are defined within the theorem or immediately before it.","section":"Theorem 4.7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially a survey of the authors' own paper [67], and the theorem numbering and proofs in Section 4 are drawn directly from that paper. The gap concerning the extension of Theorem 3.25 to the (H2) measure class is very likely fixable by citing the precise statement in [67] or by a short argument using Theorem 3.22, but the authors should be asked to address it explicitly. There is no indication of misconduct; the issue is a proof-completeness problem in the survey's presentation of its central quantitative results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth your time, with one caveat that matters. The survey does what a survey should: it organizes the recent quantitative equidistribution results for zeros of random holomorphic sections, mostly from the authors' own [67], and it frames them well around extremal plurisubharmonic functions and Bergman kernels. The expository parts are clear, the open problems are sensible, and the two theorem variants (4.2 and 4.6) are honestly labeled as such. This is a genuinely useful reference for anyone working at the interface of pluripotential theory and random polynomials.\n\nThe soft spot is real, and it sits at a load-bearing junction. Theorem 3.25, which gives the uniform rate |(1/2k)log ~B_k − φ_K| = O(log k/k), is stated only for μ a smooth volume form on K. But the proofs of Theorems 4.2, 4.4, 4.5, and 4.7 apply it to measures μ = ρλ_K with ρ^{−λ} ∈ L^1, the class in (H2). The survey never states or proves the needed extension. The proof of Theorem 4.2 also cites the wrong theorem: it says \"By (H2) and Theorem 3.24,\" but Theorem 3.24 is a rate for the sup-norm envelope φ_{K,k}, not for the L^2 Bergman kernel; getting the rate for the latter is precisely the content of Theorem 3.25. So the quantitative rates in Section 4 are not established by the results as written.\n\nThat said, this is a patchable gap rather than a fatal one. Theorem 3.22 supplies a polynomial upper bound for B_k under exactly the (H2) measure class, and the survey itself notes that the proof of Theorem 3.25 is based on that polynomial bound. In all likelihood the extension goes through with no new ideas. But as it stands the exposition is incomplete at a point where the reader needs it to be complete.\n\nMinor issue: the reference list is sloppy, with several exact duplicates ([21]=[22], [54]=[55], [57]=[58], [82]=[83], [87]=[88]). That's a day of cleanup, not a substantive flaw.\n\nBottom line: a competent, useful survey that deserves serious refereeing, but the authors should either prove the extended Bergman kernel asymptotic or state explicitly that it is a consequence of [67, Theorem 2.6] plus Theorem 3.22, and fix the mis-citation in the proof of Theorem 4.2. I would send it to referees.","headline":"Useful survey of quantitative equidistribution results from [67], but the proofs of The main theorems rely on an unstated extension of the Bergman kernel asymptotic beyond the stated hypothesis of Theorem 3.25.","tokens_in":24847,"tokens_out":3727,"would_cite":true,"duration_ms":30710,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32A60","32U05","60F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Zeros of random high-degree polynomials concentrate at the equilibrium current of a Bernstein–Markov triple, with a logarithmic error rate.","keywords":["random holomorphic sections","equidistribution of zeros","Bergman kernel function","extremal plurisubharmonic function","Bernstein-Markov measures","large deviations","pluripotential theory","random polynomials"],"falsifier":"Calculate, for a specific nondegenerate piecewise-smooth generic CR submanifold with a non-smooth admissible density, the sup-norm error $\\|\\frac{1}{2k}\\log \\widetilde B_k - V_{K,Q}\\|_{C^0(K)}$—for instance, $K$ the unit ball in $\\mathbb{C}^n$, $Q=0$, and $\\mu=(1-|z|^2)^{\\alpha}\\mathrm{Leb}_K$ with $\\alpha>-1$. If the decay is slower than $O(\\log k/k)$, Theorems 4.2, 4.4, 4.5, and 4.7 fail; if it holds, the missing link is supplied.","tokens_in":23622,"feed_emoji":"🎯","tokens_out":9750,"duration_ms":81472,"temperature":0.7,"pith_summary":"This survey lays out a program in which the zeros of random holomorphic polynomials—and, more generally, random holomorphic sections of high powers of a positive line bundle—are controlled by an equilibrium object computed from the set $K$ that carries the coefficients and a weight $Q$. The central quantitative claim is that for coefficient densities decaying at least like $(1+|z|)^{-3}$, and for $K$ a nondegenerate piecewise-smooth generic CR submanifold, the expected normalized zero divisor equals $dd^c V_{K,Q}$ with error $O(\\log k/k)$, while the random zero divisor satisfies large deviation estimates with polynomial tails. All of this flows through the Bergman (Christoffel–Darboux) kernel, whose logarithmic asymptotics must be uniform enough to replace the random log-modulus by the deterministic envelope $V_{K,Q}$. The survey also proves variants for zero intersections with algebraic curves and a subexponential large deviation bound, and it identifies the higher-dimensional correlation-scaling and hole-event problems as open.","feed_headline":"Random polynomial zeros follow equilibrium to within log k/k","feed_subtitle":"A survey derives expectation and large-deviation rates for random zero divisors over piecewise-smooth CR submanifolds.","key_machinery":"The load-bearing object is the Bergman (Christoffel–Darboux) kernel function $B_k(x)=\\sum_j |s_j(x)|^2 e^{-2kQ(x)}$ on $K$, together with its unweighted extension $\\widetilde B_k(x)=\\sup_{s\\in P_k} |s(x)|^2/\\|s\\|_{L^2(\\mu,kQ)}^2$ on all of $\\mathbb{C}^n$; this is the squared projection kernel of the space of polynomials of degree at most $k$ restricted to $K$, and its inverse is the Christoffel function of orthogonal-polynomial theory. The argument uses the uniform asymptotic $\\frac{1}{2k}\\log \\widetilde B_k \\to V_{K,Q}$—with quantitative rate $O(\\log k/k)$ for piecewise-smooth generic CR submanifolds—to show that $\\frac{1}{k}\\log|p_k|$ is, outside a small exceptional set of coefficients, uniformly close to the deterministic envelope $V_{K,Q}$. The envelope $V_{K,Q}$ is the upper envelope of plurisubharmonic functions bounded by $Q$ on $K$; its Monge–Ampère current $dd^c V_{K,Q}$ is precisely the equilibrium measure toward which the zeros converge.","core_discovery":"On its own terms, the paper's central discovery is that the one-dimensional equidistribution results for Kac polynomials and for Shiffman–Zelditch random orthogonal polynomials generalize to higher dimensions not merely as convergence but with a rate. For a Bernstein–Markov triple $(K,Q,\\mu)$ with $K$ a nondegenerate piecewise-smooth generic CR submanifold of $\\mathbb{C}^n$, $Q$ Hölder continuous, and $\\mu = \\rho\\,\\mathrm{Leb}_K$ with $\\rho^{-\\lambda}\\in L^1$, the random polynomial $p_k=\\sum a_{kj}s_j$ built from an $L^2(\\mu,kQ)$-orthonormal basis has zero divisor $[p_k=0]$ whose expectation satisfies $\\mathbb{E}_k(k^{-1}[p_k=0]) = dd^c V_{K,Q} + O((C_k+\\log k)/k)$; when the coefficient density satisfies $|f(z)|\\le C(1+|z|)^{-3}$, the error is $O(\\log k/k)$. Under the same hypotheses, the large deviation estimate holds: except on a set of coefficient vectors of probability at most $C_M k^{-M}$, the distance between $k^{-1}[p_k=0]$ and $dd^c V_{K,Q}$ is at most $C_M \\log k/k$, and a subexponential version bounds deviations by $C_\\varepsilon e^{-A\\varepsilon k}$.","pith_inferences":["If the missing Bergman-kernel rate for $\\mu=\\rho\\,\\mathrm{Leb}_K$ with $\\rho^{-\\lambda}\\in L^1$ is supplied, the $O(\\log k/k)$ expectation bound should extend to coefficient distributions satisfying only the tail condition (4.7), since the proof of Theorem 4.2 uses (H1) only through $C_k=\\log k$.","The sharp one-dimensional rate $O(k^{-1})$ and the $k^2$ large-deviation exponent suggest the $O(\\log k/k)$ rates here are likely not optimal in higher dimension; a concrete test is to compute the exact Bergman-kernel error for the unit ball and polydisk examples.","The curve-intersection theorem suggests a route to correlation scaling limits in higher dimensions: combine the Bergman-kernel asymptotics along $L$ with the Kac–Rice formula used in dimension one, which would attack Problem 4.12 directly.","For numerical polynomial system solving, the logarithmic rate means random high-degree polynomials have zeros effectively pinned to the equilibrium current; that could serve as a sampling or preconditioning principle, although the paper itself does not discuss applications."],"forward_implications":["For any nondegenerate piecewise-smooth generic CR submanifold—polygon boundaries in $\\mathbb{C}$, real polyhedra in $\\mathbb{C}^n\\supset \\mathbb{R}^n$—the expected zero distribution of Gaussian random polynomials is equidistributed to equilibrium with error $O(\\log k/k)$.","The large deviation estimate gives control of the full random current, not just its expectation: with probability at least $1-C_M k^{-M}$, the normalized zero divisor is within $C_M\\log k/k$ of the equilibrium current in the negative Sobolev metric.","When the zero set is intersected with an algebraic curve $L$, the resulting random point measure equidistributes to the slice $\\mu_L=(\\deg L)^{-1} dd^c V_{K,Q}\\wedge [L]$ almost surely, with polynomial-tail large deviation control.","The subexponential bound $\\mu_k(\\mathrm{dist}_{-2}(k^{-1}[p_k=0],dd^cV_{K,Q})\\ge\\varepsilon)\\le C_\\varepsilon e^{-A\\varepsilon k}$ holds for deviations of order one, showing concentration is exponentially fast for fixed $\\varepsilon$.","The examples of the unit ball, $[-1,1]^n$, and the unit polydisk satisfy the hypotheses, so the rates apply to explicit classical models."],"supporting_citations":[{"why":"Supplies the Bergman-kernel asymptotics and the quantitative equidistribution and large-deviation theorems the survey presents and extends.","marker":"[67]"},{"why":"Gives the one-dimensional equilibrium-distribution theorem for Gaussian random polynomials that the survey generalizes to higher dimensions.","marker":"[83]"},{"why":"Establishes almost sure equidistribution of zeros toward $dd^c V_{K,Q}$ under the Bernstein–Markov property, the qualitative baseline the survey quantifies.","marker":"[22]"},{"why":"Introduces the moment condition (H1) and supplies the Gaussian-coefficient examples plus the inequality (4.8) used in the large-deviation proof.","marker":"[6]"},{"why":"Provides the polynomial growth of Bergman kernels and quantitative envelope convergence for the case $K=X$, which motivates the rate strategy.","marker":"[41]"},{"why":"Supplies the $(C^{0,\\alpha},C^{0,\\alpha'})$-regularity of extremal psh envelopes and Fekete-point equidistribution speed used in Sections 3.2 and 3.4.","marker":"[42]"},{"why":"Gives the analytic-disc construction and Hölder regularity of envelopes for generic submanifolds that underpin Theorem 3.16 and the Bernstein–Markov theorem for piecewise-smooth sets.","marker":"[91]"},{"why":"Provides the one-dimensional large-deviation result with exponent $k^2$ that serves as the benchmark for the survey's open Problem 4.13.","marker":"[96]"}],"fun_headline_variants":["Zero equidistribution rates for random sections on CR manifolds","Random zeros on CR submanifolds reach equilibrium at log k/k","Explicit rate for random zero equidistribution in higher dimensions","Survey: zeros of random sections follow equilibrium with log k/k error"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that the log-Bergman-kernel asymptotic, proved for smooth volume measures, also holds with rate $O(\\log k/k)$ for the weighted measures $\\mu=\\rho\\,\\mathrm{Leb}_K$ with $\\rho^{-\\lambda}\\in L^1$ that condition (H2) allows; the survey quotes the smooth case only, so the quantitative claims rest on this unproved extension.","fun_headline_variants_meta":{"raw":{"variants":["Zero equidistribution rates for random sections on CR manifolds","Random zeros on CR submanifolds reach equilibrium at log k/k","Explicit rate for random zero equidistribution in higher dimensions","Survey: zeros of random sections follow equilibrium with log k/k error"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000644,"raw_usage":{"total_tokens":2925,"prompt_tokens":877,"completion_tokens":2048,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":1976}},"tokens_in":493,"tokens_out":2048,"duration_ms":13786,"temperature":1.0,"reasoning_tokens":1976,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:33:49.044364+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Calculate, for a specific nondegenerate piecewise-smooth generic CR submanifold with a non-smooth admissible density, the sup-norm error $\\|\\frac{1}{2k}\\log \\widetilde B_k - V_{K,Q}\\|_{C^0(K)}$—for instance, $K$ the unit ball in $\\mathbb{C}^n$, $Q=0$, and $\\mu=(1-|z|^2)^{\\alpha}\\mathrm{Leb}_K$ with $\\alpha>-1$. If the decay is slower than $O(\\log k/k)$, Theorems 4.2, 4.4, 4.5, and 4.7 fail; if it holds, the missing link is supplied.","supporting_citations":[{"cited_title":"M ARINESCU AND D.-V","cited_arxiv_id":null,"evidence_quote":"Supplies the Bergman-kernel asymptotics and the quantitative equidistribution and large-deviation theorems the survey presents and extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the one-dimensional equilibrium-distribution theorem for Gaussian random polynomials that the survey generalizes to higher dimensions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes almost sure equidistribution of zeros toward $dd^c V_{K,Q}$ under the Bernstein–Markov property, the qualitative baseline the survey quantifies."},{"cited_title":"D INH , X","cited_arxiv_id":null,"evidence_quote":"Provides the polynomial growth of Bergman kernels and quantitative envelope convergence for the case $K=X$, which motivates the rate strategy."},{"cited_title":"D INH , X","cited_arxiv_id":null,"evidence_quote":"Supplies the $(C^{0,\\alpha},C^{0,\\alpha'})$-regularity of extremal psh envelopes and Fekete-point equidistribution speed used in Sections 3.2 and 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the analytic-disc construction and Hölder regularity of envelopes for generic submanifolds that underpin Theorem 3.16 and the Bernstein–Markov theorem for piecewise-smooth sets."},{"cited_title":"Z EITOUNI AND S","cited_arxiv_id":null,"evidence_quote":"Provides the one-dimensional large-deviation result with exponent $k^2$ that serves as the benchmark for the survey's open Problem 4.13."}],"review_version":1}