Logarithmic fluctuations of Stationary Hastings-Levitov

Eviatar B. Procaccia; Noam Berger

REVIEW 4 major objections 3 minor 2 cited by

Logarithmic fluctuations of Stationary Hastings-Levitov

T0 review · 4 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The fluctuation field of stationary Hastings-Levitov(0) is log-correlated: variance (π/4) log t, covariance (π/4)(log t − log b) for b < t, and maximum O(log t) with high probability.

desk verdict The pi/4 variance constant is not just unproved; it is inconsistent with the paper's own Lemma A.1 and the known pi/2 linear growth rate. read the letter →

arxiv 2502.03554 v1 pith:D5EBISRF submitted 2025-02-05 math.PR math-phmath.MP

Noam Berger , Eviatar B. Procaccia This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60K3560G6030C35

keywords stationaryHastings-Levitovlog-correlatedrandomfieldconformalslitmapsDoobmartingaledecompositionfluctuationDiffusionLimitedAggregationmaximumofMarkovgenerator

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Stationary Hastings-Levitov(0) is a random growth model built by composing conformal slit maps at Poisson times, and this paper studies the fluctuations around its deterministic linear growth. The central claim is that the fluctuation field $\{M_t(x)\}$ is log-correlated: its variance at a point is $(\pi/4)\log t$, and its covariance between two points separated by $b$ is $(\pi/4)(\log t - \log b)$ for $b < t$. The paper also proves that the maximum imaginary fluctuation over the interval $[0,t]$ is at most of order $\log t$ with high probability, with failure probability $O(t^{-1/2})$. If these results are correct, the model provides a non-Gaussian, physically motivated example of a log-correlated random field with logarithmic extremes. The proof works directly on the physical process rather than on a scaling limit, and it obtains the variance and covariance from an identity expressing the martingale increments as an integral over slit maps.

What carries the argument

The argument is carried by the martingale representation $F_t(z) = z + i\pi t/2 + M_t(z)$, where $F_t$ is the backwards composition of slit maps $\varphi_x(z) = x + \sqrt{(z-x)^2 - 1}$ at Poisson-arrival times. For the variance, the key identity is $\mathbb{E}|M_t(0)|^2 = \mathbb{E}\int_0^t\int_{-\infty}^{\infty}|\varphi_x(F_s(0)) - F_s(0)|^2\,dx\,ds$; Lemma A.1 evaluates the $x$-integral at a point $iy$ as $\min\{1, \pi/(4y)\} + O(y^{-3})$, so together with the strong law $\mathrm{Im}\,F_s(0) \approx \pi s/2$ the variance becomes $(\pi/4)\int ds/s = (\pi/4)\log t$. For the maximum bound, the Markov generator of $Y_t = \mathrm{Im}\,F_t(0)$ is computed as $\mathcal{L}f(\zeta) = \int_{-\infty}^{\infty}[f(\zeta + \Delta(\zeta,x)) - f(\zeta)]\,dx$, and applying it to $f(y) = e^{\alpha y}$ yields an integral inequality whose solution bounds the exponential moments of $\mathrm{Im}\,M_t(0)$ by $\exp(\tfrac{\pi}{2}\alpha^2 e^\alpha)\,t^{\alpha^2}$. Those moments give a Chernoff tail $\mathbb{P}(\mathrm{Im}\,M_t(0) > \beta\log t) \le \mathrm{const}\cdot t^{-\beta^2/4}$, and the maximum result follows by a grid-approximation with derivative control via the half-plane Koebe distortion theorem and a planarity argument.

What would settle it

Simulate the SHL(0) process for large $t$ (for example $t = 10^6$) and compute $\mathbb{E}|M_t(0)|^2 / \log t$; if it does not converge to $\pi/4 \approx 0.785$, the main variance claim fails. Alternatively, check the uniform corridor directly: for many samples, record whether $\mathrm{Im}\,F_s(0)$ ever exits $[\pi s/2 - s^{2/3}, \pi s/2 + s^{2/3}]$ for $s > \eta$; a non-negligible probability of exit would disprove the asserted event $H_4(\eta)$ on which the lower bound depends.

Watch

Extended reading notes

Core claim

The paper's discovery is that the fluctuation field $\{M_t(x)\}_{x\in\mathbb{R}}$ of the backwards stationary Hastings-Levitov(0) map, defined as the martingale part of the decomposition $F_t(z) = z + i\pi t/2 + M_t(z)$, exhibits logarithmic spatial correlations: $\mathbb{E}|M_t(0)|^2 = (\pi/4)\log t(1+o(1))$ and $\mathrm{Cov}(M_t(0),M_t(b)) = (\pi/4)(\log t - \log b)(1+o(1))$ for $b < t$, with covariance negligible for $b > t$. In addition, the maximum of $\mathrm{Im}\,M_t$ over $x\in[0,t]$ satisfies $\mathbb{P}(\max_{x\in[0,t]}\mathrm{Im}\,M_t(x) > \beta\log t) < c t^{-1/2}$ for some $\beta > 0$ and all large $t$. These results are established for the process itself, not just for a small-particle scaling limit, through variance asymptotics obtained from an integral over slit maps and through exponential-moment bounds for the Markov generator of the imaginary part.

Load-bearing premise

The covariance and variance lower bounds rely on an unproved uniform event: that the imaginary part of the growing map at every time $s$ stays between $\pi s/2 - s^{2/3}$ and $\pi s/2 + s^{2/3}$ (for all $s$ after some threshold) with high probability, a corridor the paper asserts after equation (13) but does not prove; if this corridor fails, the variance and covariance asymptotics are not established by the given argument.

Editorial extensions

If this is right

The variance of the fluctuation field grows exactly like $(\pi/4)\log t$, confirming at the level of the full process (not only a scaling limit) the logarithmic growth of fluctuations expected for this aggregation model.
The two-point covariance decays as $(\pi/4)(\log t - \log b)$ in separation $b$, placing the field in the log-correlated universality class alongside objects like the Gaussian free field and critical models.
The maximum imaginary part over an interval of length $t$ is $O(\log t)$ with probability $1 - O(t^{-1/2})$, and the single-point right tail obeys $\mathbb{P}(\mathrm{Im}\,M_t(0) > \beta\log t) \le \mathrm{const}\cdot t^{-\beta^2/4}$, so large upward fluctuations above the logarithmic scale have polynomially decaying probability.
The generator-based method yields quantitative exponential-moment bounds, giving a route to extremal statistics (e.g., tightness of the centered maximum) that does not require Gaussianity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof would be completed by proving the uniform event $H_4(\eta)$ asserted after equation (13): that $\mathrm{Im}\,F_s(0)$ stays within $s^{2/3}$ of $\pi s/2$ for every $s>\eta$ simultaneously, with probability at least $1-\eta^{-1/3}$; a direct simulation of the trajectory could check whether this linear-growth corridor is actually maintained, and if it fails the central asymptotics may still
Because the simulated marginals of $M_t(0)$ look asymmetric, the log-correlated field here is probably not Gaussian; if the variance and covariance results hold, SHL(0) becomes a rare example of a non-Gaussian log-correlated field whose maximum still grows logarithmically, which may inform conjectures about universality of log-correlated extremes.
The paper's bound on the maximum is an upper bound only; a natural companion conjecture is matching logarithmic lower bounds (e.g., that the maximum is at least a deterministic constant times $\log t$), which would establish that the extremes grow deterministically like the standard deviation, and this would be a testable extension of the present results.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main theorems as printed are not supported by the paper's own estimates. Using Lemma A.1, the variance density is pi/(4 * ImF_s). Under the paper's own H2/H4 events, ImF_s ≈ pi*s/2, so that density is ≈ 1/(2s), not pi/(4s). The inequality in (14), which claims pi/(4 ImF_s) >= pi/(4s), is therefore false; the correct lower bound would be about 1/2 log t. The upper bound display (12) has the same substitution issue, and even granting the unproved uniform event H4, the pi/4 constant does not follow. This is a load-bearing error: Theorem 1's variance constant is likely wrong (the argument actually suggests 1/2 log t), and Theorems 2 and 3 inherit the problem. The reader's weaker concern about H4 is real but secondary; the algebraic mistake is independent and fatal to the stated results.

What the paper does well: it attacks an interesting object—the fluctuation field of stationary Hastings-Levitov(0) directly, rather than via Silvestri's scaling limit. The martingale bootstrap and the generator-based exponential moment bound are genuinely new and plausible. The covariance structure of the field is a natural question, and the overall strategy could work once the constants are corrected. The paper is also honest in building on published foundations [BPT22], and the reference list looks appropriate.

Soft spots, proportionately: Theorem 2 as printed uses min instead of max for b>t, an obvious typo but still a concrete error. More importantly, the lower bound for Theorem 1 silently depends on H4, a uniform-in-s event that is asserted but not proved. But the algebra error in (12) and (14) is the real problem. It is not a matter of missing technical detail; the inequality just goes the wrong way. This cannot be fixed by a small patch.

Who is this for? Specialists in stochastic planar growth and log-correlated fields. The method may be of interest, but the main result as stated is not reliable. If the authors correct the constant (likely to 1/2), the paper could be a solid contribution. As is, I would not cite the theorems. A serious referee could still help if the editor expects heavy revision, but the paper needs substantial rework before it is publishable.

Referee Report

4 major / 3 minor

Summary. The paper studies the stationary Hastings-Levitov(0) process in the upper half-plane and its fluctuation field M_t(x)=F_t(x)-x-iπt/2. It claims three theorems: Theorem 1 gives E|M_t(0)|^2=(π/4)log t(1+o(1)); Theorem 2 gives a logarithmic covariance formula; Theorem 3 gives a high-probability bound of order β log t for max_{x∈[0,t]} Im M_t(x), with failure probability O(t^{-1/2}). The proofs use the martingale decomposition from [BPT22], a bootstrap on Im F_s(0), a generator computation for exponential moments, and a conformal-geometry argument for the maximum.

Significance. If correct, the paper would provide a non-Gaussian, physically motivated example of a logarithmically correlated random field with precise variance growth and logarithmic maximal growth. The paper does not fit free parameters, and the martingale representation is anchored in the independently published construction of [BPT22]. However, the central variance constant in Theorem 1 is not established by the submitted proof because of an algebraic error in the lower bound, the uniform event H4 is asserted without proof, and Theorem 2 as printed is false in the regime b>t. The significance is therefore conditional on a substantial reworking of the main arguments.

major comments (4)

[Section 2, Eq. (14)] The displayed lower bound is algebraically false. Under the event H4, Im F_s(0) ≤ πs/2 + s^{2/3}, so Lemma A.1 gives ∫ |φ_x(F_s(0))-F_s(0)|^2 dx ≥ π/(4 Im F_s(0)) - c/(Im F_s(0))^3 ≥ 1/(2s) - O(s^{-4/3}). Since 1/(2s) < π/(4s), the inequality '≥ π/(4 P[H4(η)] ∫_η^t ds/s' does not follow. At best this route yields E|M_t(0)|^2 ≥ (1/2 - o(1)) log t, which is strictly weaker than the claimed π/4 log t. The proof of Theorem 1's lower bound is therefore not valid.
[Section 2, Eq. (13)] The uniform event H4(η) is used with the assertion P(H4(η)) ≥ 1 - η^{-1/3}, but no proof is given. The preceding bound P(H2(s)) ≥ 1 - c/s^{1+1/6} is pointwise in s; it cannot be turned into a bound for the uncountable intersection over all s > η without a maximal inequality or a dyadic/continuity argument. This unproved uniform-control estimate is load-bearing for the lower bound in Theorem 1 and is also invoked in the bootstrap argument.
[Theorem 2 (Section 1.1)] As printed, Theorem 2 is false. For b > t, min{log t - log b, 0} = log t - log b < 0, so the right-hand side is negative and, for fixed t, tends to -∞ as b → ∞. This contradicts Lemma 3.3, which gives |Cov(M_t(0),M_t(b))| ≤ c√(t/b) log t = o(1) in that regime. The intended expression is presumably max{log(t/b), 0}, or the theorem must be restricted to b < t with an explicit relation between b and t. The phrase 'the little-o notation is with respect to b → ∞' is also ill-posed because t is free; if t grows with b, the error may depend on t.
[Section 3, Eq. (32)] The lower-bound estimate in Lemma 3.4 is not justified for the stated range 'any t > b'. Substituting (31) into (32) gives an error term of order sqrt((log t - log(b log^3 b))(log t + b^2)) times (1/(1+(b log^3 b)^2) - 1/(1+t^2)). The displayed simplification to O(log log b) fails when t is large relative to b log^3 b; for example, with log t = b^2 the error is of order b/log^6 b, which is not O(log log b). Thus Lemma 3.4, and with it the covariance asymptotics in the intended range of Theorem 2, require a precise growth condition on t relative to b and a proof of the estimate in (31).

minor comments (3)

[Section 4, Eq. (35)] On the event H2(s), one only has 1 + Y_s ≥ 1 + πs/2 - s^{2/3}, so the displayed inequality e^{α/(1+Y_s)} ≤ e^{α/(1+πs/2)} is not literally correct; the s^{2/3} correction should appear in the denominator. The asymptotic conclusion is likely unaffected, but the displayed inequality should be corrected.
[Section 3, Lemma 3.2] The event H3.1(s) is used before being defined clearly for all z; please state explicitly whether it is required for all z ∈ H or only for the specific endpoints appearing in the argument.
[Section 3, after Eq. (22)] There is a typo: 'sonstant' should be 'constant'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: [BPT22] supplies independent published inputs, and the new variance/covariance/maximum bounds are derived from the martingale structure, not assumed.

full rationale

The derivation is not circular. The SHL(0) process, the decomposition Ft(z)=z+iπt/2+Mt(z), the martingale property, and the integral bounds quoted from [BPT22, Lemma 3.3] come from a separately published paper with stated proofs; despite overlapping authorship, these are parameter-free external results whose assumptions do not include the present paper's conclusions, so the shared authorship does not raise the circularity score. The proof of Theorem 1 is a genuine two-stage argument: an initial crude diffusive bound yields the high-probability event H2, after which Lemma A.1, a direct integral evaluation, is applied to obtain the constant π/4. Theorem 2 uses martingale orthogonality, Cauchy-Schwarz, and the established variance estimate; Theorem 3 uses exponential moments, conformal distortion, and BPT22 path results. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors is invoked to force a choice. Two non-circular rigor issues should be flagged separately: the lower-bound inequality in (14) appears algebraically false, since under H4 one may have ImF_s ≈ πs/2 + s^{2/3}, making π/(4ImF_s) ≈ 1/(2s) < π/(4s), so the displayed "≥ π/(4s)" does not follow; and the uniform bound P(H4(η)) ≥ 1 − η^{-1/3} is asserted without proof. These are correctness concerns, not circularity, and therefore do not affect the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the SHL(0) martingale representation and variance identity from [BPT22], the Feller/generator description of the imaginary part, standard conformal distortion and large-deviation tools, and an asserted uniform linear-growth estimate. There are no fitted free parameters and no invented entities.

assumptions (5)

domain assumption SHL(0) admits the decomposition F_t(z) = z + i pi t/2 + M_t(z), with M_t a zero-mean martingale satisfying the variance identity E|M_t(0)|^2 = E integral_0^t integral_{-inf}^{inf} |phi_x(F_s(0))-F_s(0)|^2 dx ds.
Taken from [BPT22]; all variance, covariance, and maximum bounds are built on this representation and on Lemma 3.3 of [BPT22].
domain assumption The imaginary part Y_t = Im F_t(0) is a Feller process with generator Lf(zeta) = integral_{-inf}^{inf} [f(zeta+Delta(zeta,x))-f(zeta)] dx.
Used in Section 4 to derive the exponential moment bound; if the process is not Feller or the generator is not this operator, the Chernoff estimates fail.
standard math Theorem 5 (Koebe 1/4 theorem for half-plane): if f:H->C is conformal, then |f'(w)| <= 16|f'(z)| for w in B(z, Im z/2).
Used in Section 5 to pass from grid-point estimates to a uniform bound on [0,t].
standard math Hoeffding's inequality, Gronwall's inequality, Dynkin's formula, and the negative correlation of increasing and decreasing functions of Y_s.
Used throughout; these are standard tools.
ad hoc to paper The uniform event H4(eta) has probability at least 1-eta^{-1/3}; i.e. Im F_s(0) stays within s^{2/3} of pi s/2 for all s > eta.
Asserted in the proof of Theorem 1's lower bound after (13) without a displayed derivation; load-bearing for converting the variance integral into log t.

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Cite this review

Pith. "Pith review of Logarithmic fluctuations of Stationary Hastings-Levitov." pith.science (2026). https://pith.science/paper/D5EBISRF

@misc{pith2026250203554,
  author       = {Pith},
  title        = {Pith review of: Logarithmic fluctuations of Stationary Hastings-Levitov},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D5EBISRF}},
  note         = {Machine review of arXiv:2502.03554}
}

abstract

We prove that the fluctuation field $\{M_t(x)\}_{x\in\mathbb{R}}$ of stationary Hastings-Levitov$(0)$ exhibits logarithmic spatial correlations. Moreover, by studying the infinitesimal generator of the imaginary part of $M_t(0)$, we show that for some $\beta>0$, $\max_{x\in[0,t]}\text{Im} M_t(x)<\beta\log t$ with high probability, as $t\to\infty$.

Figures

Figures reproduced from arXiv: 2502.03554 by the authors.

**Figure 1.** Computer simulation of {F˜ t(x)}x∈R where Mt(z) is a zero mean martingale, which can be realized as the limit of the martingale part in the Doob decomposition (4) F m t (z) = z + D m t (z) + Mm t (z). In this paper we study the field {Mt(x)}x∈R and prove it observes logarithmic spatial correlation and study the maximum of the field. -3000 -2000 -1000 0 1000 2000 3000 -25 -20 -15 -10 -5 0 5 10 [PITH_FULL_IMAGE:figur… view at source ↗

**Figure 2.** [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Mt(0) empirical distribution histogram Theorem 3. There exists a c > 0 such that for any β > 0 large enough, for every t > 0 large enough, (6) P max x∈[0,t] ImMt(x) > β log t < c t 1/2 . 2. Proof of Theorem 1 First we improve the law of large numbers estimate from [BPT22]. Lemma 2.1. There exist ξ2.1, c2.1 > 0 such that for every z ∈ H, P [ImFt(z) < Imz + ξ2.1t] < e−c2.1t . Proof. W.l.o.g we assume that t m ∈ N,… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Illustration of bounding set (54), with zl = ν 2t F2t(− 1 2 t) (t˜l) and zr = ν 2t F2t( 3 2 t) (t˜r). Proof. Using Taylor expansion of √ · at y → ∞ (56) [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

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Forward citations

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Reference graph

Works this paper leans on

12 extracted references · 9 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Stationary eden model on cayley graphs

Ton \'c i Antunovi \'c and Eviatar B Procaccia. Stationary eden model on cayley graphs. The Annals of Applied Probability , pages 517--549, 2017

work page 2017

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Stretched idla

Noam Berger, Jacob J Kagan, and Eviatar B Procaccia. Stretched idla. ALEA Lat. Am. J. Probab. Math. Stat , 11(1):471--481, 2014

work page 2014

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Noam Berger, Eviatar B Procaccia, and Amanda Turner. Growth of stationary hastings--levitov. The Annals of Applied Probability , 32(5):3331--3360, 2022

work page 2022

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Matthew B Hastings and Leonid S Levitov. Laplacian growth as one-dimensional turbulence. Physica D: Nonlinear Phenomena , 116(1-2):244--252, 1998

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G. F. Lawler and V. Limic. Random walk: a modern introduction . Cambridge Univ Pr, 2010

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Paul Meakin. Diffusion-controlled deposition on fibers and surfaces. Physical Review A , 27(5):2616, 1983

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Yingxin Mu, Eviatar Procaccia, and Yuan Zhang. Scaling limit of dla on a long line segment. Transactions of the American Mathematical Society , 375(12):8769--8806, 2022

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J. Norris and A. Turner. Hastings--levitov aggregation in the small-particle limit. Communications in Mathematical Physics , pages 1--33, 2012

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