Local Weyl law and length-minimising loops on hyperbolic surfaces

Daniel Meriaz

REVIEW 3 major objections 3 minor 12 references

For large random hyperbolic surfaces, the pointwise local Weyl law has variance (τ tanh(πτ))/(4π²(g−1)L)‖f‖², matching the random wave prediction.

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2026-07-31 23:08 UTC pith:VXBKRU3Z

load-bearing objection A credible, significant computation of the pointwise local Weyl law variance on random hyperbolic surfaces, with a real but likely fixable gap in one stationary-phase step and several typos. the 3 major comments →

arxiv 2607.24060 v1 pith:VXBKRU3Z submitted 2026-07-27 math-ph math.GTmath.MPmath.SP

Local Weyl law and length-minimising loops on hyperbolic surfaces

Daniel Meriaz This is my paper

classification math-ph math.GTmath.MPmath.SP MSC 58J5030F6053C2211F72

keywords local Weyl lawrandom hyperbolic surfacesWeil-Peterssonrandom wave modeleigenfunction variancegeodesic loopslength-minimising loopspre-trace formula

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.

This paper establishes a precise asymptotic for the variance of a smoothed local Weyl law on random hyperbolic surfaces of large genus. The variance is proportional to (τ tanh(πτ))/(4π²(g−1)L)‖f‖², exactly the value predicted by the random wave model, up to explicit error terms. The proof reduces the statistic to a sum over geodesic loops, identifies the dominant diagonal contribution, and shows that the remaining off-diagonal terms are negligible. A new geometric tool, the notion of length-minimising geodesic loops, is introduced, and it is proven that such loops are always simple and intersect trivially, which controls the off-diagonal terms. The paper also shows that if eigenfunctions were independent Gaussians, the variance would agree with the theorem up to the stated errors.

Core claim

Theorem 1.1 states that as the genus g tends to infinity, the variance over Weil–Petersson random surfaces of the smoothed eigenfunction count N(X,z) equals (1/(4π(g−1)))·(τ tanh(πτ)/(πL))‖f‖²_{L²(R)} plus error terms of size O_{f,τ}(1/(L²g)) and O_{f,L,τ}(1/g²). Equivalently, the fluctuations of the local Weyl law are those of a Gaussian random wave field. The proof splits the second moment of the oscillatory part into diagonal and off-diagonal pairs of geodesic loops; the diagonal pairs give the main term through an exact integral involving the complete elliptic integral of the first kind, and the off-diagonal pairs are shown to be O(1/g²) using new results on length-minimising loops.

What carries the argument

Three pieces carry the argument. First, a weight function H(ℓ) encodes the space-averaged diagonal contribution of closed geodesics of length ℓ, turning the diagonal term into a weighted sum over the length spectrum; an integration formula for Weil–Petersson volumes over moduli space then reduces the expectation to a single integral of H against a volume ratio. Second, an exact integral identity involving the complete elliptic integral K and the cosine transform evaluates that integral, producing the constant τ tanh(πτ). Third, length-minimising loops are geodesic loops that are shortest outside a given subgroup of the fundamental group; the paper proves every such loop is simple and that su

Load-bearing premise

The theorem stands on the interchange in Lemma 4.18, where the product of two test functions is replaced by its value at ℓ with an error term that is then integrated against a kernel over all y,v ≥ 0; the paper does not supply a uniform-in-L bound for this error integral, and if that integral contributes at order 1/L rather than 1/L², the stated leading constant would acquire an uncontrolled L-dependent correction.

What would settle it

Evaluate the remainder integral in Lemma 4.18 with the explicit error O_f((y+v)/L) and test numerically whether the y,v integral converges to O(1/L²); if it instead saturates at O(1/L), the leading constant is not established.

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

If this is right

The pointwise local Weyl law on large random hyperbolic surfaces is asymptotically deterministic: its variance decays like 1/g, so the smoothed eigenfunction count concentrates on its mean.
If eigenfunctions are replaced by independent Gaussian variables of the same variance, the variance of the statistic agrees with the theorem up to the stated error terms, confirming the Gaussian wave picture for this statistic.
The length-minimising loop theorems supply a general method to control statistics that depend on pairs of short geodesic loops, which appear in other spectral and geometric questions.
The explicit constant gives a precise target for numerical simulation of eigenfunctions on random surfaces of large genus.
The exploring loops construction yields a sequence of simple, trivially intersecting loops that weakly fill the surface, providing a new topological decomposition.

Where Pith is reading between the lines

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

The simplicity theorem for length-minimising loops likely extends to any negatively curved manifold, since its proof uses only the arc-loop-arc decomposition and length minimality; if so, the off-diagonal control would generalise.
A natural extension would be to let L grow with g, e.g. L ~ c log g; the current error O(1/(L²g)) suggests a window where the random wave prediction remains valid, beyond which the variance may change.
The proof leaves a potential gap: the stationary-phase/uniform-asymptotics step in Lemma 4.18 lacks an explicit uniform bound for the remainder integral; if that integral is not O(1/L²), the leading constant could acquire a logarithmic L-dependence.
The statement of Lemma 4.4 appears misprinted; the proof suggests the intended identity, but the main constant depends directly on it, so a corrected statement is needed before relying on the constant.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A credible, significant computation of the pointwise local Weyl law variance on random hyperbolic surfaces, with a real but likely fixable gap in one stationary-phase step and several typos.

read the letter

The paper gives the first pointwise local-Weyl-law variance asymptotics for Weil–Petersson random surfaces. If Theorem 1.1 is right, it confirms Berry’s random wave prediction at the second-moment level. I think it is likely right, but the printed proof has a gap in the analytic core that needs to be filled before I would bet on it.

What is genuinely new: the variance formula for N(X,z) with the explicit leading constant τ tanh(πτ)/(4π²(g−1)L) ‖f‖², the length-minimising loop theorem (they are simple and mutually trivially intersecting), and the exploring loops. The strategy is sound: Selberg pre-trace, split into diagonal and off-diagonal, Mirzakhani integration, stationary phase for the diagonal main term, and Monk–Thomas for the off-diagonal. The geometric part, especially the loop collar argument giving a lower bound on injectivity radius when two short primitive loops exist, is clean and clever. The paper is honest about using Appendix A only as a post-hoc consistency check, not as an input.

The soft spots are concentrated in Section 4.5. The step in Lemma 4.18 where Ŷ(ℓ + y/L)Ŷ(ℓ + (y+v)/L) is replaced by Ŷ(ℓ)² with error O((y+v)/L), then integrated against the elliptic kernel over y,v, is not fully justified. The paper asserts exponential decay in y,v, and the kernel does decay like e^{-(y+v)/2}, and the support of Ŷ forces y+v ≲ L, so the tail is exponentially small. Still, a uniform-in-L bound on the doubled error integral is not written down. I believe it is true and fillable, but as printed it is a real gap in the proof of the main term. The other flags are minor: the proof of Theorem 1.1 at the end of Section 3 writes O(1/(Lg)) where the statement and Proposition 3.3 give O(1/(L²g)) — clearly a typo. Lemma 4.4 is not misprinted: the s=0 value π³/4 is exactly the limit of the s≠0 formula. There are also small notation slips (e.g., H vs T in Lemma 4.5).

Overall: this is a substantial, well-researched paper with a plausible main theorem and a novel geometric component. It deserves a serious referee. I would send it to review, and I would want the referee to push for a complete proof of the stationary-phase estimate in Lemma 4.18 before accepting.

Referee Report

3 major / 3 minor

Summary. The paper studies the variance of a smoothed pointwise local Weyl law N(X,z)=Σ h(r_j)|φ_j(z)|² on Weil–Petersson random hyperbolic surfaces of large genus, with the basepoint z uniformly distributed. Theorem 1.1 claims that this variance is (4π(g−1))⁻¹ · τ tanh(πτ)/(πL) · ‖f‖²_{L²} plus O_{f,τ}(1/(L²g)) and O_{f,L,τ}(1/g²), in agreement with Berry's random wave model. The proof uses Selberg's pre-trace formula to write N as a constant plus a sum over based geodesic loops, splits loop pairs into diagonal and off-diagonal terms, evaluates the diagonal term via Mirzakhani's integration formula and a stationary-phase argument (Section 4), and bounds the off-diagonal term using new geometric results on length-minimising loops (Theorem 5.3, Theorem 5.5, Theorem 1.2) plus work of Monk–Thomas (Section 6). Appendix A checks compatibility with an i.i.d. Gaussian model.

Significance. If Theorem 1.1 is correct, it is a substantial parameter-free confirmation of Berry's random wave model for pointwise eigenfunction densities on large random hyperbolic surfaces. The leading constant is derived, not fitted, from Selberg's pre-trace formula and Mirzakhani's integration formula, and the Gaussian comparison in Appendix A is explicitly post-hoc, so it does not enter the main proof. The new geometric notion of length-minimising loops and the simplicity theorems are of independent interest. However, the proof as printed has a missing factor π in the central constant, a gap in the uniform-asymptotics step in Lemma 4.18, and an incorrect argument in the proof of Theorem 5.3; these issues are technical rather than conceptual and appear repairable.

major comments (3)

[§4.5, Lemma 4.18, around (4.20)] The step replacing f̂((ℓ+y)/L) f̂((ℓ+(y+v))/L) by f̂(ℓ)² + O_f((y+v)/L) and then integrating against K(√((e^y−1)/(e^{y+v}−1)))/√(e^{y+v}−1) needs a uniform bound. The text cites 'exponential decay in both the y,v variables' for the integral of (y+v) times this kernel. Near v=0 the kernel is not exponentially decaying in v; it behaves like ~ const·log(1/v)/√(e^y−1), and the required O(1) bound for the error integral is not demonstrated. Please provide a complete estimate of J(L)=∫∫ ((y+v)/L) K(...)/√(...) dy dv over the relevant support, with explicit logarithmic integrability near v=0. This is load-bearing for Proposition 4.10 and hence for Theorem 1.1.
[§4.5, Lemma 4.18 and proof of Proposition 4.10] There is a missing factor π in the main constant. Lemma 4.18 states ∫ F_{1,1}^{(0)} ... = τ tanh(πτ)/(4L) ‖f‖² + O(1/L²), but Proposition 4.10 and Theorem 1.1 require τ tanh(πτ)/(4πL) ‖f‖². The proof of Proposition 4.10 follows Lemma 4.18 and uses 1/(4L), so the printed chain of arguments would be off by π. The calculation after Plancherel and Lemma 4.4 gives the factor 1/(4πL), so this is a correction of a typo, but it must be fixed both in Lemma 4.18 and in the proof of Proposition 4.10.
[§5.2, proof of Theorem 5.3] In the case p=z, the proof states δ = b·η₂ ∈ G, but the ALA decomposition gives δ = η₁·b·η₂, and η₂ is not equal to η. The conclusion can be repaired: η₁, b, and η₂ are all loops based at z of length strictly less than ℓ(δ), so each lies in G, and therefore δ ∈ G. As printed, however, the argument is incorrect. Since Theorem 5.3 underlies Theorem 1.2 and the off-diagonal estimate in Proposition 3.4, this proof must be corrected.

minor comments (3)

[§3, proof of Theorem 1.1] The error term is printed as O_{f,τ}(1/(Lg)) in the last display, while Theorem 1.1 and Proposition 3.3 give O_{f,τ}(1/(L²g)). This appears to be a typo.
[§4.5, proof of Lemma 4.18] The phrase 'exponential decay in both the y,v variables' is misleading near v=0; the kernel is logarithmically integrable there rather than exponentially decaying. The estimate is plausible, but a precise split of the integration domain and a bound using Lemma 4.2 are needed.
[Appendix A] The notation N(X,z) is reused for the Gaussian surrogate model, creating possible confusion with the original local Weyl law. Consider using a different symbol, e.g. N_G(X,z).

Circularity Check

0 steps flagged

No significant circularity: the variance is derived from Selberg's pre-trace formula, Mirzakhani's integration formula, and explicit integral identities, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claim, Theorem 1.1, is a variance asymptotic for the local Weyl law on Weil--Petersson random hyperbolic surfaces. The derivation chain is: pre-trace formula expresses N(X,z) as a constant plus a sum over geodesic loops; the variance is reduced to expectations of diagonal and off-diagonal pair sums; Mirzakhani's integration formula converts the diagonal expectation into a weighted integral of the function H over geodesic lengths; and Proposition 4.10 evaluates the main integral using exact identities (notably Lemma 4.4) and stationary-phase/uniform-asymptotic arguments. No parameter is fitted to the target variance, and the leading constant tau tanh(pi tau)/(4 pi L) ||f||^2 emerges from explicit integral evaluation rather than from matching data. The off-diagonal bound uses the geometric results of Section 5 and the external theorem of Monk--Thomas [MT22]; the Monk--Thomas citation is not self-citation (the present author is not an author of [MT22]), and it is used as an independent bound on expected numbers of embedded one-holed tori/pairs of pants. The Gaussian comparison in Appendix A is explicitly framed as a consistency check: it uses Theorem 1.1 to compare the variance of a Gaussian model with the already-proved variance, so it is not an input to the main derivation. The analytic issue raised about Lemma 4.18 -- namely, whether the error from replacing the f-hat factors by f-hat(ell)^2 is integrable uniformly in L -- is a correctness/rigor concern about an estimate, not a circularity: the claim does not reduce by definition to its input, and no equation is produced where the output is equal to a fitted value by construction. The manuscript contains no self-definitional step, no fitted input called a prediction, and no load-bearing self-citation chain. Accordingly the appropriate circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

No free parameters are fitted: f, L, τ are inputs. The central claim rests on standard spectral identities and external moduli-space results; no ad hoc entity is introduced. The new geometric notions are definitions with proven theorems, not ungrounded postulates.

axioms (7)

standard math Selberg pre-trace and trace formulae (Theorem 2.2, Lemma 2.3)
Basis for expressing the local Weyl law as a loop sum; standard result cited to Selberg.
domain assumption Mirzakhani integration formula (Theorem 2.6)
Used to integrate H(ℓ) over moduli space; external theorem from [Mir07].
domain assumption Weil–Petersson volume asymptotics (2.7)–(2.10)
Needed for expectations of simple-separating and sns sums; cited to Mirzakhani–Petri, Mirzakhani–Zograf, Nie–Wu–Xue.
domain assumption Rudnick's GOE variance bound [Rud23]
Used in Lemma 3.2 and Appendix A to estimate N(X) variance as O(1/g²).
domain assumption Monk–Thomas expected number of short embedded χ=−1 subsurfaces (Lemma 6.5)
Used to bound vol X_L in the off-diagonal estimate.
standard math Integral-table identities for elliptic integrals and cosine transforms (GR07 6.153, Oberhettinger 1.4.43/1.4.44)
Exact evaluation in Lemma 4.4; one statement appears misprinted but the derivation shows the intended identity.
domain assumption Buser's geodesic-loop geometry (finite transverse self-intersections, collar bound)
Used in Section 5 and Lemma 5.11; standard hyperbolic geometry.

pith-pipeline@v1.3.0-alltime-deepseek · 34638 in / 26138 out tokens · 205578 ms · 2026-07-31T23:08:06.137187+00:00 · methodology

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read the original abstract

We study the variance of a local Weyl law over a fixed smooth energy window, when averaged over large Weil--Petersson hyperbolic surfaces. Our results are consistent with the predictions of Berry's random wave model. Our approach allows to explicitly integrate certain test functions which depend on lengths of based geodesic loops, and relate them to the associated lengths of the closed geodesics in their free-homotopy class. We thus utilise the work of Mirzakhani, with exact stationary phase arguments, to identify correct main and error terms, making explicit the asymptotic behaviour of the variance of the local Weyl law. Furthermore, we introduce the geometric notion of length-minimising geodesic loops and sequences, based at a point. We prove a complete characterisation of the topology of these, namely that they are simple. This forms a key ingredient in our study, and yields a new streamlined argument to bound the contributions of remainder terms which depend on lengths of pairs of different short primitive geodesic loops. To illustrate the generality of our results, we further introduce a family of "exploring" loops based at a point, which might be of independent interest.

Figures

Figures reproduced from arXiv: 2607.24060 by Daniel Meriaz.

**Figure 1.** Figure 1: Example of diagonal and off-diagonal pairs of geodesic loops based at a point z. They are freely homotopic to (powers of) the closed geodesic γ. Diagonal terms. The main term of Theorem 1.1 arises from the diagonal terms, whereas the off-diagonal terms yield contributions to the O(g −2 ) remainder terms. In the diagonal terms, on any fixed surface X, one first groups the pairs by the associated closed geod… view at source ↗

**Figure 2.** Figure 2: The second shortest primitive loop γ2 is simple and intersects the systole γ1 trivially. As an illustration of the generality of the definitions of length-minimising loops and sequences, we introduce the family of exploring loops (c.f. Definition 5.8), which might be of independent interest. These are a length-minimising sequence γ1, . . . , γn of geodesic loops, where n ≤ 2g − 1. They iteratively “explore… view at source ↗

**Figure 3.** Figure 3: The ALA-decomposition of a loop with two self-intersections 5.1. Arc-Loop-Arc decomposition. We begin with the following simple decomposition of a loop into arcs. Notation 1 (Arc-Loop-Arc decomposition). Let c : [0, 1] → X be a non-simple arc with a finite number of self-intersections. Define the time of first self-intersection t := sup{t ′ > 0 : ∀s ∈ [0, t′ ), c(s) ̸= c(t ′ )} and s ∈ [0, t) be the unique… view at source ↗

**Figure 4.** Figure 4: Example of the loops appearing in the proof, when δ has one self-intersection. □ [PITH_FULL_IMAGE:figures/full_fig_p036_4.png] view at source ↗

**Figure 5.** Figure 5: Length-minimising loops γ and δ intersecting at p. The contradiction follows as both δ1 and γ ′ (Figures 5b and 5c respectively) have lengths < ℓ(γ), and γ ∼ ¯δ1·γ ′ . Hence the loops δ1 = β1·α¯1 and γ ′ := β1·α2 both have lengths < ℓ(γ). As such, they are both homotopic to elements in Gi−1. Thus γ = α1·α2 ∼ α1·β¯ 1·β1·α2 = ¯δ1·γ ′ . Therefore γ ∈ Gi−1, contradicting its definition. We have reduced all po… view at source ↗

**Figure 6.** Figure 6: Illustration of the two different possibilities for intersections of γ2 with ∂Nε(γ1). The grey region is Σ(γ1). By the above, the loop γn can intersect either one or two boundary components of Σ(Tn−1). We note that by Definition 5.7, if γn intersects two different boundary components, then they are not freely homotopic in X (up to orientation). Indeed, the cylinder between them would then be in Σ(Tn−1). A… view at source ↗

discussion (0)

Reference graph

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