Macroscopic Hausdorff dimension of the level sets of the Airy processes

Fei Pu; Sudeshna Bhattacharjee

REVIEW 2 major objections 4 minor 1 cited by

Macroscopic Hausdorff dimension of the level sets of the Airy processes

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Pith's one-line read The paper proves that the upper and lower level sets of both Airy processes have exact macroscopic Hausdorff dimensions: $1-\gamma^{3/2}$ for peaks and $1-\gamma^{3}$ for valleys.

desk verdict Exact Airy level-set dimensions, likely correct, but the written proof has a repairable threshold typo and leans on unpublished companion estimates. read the letter →

arxiv 2501.00772 v2 pith:SNJ5QCHT submitted 2025-01-01 math.PR

Sudeshna Bhattacharjee , Fei Pu This is my paper · ORCID

classification math.PR MSC 60A1060F0560H15

keywords AiryprocesseslevelsetsmacroscopicHausdorffdimensionmultifractalitylastpassagepercolationtailprobabilitiesTracy-WidomdistributionKPZuniversalityclass

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

The Airy$_1$ and Airy$_2$ processes are the stationary random curves that appear as the scaling limits of one-dimensional random growth models in the KPZ universality class, and their extreme values are governed by logarithmic gauge functions. This paper determines, for every threshold $\gamma \in (0,1)$, the macroscopic Hausdorff dimension of the sets of times where these processes are abnormally high or abnormally low. The result is that, almost surely, the upper level sets have dimension $1-\gamma^{3/2}$ and the lower level sets have dimension $1-\gamma^3$, with identical formulas for Airy$_1$ and Airy$_2$. Since $\gamma$ can be chosen freely, the level sets form a continuous multifractal spectrum, giving a quantitative answer to how the rare extremes of the KPZ fixed point are distributed over exponentially large time scales.

What carries the argument

The load-bearing object is the $\theta$-thickness criterion: a set $E$ is $\theta$-thick if, for all large $n$, $E$ meets every interval of length $e^{\theta n}$ whose left endpoint belongs to a grid of spacing $e^{\theta n}$ inside $[e^n,e^{n+1}]$, and a $\theta$-thick set has macroscopic Hausdorff dimension at least $1-\theta$. The paper verifies this condition for each level set by estimating the probability that a small interval contains no extreme point. For Airy$_1$ those estimates use association (the FKG property) of the process, the super-exponential decay of its covariance, and the sharp one-point tail rates of the GOE Tracy--Widom distribution. For Airy$_2$, whose covariance decays only polynomially, the estimates instead come from exponential last-passage percolation via sharp bounds on the probability that the maximum or minimum of a rescaled passage time over a long interval stays below or above a threshold. The matching upper bounds come from a standard macroscopic-dimension upper-bound theorem fed by the same interval tail probabilities.

What would settle it

Compute the exact logarithmic asymptotics of the one-interval tails that the proof uses: if $P\{\max_{s\in[0,1]} A_1(s) > x\}$ turns out to decay as $e^{-c x^{3/2}}$ with $c \neq 4\sqrt{2}/3$, or if $P\{\min_{s\in[0,1]} A_2(s) < -x\}$ turns out to decay as $e^{-c x^3}$ with $c \neq 1/12$, then the dimension exponents $1-\gamma^{3/2}$ and $1-\gamma^3$ would be off.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1 and Theorem 1.2. For $U_1(\gamma)=\{t>e: A_1(t)> \frac{\gamma}{2}((3\log t)/2)^{2/3}\}$ and $U_2(\gamma)=\{t>e: A_2(t)> \gamma((3\log t)/4)^{2/3}\}$, it holds almost surely that $\operatorname{Dim}_H(U_1(\gamma))=\operatorname{Dim}_H(U_2(\gamma))=1-\gamma^{3/2}$; for $L_1(\gamma)=\{t>e: A_1(t)<-\gamma(3\log t)^{1/3}\}$ and $L_2(\gamma)=\{t>e: A_2(t)<-\gamma(12\log t)^{1/3}\}$, it holds almost surely that $\operatorname{Dim}_H(L_1(\gamma))=\operatorname{Dim}_H(L_2(\gamma))=1-\gamma^3$. Here $\operatorname{Dim}_H$ is the Barlow--Taylor macroscopic Hausdorff dimension, which measures how well a subset of $\mathbb{R}$ fills the exponentially growing shells $[e^n,e^{n+1}]$. The paper obtains the upper bounds from a general theorem on macroscopic dimensions together with sharp interval tail probabilities, and the lower bounds by verifying a thickness condition that forces the level set to intersect every small cell of an exponentially spaced grid. The result makes precise that, with respect to the natural $(\log t)^{2/3}$ and $(\log t)^{1/3}$ gauges, the peaks and valleys of both Airy processes are multifractal.

Load-bearing premise

The whole result leans on sharp estimates of how unlikely it is for the Airy processes, or their last-passage-percolation approximations, to reach very high or very low values; these estimates are quoted from two not-yet-published companion papers, and if any of them gives a wrong rate, the corresponding level-set dimension formula fails.

Editorial extensions

If this is right

Almost surely, for every $\gamma\in(0,1)$, the level-set dimensions are known exactly rather than merely bounded: peaks are $1-\gamma^{3/2}$ and valleys are $1-\gamma^3$, for both Airy$_1$ and Airy$_2$.
The two Airy processes share the same level-set dimensions because the leading tail exponents of their one-point laws coincide; the asymmetry between the peak exponent $3/2$ and the valley exponent $3$ mirrors the asymmetric upper and lower tails of the Tracy--Widom distributions.
As $\gamma$ varies in $(0,1)$, the dimensions trace a continuous multifractal spectrum, decreasing from nearly $1$ to nearly $0$; level sets at larger thresholds are macroscopically smaller in a precisely quantified way.
The proof's two ingredients, a sharp interval tail estimate and a decorrelation or last-passage-percolation input, are the only model-specific parts, so the same scheme can produce level-set dimensions for any stationary KPZ-class process once those two ingredients are known.

Reading between the lines

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

Our inference: the thickness criterion actually shows the level sets are present in every grid cell at the relevant scales, not merely that their dimension is large; this stronger 'no large empty cells' property could be useful for studying the occupation and intersection behavior of the Airy processes.
Our inference: because the theorem stops at $\gamma<1$ and monotonicity forces dimension $0$ at $\gamma=1$, the natural follow-up is to determine the correct gauge, for instance whether the critical level sets are countable, exactly at the threshold.
Our inference: the exponential-LPP route used for Airy$_2$ suggests the same level-set dimensions should hold for the KPZ fixed point and for other limiting processes built from the same LPP weights; a numerical box-counting test at a single $\gamma$ would provide a quick check of this universality.

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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 short version: this is a real result—exact macroscopic Hausdorff dimensions for upper and lower level sets of Airy1 and Airy2, with formulas 1 - gamma^{3/2} and 1 - gamma^3. The proof adapts the Khoshnevisan-Kim-Xiao thickness method to the Airy processes, and it works. I'd send this to a good referee.

What's genuinely new: no one had these dimensions before. The paper's real added value beyond the KKX framework is the set of quantitative LPP tail estimates, Propositions 5.4 and 5.5. Those are nontrivial and drive the Airy2 bounds. The Airy1 upper bound uses association and super-exponential covariance decay; the lower bound uses the same thickness argument. The structure is coherent.

Now the soft spots, in proportion. Equation (3.16) is wrong as written: the event {inf A1(s) >= -gamma (3n)^{1/3}} does not imply {min_j A1(z_j) >= -gamma (3n)^{1/3}}; you need the (n+1) threshold because log s <= n+1 on the shell. The authors used the correct version in the Airy2 proof and in the Airy1 upper-set proof, so it's a genuine typo, not a conceptual gap. Easily repaired, and the summability in (3.15) still works with n+1.

The bigger issue is external dependence. The crucial lower-tail estimates—Lemma 5.1, Lemma 3.12 of [7], and the Airy1 association and tail inputs from [25]—are not proved here. The authors cite their own preprints, which are parameter-free but not yet formally published. That makes the main theorem conditional. I don't think that's disqualifying; the preprints are specific and checkable, and this is a standard division of labor in the field. But a referee should either verify those imports or require that the companion papers be posted in final form.

One thing the reader's report underweights: the paper is honest about what it needs. It doesn't hide the imports, and the claimed dependence is exactly where you'd expect it. The true load-bearing constant is the 1/12 in Proposition 5.5; if that rate were different, the lower-bound dimension in (1.6) would shrink. That's a real fragility, but it's also the kind of thing peer review can settle.

Who this is for: people working on KPZ universality and multifractal geometry of random processes. It's a within-subfield advance, not a breakthrough. It deserves a serious referee. My recommendation: send it out, with a request to fix (3.16) and to clarify the status of [7] and [25] in the final version.

Referee Report

2 major / 4 minor

Summary. The paper studies the macroscopic Hausdorff dimension of the upper and lower level sets of the Airy1 and Airy2 processes, where the level thresholds are chosen according to the known almost-sure growth rates. The main results, Theorems 1.1 and 1.2, assert that for every gamma in (0,1), the upper level sets have dimension 1 - gamma^{3/2} and the lower level sets have dimension 1 - gamma^3, almost surely. The proofs follow the framework of Khoshnevisan, Kim, and Xiao [18]: upper bounds are obtained from tail estimates for the extrema over unit intervals, while lower bounds are obtained by verifying that the level sets are theta-thick for theta in the appropriate range and applying the thickness criterion. The Airy1 argument uses association and super-exponential covariance decay; the Airy2 argument uses new quantitative tail estimates for the maximum and minimum of the Airy2 process, which are derived in Section 5 from exponential last-passage percolation estimates.

Significance. Assuming the imported estimates are correct, the paper provides the first exact macroscopic Hausdorff dimensions for the level sets of Airy processes, thereby establishing multifractality of the peaks and valleys with respect to the natural gauge functions. The formulas are explicit and parameter-free, and the proofs are structured in a way that separates the probabilistic input (tail and dependence estimates) from the geometric argument. A particular strength is that the lower bounds are obtained through the thickness criterion with explicit Borel-Cantelli summability, rather than through abstract existence arguments. However, the paper is not self-contained: several of the load-bearing tail and association estimates are imported from unpublished preprints by the same research group.

major comments (2)

[Section 3, Eq. (3.16)] The first inequality in (3.16) is false as written. On the interval [x_{i,n}, x_{i,n}+e^{n theta}) one has log s <= n+1, so the event {inf_s A1(s)/(log s)^{1/3} >= -gamma 3^{1/3}} is contained in {inf_s A1(s) >= -gamma(3(n+1))^{1/3}}, not in {inf_s A1(s) >= -gamma(3n)^{1/3}}. In fact the reverse containment holds for the n-threshold. The correct threshold n+1 is used later in the Airy2 proof (Section 4). Replacing n by n+1 in (3.16) and in the estimates (3.17)-(3.18) preserves the summability condition (3.15) and does not change the final dimension, so the error is repairable; nonetheless, the displayed inequality is a genuine mistake in a central step of the proof of (1.5).
[Section 5, Propositions 5.4 and 5.5] The proofs of the Airy2 bounds rely on several quantitative LPP estimates that are not proved in this paper: [7, Lemmas 2.7, 3.6, 3.9, 3.12], [25, Theorem 1.2 and Proposition 6.1], [5, Theorem 1.4], and [2, Proposition 2.1]. In particular, the constant 1/12 in [7, Lemma 3.12] is used in Proposition 5.5 to obtain the exact rate e^{-(1/12)(1+epsilon)x^3}, which is essential for the Borel-Cantelli summability in the proof of Theorem 1.2(1.6); if that constant were smaller, the lower-bound dimension would shrink below 1 - gamma^3. Since these are preprints by the same research group and are not yet formally published, the main theorems are conditional on the correctness of these external results. The authors should either include proofs of the imported lemmas in an appendix or verify their status so that the paper is self-contained enough for the claims to be assessed.

minor comments (4)

[Section 3, after Eq. (3.16)] In the line `inf_{s in [x_{i,n}, x_{i,n}+e^{n theta})} A1(s)/(log x)^{1/3}`, the denominator should be `(log s)^{1/3}`, not `(log x)^{1/3}`.
[Section 5, proof of Proposition 5.5] The sentence `We consider the points z_j as defined in the proof of Proposition 5.5` should refer to Proposition 5.4, not Proposition 5.5.
[Section 5, definition of event E] In the event E, the term `E(T_{v,z_j})` should be `E(T_{v,u_N(z_j)})` to match the notation used elsewhere in the proof.
[Section 1, Theorems 1.1 and 1.2] The theorems are phrased as holding almost surely for every gamma in (0,1), but the proofs establish the bounds for each fixed gamma on an event that may depend on gamma. A short density or monotonicity argument should be added to justify the simultaneous statement for all gamma.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: exact dimension formulas are derived from independent LPP tail estimates and the KKX18 thickness criterion, not from the conclusions themselves.

full rationale

The derivation chain is: Theorem 1.1 and Theorem 1.2 are obtained by applying the Khoshnevisan–Kim–Xiao macroscopic Hausdorff dimension criteria (upper bound via tail of supremum/infimum, lower bound via θ-thickness) to estimates on the Airy processes. The Airy1 upper tail (3.3), association and covariance decay are imported from [25, Prop 6.1, Thm 1.2] and [6]; the Airy1 lower-tail Lemma 3.1 is derived in the paper from [7, Lemma 2.7]; the Airy2 tails are derived from Propositions 4.1 and 4.2, whose proofs (Propositions 5.4 and 5.5) are carried out in the paper using [7, Lemmas 3.6 and 3.12] and standard LPP estimates. None of these inputs is a restatement of the level-set dimensions; all are parameter-free statements about one-point/two-point distributions, extrema of LPP passage times, or geodesic fluctuations whose assumptions do not include the target results. The exact constants 4/3, 1/12, etc. enter through those estimates, but they are not fitted to the level sets and are not derived from the dimension claims. The typo in (3.16) (n instead of n+1) is a minor repairable slip and does not change the summability argument. The heavy reliance on unpublished companion preprints by the same research group is a verification/correctness risk, not evidence of circular reasoning.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free fitted parameters and no invented entities. It relies on a network of external results: stationarity and association of Airy1, sharp tail asymptotics, covariance decay, and LPP limits and estimates. These are treated as inputs here; they are not derived in this paper. The auxiliary constants epsilon, delta, and eta in the proofs are chosen small to make Borel-Cantelli sums converge and do not affect the final dimensions.

assumptions (6)

domain assumption Airy1 is stationary and associated; Airy2 is stationary (from [25, Theorem 1.2] and [24]).
Needed for inequalities (2.3)-(2.5) and for the thickness arguments in Section 3.
domain assumption Sharp one-point tail asymptotics (3.1)-(3.2) for Airy1, its max tail (3.3), and the analogous Airy2 one-point tail.
Used to compute the exponents in the level-set gauge functions.
domain assumption Super-exponential covariance decay (3.4) for Airy1 from [6, Theorem 1.1].
Used to control the association correction term in (3.12) and (3.17).
domain assumption Weak convergence of exponential LPP to Airy processes (Theorems 2.3 and 2.4) from [8, Theorem 3.8] and [7, Theorem 1.4].
Transfers LPP tail estimates to the Airy2 process and to the Airy1 minimum.
domain assumption LPP tail estimates Lemmas 5.1, 5.2, 3.6, 3.12 from [7], [8, Theorem 4.2], [2, Proposition 2.1], and [5, Theorem 1.4].
These supply the quantitative maximum/minimum bounds behind Propositions 4.1, 4.2, and Lemma 3.1.
standard math Borel-Cantelli and FKG/Lebowitz inequalities for associated variables.
Standard probabilistic tools used throughout the proofs.

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Cite this review

Pith. "Pith review of Macroscopic Hausdorff dimension of the level sets of the Airy processes." pith.science (2026). https://pith.science/paper/SNJ5QCHT

@misc{pith2026250100772,
  author       = {Pith},
  title        = {Pith review of: Macroscopic Hausdorff dimension of the level sets of the Airy processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SNJ5QCHT}},
  note         = {Machine review of arXiv:2501.00772}
}

abstract

We study the Macroscopic Hausdorff dimension of the upper and lower level sets of the Airy processes, following the general method developed in Khoshnevisan et al. \cite{KKX17}. For the Airy$_1$ process, the approach to macroscopic Hausdorff dimension of level sets hinges on some inequalities for its joint probabilities, while for the Airy$_2$ process, we make use of some quantitative estimates on the tail probabilities of its maximum and minimum over an interval.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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Limit theorems for extrema of Airy processes

Basu, R. and Bhattacharjee, S.: Limit theorems for extre ma of Airy processes. arXiv:2406.11826 (2024)

work page Pith review arXiv 2024

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and Xiao, Y.: Intermittency a nd multifractality: a case study via parabolic stochastic PDEs

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work page 2017

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and DiFranco, J.: Asymptotics o f Tracy-Widom distributions and the total integral of a Painlev´ e II function

Baik, J., Buckingham, R. and DiFranco, J.: Asymptotics o f Tracy-Widom distributions and the total integral of a Painlev´ e II function. Comm. Math. Phys. 280 (2008), no. 2, 463–497

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Bal´ azs, M., Basu, R. and Bhattacharjee, S.: Geodesic tr ees in last passage percolation and some related problems. arXiv.2308.07312 (2024). 19

arXiv 2024

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Barlow, M. T. and Taylor, S. J.: Fractional dimension of s ets in discrete spaces. With a reply by J. Naudts J. Phys. A 22 (1989), no.13, 2621–2628

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Barlow, M. T. and Taylor, S. J.: Deﬁning fractal subsets o f Zd. Proc. London Math. Soc. (3) 64 (1992), no.1, 125–152

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and Basu, R

Baslingker, J. and Basu, R. and Bhattacharjee, S. and Kri shnapur, M.: Optimal tail estimates in β-ensembles and applications to last passage percolation. a rXiv.2405.12215 (2024)

work page arXiv 2024