Spectral gap with polynomial rate for random covering surfaces

Davide Macera; Joe Thomas; Will Hide

REVIEW 3 major objections 3 minor 25 references

Spectral gap with polynomial rate for random covering surfaces

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A uniformly random degree-n cover of any closed hyperbolic surface has no new Laplacian eigenvalue below 1/4 - c n^{-b}, with probability tending to 1.

desk verdict The polynomial spectral gap claim is likely right in spirit, but the main probability estimate does not follow from the quoted theorem as stated. read the letter →

arxiv 2505.08479 v1 pith:UEKPF6YU submitted 2025-05-13 math.SP math.DGmath.OAmath.PR

Will Hide , Davide Macera , Joe Thomas This is my paper · ORCID

classification math.SPmath.DGmath.OAmath.PR MSC 58J5035P15

keywords randomcoveringsurfaceshyperbolicspectralgapLaplacianeigenvaluesstrongconvergenceSelbergtransformpolynomialerrorratepermutationrepresentations

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

This paper proves that a uniformly random degree-n cover of any fixed closed hyperbolic surface has an optimal spectral gap with a polynomial error rate: there exist constants b, c > 0, depending only on the genus of the base surface, such that the first new Laplacian eigenvalue satisfies $\lambda^{\mathrm{new}}_1(X_n) \geq \frac14 - c n^{-b}$ with probability tending to 1. The value $\frac14$ is the bottom of the Laplacian spectrum on the hyperbolic plane, so it is the best possible limiting position for the gap. The proof phrases the spectral gap as the operator norm of a Selberg-transformed geometric ball cutoff, approximates that operator by a matrix-valued group-algebra polynomial, and feeds it into a recent effective strong-convergence theorem for random permutation representations of surface groups. This gives the first polynomial error rate for spectral gaps of random surfaces, improving on earlier rates of order $\log\log n / \log n$.

What carries the argument

The load-bearing object is the Selberg transform, the spherical transform that converts a radial kernel on the hyperbolic plane into a Fourier multiplier evaluated at the Laplacian spectral parameter $r$ (with eigenvalue $\lambda = \frac14 + r^2$). The paper works with the geometric ball cutoff $k_1(z,w) = \mathbf{1}_{d(z,w) \leq 1}$ and its integral operator $P_{k_1}$; on the new spectrum of a cover, the operator norm of $P_{k_1}$ equals $h_1\!\left(i\sqrt{\frac14 - \lambda^{\mathrm{new}}_1(X_n)}\right)$, where $h_1$ is the Selberg transform of $k_1$, so an upper bound on that norm translates directly into a lower bound on $\lambda^{\mathrm{new}}_1(X_n)$. To compare the cover's operator norm with the corresponding operator on the universal cover, the paper decomposes $P_{k_1}$ as a finite sum $\sum_{\gamma \in S(1)} a_{\gamma,1} \otimes \rho_n(\gamma^{-1})$ over short elements of the surface group, approximates each compact operator $a_{\gamma,1}$ by a rank-$r$ operator $b^{(r)}_{\gamma,1}$ with error $O(r^{-1/2})$, and assembles these blocks into a self-adjoint matrix-valued group-algebra polynomial to which an effective strong-convergence theorem applies. The Selberg transform is what monitors the spectral parameter near $\frac14$, and the finite-rank reduction is what lets the infinite-dimensional operator be treated by finite-dimensional probabilistic bounds.

What would settle it

Compute the precise dependence of $c_d$ and $b$ in the effective strong-convergence theorem quoted as Theorem 2.2 (the cited paper's Theorem 6.1), and evaluate the failure probability bound at $\varepsilon = (r\log n/n)^{1/b}$ with $r = n^a$ for some small $a>0$; if the only bound is $c_d n^{-\varepsilon b}$, then $n^{-\varepsilon b} = \exp\left(-b r^{1/b} n^{-1/b} (\log n)^{1+1/b}\right) \to 1$, so the failure probability does not tend to 0 and the proof of Theorem 1.1 cannot be completed as written.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for any closed hyperbolic surface $X$, there are constants $b, c > 0$ depending only on the genus of $X$ such that a uniformly random degree-$n$ cover $X_n$, sampled uniformly among the finitely many labelled degree-$n$ covers, satisfies $\lambda^{\mathrm{new}}_1(X_n) \geq \frac14 - c n^{-b}$ with probability tending to 1 as $n \to \infty$. The theorem concerns only the new spectrum, the part of $L^2(X_n)$ orthogonal to lifts of functions from the base. Because $\frac14$ is the bottom of the $L^2$ spectrum of the Laplacian on the hyperbolic plane, no weaker lower bound can be expected asymptotically; the content is that the gap approaches this optimal value at a polynomial rate in $n$. Along the way the paper quantifies the failure probability in the strong-convergence input as $O((\log n)^{-1/b})$, so the high-probability statement is itself quantitative.

Load-bearing premise

The proof's central step assumes that the quoted effective strong-convergence theorem gives a failure probability tending to 0 when the spectral slack $\varepsilon$ is as small as $(r\log n / n)^{1/b}$ and the matrix dimension grows polynomially in $n$; as the theorem is stated in the paper (failure probability bounded by $c_d n^{-\varepsilon b}$), that quantity approaches 1, not 0, in this regime, so the high-probability conclusion depends on unstated information about how $c_d$ depends on $d$ and how $b$ depends on $\varepsilon$.

Editorial extensions

If this is right

For any fixed closed hyperbolic base surface, with probability tending to 1, the random degree-$n$ cover has no new Laplacian eigenvalues in $[0, \frac14 - c n^{-b})$, so the first new eigenvalue converges to $\frac14$ at the polynomial rate $c n^{-b}$.
This is the first polynomial error rate for spectral gaps in any random surface model; previously the best available rates for covers were of order $\log\log n / \log n$ or worse.
Because the constants depend only on the genus of the base, the same quantitative bound applies uniformly across all degree-$n$ covers sampled from a given base surface.
Taking $n$ growing along a sequence and passing to the corresponding covers gives a tower of closed hyperbolic surfaces whose Laplacian first eigenvalue approaches $\frac14$ at a polynomial rate in the cover degree.
The paper's failure-probability bound, $O((\log n)^{-1/b})$, makes the convergence statement quantitative: not merely 'probability tends to 1' but an explicit decay rate for the exceptional covers.

Reading between the lines

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

The same Selberg-transform-and-finite-rank scheme should apply to other radial observables built from the Laplacian on random covers, such as heat-kernel traces or spectral window counts near $\frac14$, yielding polynomial-rate estimates wherever the strong-convergence theorem provides the comparison.
The expected fluctuation scale for $\lambda^{\mathrm{new}}_1$ is $n^{-2/3}$ (by analogy with random regular graphs and Tracy-Widom statistics), so the exponent $b$ in the theorem is likely far from optimal; extracting the actual $b$ from the strong-convergence argument would show how much room remains.
If the effective strong-convergence theorem also holds for finite-area non-compact hyperbolic surfaces, the same proof structure would give polynomial-rate spectral gaps in that setting, extending the result beyond closed base surfaces.

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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

Colleague,

This note aims to be the first to get a polynomial error rate n^{-b} for the spectral gap of random degree-n covers of a fixed closed hyperbolic surface, improving the previous log log n / log n rate. If the proof worked, it would be a nice quantitative advance. The setup is clean: use the Selberg transform of a ball cutoff, approximate the resulting operators by finite rank, and feed them into the effective strong convergence theorem of Magee–Puder–van Handel.

The problem is the one step where the theorem is used. The paper’s Theorem 2.2, quoted from MPvH, says the bad event has probability at most c_d n^{-ε b}. In Section 3 the authors set ε = (r log n / n)^{1/b} and claim failure probability O_X(1/(log n)^{1/b}). That does not follow. With any r = n^a, 0

What is good: the Selberg-transform machinery is applied correctly; the finite-rank approximation lemmas are taken from the first author's earlier work and are used honestly; the introduction situates the result well. The structure of the proof is exactly what you'd want if the input theorem had the right quantitative form.

The central claim might be repairable if the actual MPvH paper contains a stronger estimate, for example a bound like c_d exp(-c n ε^b) or c_d n^{-c ε^b n^α} with the right regime. But as written, the quoted theorem cannot support the argument. The authors need to state the precise MPvH formulation and verify the asymptotics. This is not about constants; the exponent in the probability fails to vanish.

My recommendation: send to a referee, because the target result is significant and the flaw is identifiable and likely fixable, but the paper should not be accepted in this form.

Referee Report

3 major / 3 minor

Summary. The paper claims a polynomial-rate spectral gap for uniformly random degree-n covers of a fixed closed hyperbolic surface: for every such surface X there exist b,c>0 such that a uniformly random degree-n cover X_n has λ_1^new(X_n) ≥ 1/4 − c n^{-b} with probability tending to 1. The proof uses the Selberg transform of a ball kernel to convert the spectral gap question into an operator norm problem on the new subspace, approximates the relevant operators by finite-rank operators, and applies an effective strong-convergence theorem of Magee, Puder, and van Handel for random permutation representations of surface groups. If the high-probability comparison with the regular representation is valid, the final polynomial error rate follows from a simple lower bound on the Selberg transform.

Significance. A polynomial error rate for the new spectral gap of random covers would be a substantial quantitative improvement over the previously known rates of order log log n/log n, and it would move the study of these surfaces closer to the conjectured n^{-2/3+o(1)} fluctuation scale. The paper is concise and the reduction via the Selberg transform and finite-rank approximation is elegant. The main theorem is crisp and falsifiable, and the method is a natural application of the recent breakthrough in strong convergence. However, as written the central probability estimate is not justified: the quoted effective theorem, as stated, does not give a vanishing failure probability in the regime used in the proof, and the growth of the constant c_d with the matrix dimension is left uncontrolled. These issues are load-bearing for the proof of Theorem 1.1.

major comments (3)

[Section 3, paragraph beginning 'We now apply Theorem 2.2'] Theorem 2.2 is applied with ε = (r log n/n)^{1/b}, and the proof immediately asserts success with probability 1 − O_X((log n)^{-1/b}). This does not follow from the displayed bound P ≤ c_d n^{-ε b}. Indeed, with r = n^a one has n^{-ε b} = exp(−b n^{-(1-a)/b} (log n)^{1+1/b}), which tends to 1 as n → ∞, not to 0. Thus the displayed bound does not even tend to zero at the chosen ε, and it certainly does not imply the claimed O_X((log n)^{-1/b}) failure probability. This high-probability comparison is the step that connects the finite-rank operator to the regular representation and from which the polynomial rate is extracted, so Theorem 1.1 is unsupported as written. The authors need to quote the exact ε-dependence in [MPvH25, Theorem 6.1] (e.g., whether the failure probability is of the form c_d n^{-c ε^2}, c_d exp(−c n ε^b), or something else) and then verify that the chosen ε produces a vanishing failure probability.
[Section 3, same paragraph] The prefactor c_d is never quantified. In the same application, the matrix dimension satisfies d ≤ r|S(1)|, so d grows like n^a after the choice r = n^a, while Theorem 2.2 only states that c depends on g and the word length and writes c_d. If c_d grows rapidly with d, then even a corrected tail bound might not be o(1). A valid derivation must either specify the growth of c_d in d or impose a restriction on a that ensures c_d times the tail bound tends to zero. As the manuscript stands, the high-probability event is not established even if the exponent in Theorem 2.2 were corrected.
[Theorem 1.1 and final paragraph of Section 3] The theorem and abstract claim that b,c depend only on the genus of X, but the proof records constants A, C, a diameter bound, and a word-length bound that depend on the specific base surface X, and the final 'const' is not argued to be uniform over all closed hyperbolic surfaces of a fixed genus. No argument is supplied that these constants can be bounded in terms of the genus alone. The authors should either weaken the statement to say the constants depend on X or provide the missing uniformity argument.

minor comments (3)

[Section 3, proof of Theorem 1.1] The symbol r is used both for the truncation rank in Lemma 3.3 and for the auxiliary parameter r = n^a. Since the rank must be an integer, the proof should take r = floor(n^a) and note that the estimates are unaffected.
[Lemmas 3.2 and 3.3] Both lemmas are only sketched, with details deferred to [Hid23, Lemmas 5.1 and 5.2]. Since the constants in these lemmas enter the main quantitative estimates, the paper would be more self-contained if the proofs were given in full or the precise constant dependencies were stated.
[Theorem 2.2] The displayed exponent in the denominator of the failure probability is written as ε b; because b also appears in the main theorem and the subsequent application, the authors should make the intended exponent unambiguous and check it against the source theorem.

Circularity Check

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No circularity: the proof is a direct application of an external effective strong-convergence theorem, with only auxiliary self-citations that are not load-bearing.

full rationale

The paper's derivation chain does not reduce to its own inputs by construction. Lemma 3.1 connects the new spectral gap to the Selberg transform norm using standard functional calculus; it is not equivalent to the claimed bound. Lemmas 3.2 and 3.3 are quantitative approximation and support estimates; although they cite the first author's earlier work [Hid23], the present paper includes proof sketches, and their role is only to control error terms, not to encode the final spectral gap. The central probabilistic input is Theorem 2.2, quoted from Magee, Puder and van Handel [MPvH25], which are not authors of this paper and which is an independent external result about strong convergence of random permutation representations. The final estimate is obtained by choosing r = n^a, not by fitting a parameter to the desired conclusion. The skeptic's concern that the displayed failure probability in Theorem 2.2 may not vanish for the chosen epsilon is a quantitative correctness issue about the applicability of an external theorem, not a circularity: no equation in this paper is defined in terms of the theorem it derives, no fitted value is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the argument. Self-citation appears only for routine auxiliary lemmas, and those lemmas are not the source of the optimal rate. Therefore the manuscript exhibits no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The core of the proof is imported from [MPvH25]. The paper contributes a short combination of known approximation arguments (from [Hid23]) with that black box.

free parameters (4)

a (exponent in r = n^a) = unspecified, small positive
In the proof of Theorem 1.1, r is set to n^a for a small a>0 to balance the finite-rank error 1/sqrt(r) and the strong-convergence error (r log n/n)^{1/b}; existence of such an a suffices.
constant b = unspecified
Exponent in Theorem 1.1 inherited from the exponent b in [MPvH25, Theorem 6.1]; no numerical value is given.
constant c = unspecified
Constant in Theorem 1.1; the proof gives an existential constant depending on X.
t (kernel radius) = 1
The ball cutoff k_t is evaluated at t=1; any fixed t>0 would work, so this is a convenient hand choice.

assumptions (4)

domain assumption Theorem 2.2 ([MPvH25, Theorem 6.1]): effective strong convergence with failure probability c_d n^{-ε b}
The entire proof of Theorem 1.1 rests on this quantitative bound; the paper does not prove it and only quotes it.
standard math Selberg transform and spectral decomposition of the Laplacian on hyperbolic surfaces
Standard background used in Section 2.1.
standard math L^2(F)⊗ℓ^2(Γ_g) is isometrically isomorphic to L^2(H) via f⊗δ_γ ↦ f∘γ^{-1}
Used in Section 3 to identify the operator norm with the plane operator norm.
standard math Liebeck-Shalev theorem: uniform random covers are connected with probability tending to 1
Ensures λ^new_1 is well-defined and positive.

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Cite this review

Pith. "Pith review of Spectral gap with polynomial rate for random covering surfaces." pith.science (2026). https://pith.science/paper/UEKPF6YU

@misc{pith2026250508479,
  author       = {Pith},
  title        = {Pith review of: Spectral gap with polynomial rate for random covering surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEKPF6YU}},
  note         = {Machine review of arXiv:2505.08479}
}

abstract

In this note we show that the recent work of Magee, Puder and van Handel [MPvH25] can be applied to obtain an optimal spectral gap result with polynomial error rate for uniformly random covers of closed hyperbolic surfaces. Let $X$ be a closed hyperbolic surface. We show there exists $b,c>0$ such that a uniformly random degree-$n$ cover $X_{n}$ of $X$ has no new Laplacian eigenvalues below $\frac{1}{4}-cn^{-b}$ with probability tending to $1$ as $n\to\infty$.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Nalini Anantharaman and Laura Monk, Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps , arXiv:2304.02678 (2023)

arXiv 2023

[2] [2]

, Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II , arXiv:2502.12268 (2025)

arXiv 2025

[3] [3]

1339, Springer, Berlin, 1988, pp

Peter Buser, Marc Burger, and Jozef Dodziuk, Riemann surfaces of large genus and large _1 , Geometry and analysis on manifolds ( K atata/ K yoto, 1987), Lecture Notes in Math., vol. 1339, Springer, Berlin, 1988, pp. 54--63. 961472

work page 1987

[4] [4]

Charles Bordenave and Benoit Collins, Norm of matrix-valued polynomials in random unitaries and permutations, arXiv preprint arXiv:2304.05714 (2023)

arXiv 2023

[5] [5]

Nicolas Bergeron, The spectrum of hyperbolic surfaces, Springer, 2016

work page 2016

[6] [6]

Oriol Bohigas, Marie-Joya Giannoni, and Charles Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Physical review letters 52 (1984), no. 1, 1

work page 1984

[7] [7]

Bridson and Andr \'e Haefliger, Metric Spaces of Non-Positive Curvature , Grundlehren Der Mathematischen Wissenschaften , vol

Martin R. Bridson and Andr \'e Haefliger, Metric Spaces of Non-Positive Curvature , Grundlehren Der Mathematischen Wissenschaften , vol. 319, Springer, Berlin, Heidelberg, 1999

work page 1999

[8] [8]

Tropp, and Ramon van Handel , A new approach to strong convergence, Annals of Mathematics, to appear (2024)

Chi-Fang Chen, Jorge Garza-Vargas , Joel A. Tropp, and Ramon van Handel , A new approach to strong convergence, Annals of Mathematics, to appear (2024)

work page 2024