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arxiv: 2605.22059 · v1 · pith:33QTKEB5new · submitted 2026-05-21 · 🧮 math.GT · math.NT

Closed geodesics in short intervals for random hyperbolic surfaces

Zeev Rudnick This is my paper

Pith reviewed 2026-05-22 02:47 UTC · model grok-4.3

classification 🧮 math.GT math.NT
keywords closed geodesicshyperbolic surfacesshort intervalsvarianceWeil-Petersson measurelarge genusGOE statisticsLaplace spectrum
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The pith

In the large genus limit, the variance of weighted closed geodesics in short intervals [X, X+H] on random hyperbolic surfaces approaches 2H log X.

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

The paper studies the distribution of closed geodesics on hyperbolic surfaces of large genus chosen randomly according to the Weil-Petersson measure. It defines the random variable Ψ_M(X;H) that counts primitive closed geodesics with lengths in an interval of width H = o(X), weighted by their lengths. The main result establishes that the variance of this count tends to 2H log X as genus tends to infinity, for X going to infinity. This parallels the Chebyshev function for primes in short intervals but incorporates an extra factor of 2 from the much higher density of Laplace eigenvalues on these surfaces together with their expected GOE statistics.

Core claim

Viewing the surface M as a random point in moduli space equipped with the Weil-Petersson measure, the random variable Ψ_M(X;H) counting closed geodesics with norms in [X, X+H], weighted by primitive length, satisfies lim_{g→∞} Var(Ψ_M(X;H)) ∼ 2 H log X when X→∞ and H=o(X). The factor of 2 arises because the Laplace spectrum of generic hyperbolic surfaces obeys GOE statistics and has higher spectral density than the zeros of finite-degree L-functions.

What carries the argument

The random variable Ψ_M(X;H) counting primitive closed geodesics in short intervals, whose variance is computed from the expected GOE form factor of the Laplace spectrum under the Weil-Petersson measure on moduli space.

If this is right

  • The variance of closed-geodesic counts in short intervals follows a random-matrix prediction specific to the geometric setting.
  • For automorphic L-functions of degree d>1 the corresponding short-interval variance is H log X under the Riemann Hypothesis, without the extra factor of 2.
  • The higher density of Laplace eigenvalues on hyperbolic surfaces produces a larger variance than the classical prime-number case.
  • Similar variance computations become accessible for other counting problems on random hyperbolic surfaces.

Where Pith is reading between the lines

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

  • The same GOE-based approach may apply to variance results for other geometric invariants on moduli space.
  • Numerical experiments on random surfaces of moderate genus could provide evidence for or against the predicted 2H log X growth.
  • The result suggests that short-interval statistics in hyperbolic geometry are governed by universal spectral features rather than arithmetic specifics.

Load-bearing premise

The Laplace spectrum of a generic hyperbolic surface obeys GOE statistics and the surfaces are distributed according to the Weil-Petersson measure on moduli space.

What would settle it

Numerical sampling of many high-genus surfaces, computing the empirical variance of Ψ_M(X;H) for large X and small H, and finding a constant other than 2 in front of H log X.

read the original abstract

We study the distribution of closed geodesics in short intervals on random hyperbolic surfaces of large genus, and compare it with the classical problem of primes in short intervals. Viewing the surface $M$ as a random point in moduli space equipped with the Weil--Petersson measure, we investigate the random variable $\Psi_M(x;H)$ counting closed geodesics with norms in the interval $[X, X+H]$, weighted by primitive length, where $H=o(X)$. This is analogous to the Chebyshev function in prime number theory. Our main result establishes that in the large genus limit, \[ \lim_{g\to \infty}\mathrm{Var}(\Psi_M(X;H)) \sim 2\,H \log X, \] when $X\to \infty$, $H=o(X)$. Goldston and Montgomery related the variance for primes in short intervals to the form factor associated with zeros of the Riemann zeta function, and conjectured that it is asymptotic to \[ H\log(X/H). \] We show that for automorphic L-functions of degree $d>1$, the early-time GUE form factor already follows from the Riemann Hypothesis, thereby recovering the variance $H\log X$ in the very short interval regime predicted by Bui, Keating and Smith. In the geometric setting, the appearance of $\log X$ reflects the much higher spectral density of Laplace eigenvalues relative to zeros of finite-degree $L$-functions, while the additional factor of $2$ is explained by the expected GOE statistics for the Laplace spectrum of generic hyperbolic surfaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper studies the distribution of closed geodesics in short intervals on random hyperbolic surfaces of large genus, viewed as random points in moduli space with the Weil-Petersson measure. It defines the random variable Ψ_M(X;H) as the weighted count of closed geodesics with norms in [X, X+H] for H=o(X), analogous to the Chebyshev function. The main result states that lim_{g→∞} Var(Ψ_M(X;H)) ∼ 2 H log X as X→∞. It derives the variance H log X for automorphic L-functions of degree d>1 under RH via the early-time GUE form factor, and attributes the extra factor of 2 in the geometric setting to expected GOE statistics for the Laplace spectrum of generic surfaces.

Significance. If established, the result would give a precise asymptotic variance for short-interval counts of closed geodesics on random surfaces, linking geometric statistics under WP measure to random matrix predictions (GOE) and contrasting with the prime-number and L-function cases. The derivation of the GUE form factor directly from RH for higher-degree L-functions is a concrete contribution that recovers the Bui-Keating-Smith prediction.

major comments (2)
  1. [Abstract, paragraph on main result] Abstract, paragraph on main result: the asymptotic Var(Ψ_M(X;H)) ∼ 2 H log X is load-bearing on the precise prefactor 2, which is attributed to 'expected GOE statistics for the Laplace spectrum of generic hyperbolic surfaces.' No derivation of the two-point correlation function (or form factor) from the Weil-Petersson measure is indicated; the paper instead invokes this as an external expectation, unlike the explicit derivation from RH for the L-function case.
  2. [Abstract] Abstract: the statement that the large-genus limit commutes with the short-interval limit (X→∞, H=o(X)) is asserted without visible error terms, uniformity statements, or verification that the limits may be interchanged. Full details of the variance computation via the Selberg trace formula or spectral pair correlation are required to support the claim.
minor comments (1)
  1. [Abstract] Notation for Ψ_M(X;H) and the precise weighting by primitive length could be stated more explicitly in the abstract to avoid ambiguity with unweighted counts.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. Below we respond point by point to the major comments. We clarify the role of the GOE expectation in the geometric setting and the structure of the limit statements, while noting where additional discussion can be added in revision.

read point-by-point responses

  1. Referee: [Abstract, paragraph on main result] Abstract, paragraph on main result: the asymptotic Var(Ψ_M(X;H)) ∼ 2 H log X is load-bearing on the precise prefactor 2, which is attributed to 'expected GOE statistics for the Laplace spectrum of generic hyperbolic surfaces.' No derivation of the two-point correlation function (or form factor) from the Weil-Petersson measure is indicated; the paper instead invokes this as an external expectation, unlike the explicit derivation from RH for the L-function case.

    Authors: The manuscript explicitly distinguishes the two settings. For automorphic L-functions we derive the early-time GUE form factor directly from the Riemann Hypothesis, recovering the variance H log X. In the geometric case the factor of 2 is attributed to the expected GOE statistics of the Laplace spectrum on generic hyperbolic surfaces, a standard conjecture in the literature (supported by numerical evidence and heuristic arguments in the field). The paper does not claim to derive the two-point correlation function from the Weil-Petersson measure; that would constitute a separate and deeper result on the moduli space. We can expand the discussion in the introduction and add further references to make this distinction and the conditional nature of the prefactor clearer. revision: partial

  2. Referee: [Abstract] Abstract: the statement that the large-genus limit commutes with the short-interval limit (X→∞, H=o(X)) is asserted without visible error terms, uniformity statements, or verification that the limits may be interchanged. Full details of the variance computation via the Selberg trace formula or spectral pair correlation are required to support the claim.

    Authors: The abstract is necessarily brief. The full manuscript carries out the variance computation via the Selberg trace formula and the resulting spectral pair correlation in the large-genus limit; the short-interval regime H = o(X) is handled after the genus tends to infinity. The order of limits is therefore genus first, then X → ∞ with H = o(X). Explicit error terms that would justify interchanging the limits in either order are not provided and would require a more quantitative version of the large-genus asymptotics. We can add a short remark in the introduction or in the statement of the main theorem clarifying the order of limits and pointing to the relevant sections of the proof. revision: partial

standing simulated objections not resolved
  • Deriving the two-point correlation function (or form factor) explicitly from the Weil-Petersson measure on moduli space

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central result on the variance limit under the Weil-Petersson measure is obtained by direct analogy to the Goldston-Montgomery relation between short-interval variance and the form factor, with the log X term arising from the higher density of Laplace eigenvalues and the prefactor 2 attributed to the external expectation of GOE pair correlation for generic surfaces. This expectation is invoked as a known random-matrix feature rather than fitted or self-defined inside the paper's equations. The parallel derivation for automorphic L-functions recovers H log X from the Riemann Hypothesis (a stated conjecture, not an internal input), without reducing the geometric claim to a self-citation chain or ansatz smuggled from prior work by the same authors. No load-bearing step equates the output variance to a fitted parameter or renames a known empirical pattern as a new derivation; the argument remains self-contained against the listed assumptions and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the Weil-Petersson random model for surfaces, the assumption that generic surfaces have GOE spectral statistics, and the Riemann Hypothesis for the auxiliary L-function statements.

axioms (3)
  • domain assumption Weil-Petersson measure equips the moduli space of genus-g hyperbolic surfaces with a probability measure
    Used to define the random surface M and the random variable Ψ_M
  • domain assumption Laplace spectrum of a generic hyperbolic surface obeys GOE statistics
    Invoked to explain the factor of 2 in the variance
  • domain assumption Riemann Hypothesis holds for automorphic L-functions of degree d>1
    Used to obtain the early-time GUE form factor and recover the variance H log X

pith-pipeline@v0.9.0 · 5807 in / 1584 out tokens · 55875 ms · 2026-05-22T02:47:30.260989+00:00 · methodology

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Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    PhD thesis, UCL, 2026

    Acosta Reche, A. PhD thesis, UCL, 2026. 4

    work page 2026

  2. [2]

    M.; Doron, E.; Keating, J.P; Kitaev, A

    Argaman, N.; Dittes, F. M.; Doron, E.; Keating, J.P; Kitaev, A. Yu; Sieber, M; Smilansky, U;. Correlations in the actions of periodic orbits derived from quantum chaos Phys. Rev. Lett. 71 (1993), no. 26, 4326–4329. 7

    work page 1993

  3. [3]

    and Steiner, F

    Aurich, R. and Steiner, F. Energy-level statistics of the Hadamard-Gutzwiller ensemble. Phys. D 43 (1990), no. 2-3, 155–180. 21

    work page 1990

  4. [4]

    On Multiplicities in Length Spectra of Semi-Arithmetic Hyperbolic Surfaces

    Belolipetsky, Mikhail; Cosac, Gregory; D´ oria, Cayo; Teixeira Paula, Gisele. On Multiplicities in Length Spectra of Semi-Arithmetic Hyperbolic Surfaces. Commun. Math. Phys. 407, 76 (2026). 4

    work page 2026

  5. [5]

    Bogomolny, E.B., Georgeot, B., Giannoni, M.-J., Schmit, C.: Arithmetical chaos. Phys. Rep. 291(5–6), 219–324 (1997). 4

    work page 1997

  6. [6]

    Bogomolny, E.B., Schmit, C.: Multiplicities of periodic orbit lengths for non- arithmetic models. J. Phys. A 37(16), 4501–4526 (2004). 4

    work page 2004

  7. [7]

    Bohigas, M.-J

    O. Bohigas, M.-J. Giannoni, and C. Schmit, in ”Quantum Chaos and Statisti- cal Nuclear Physics”, edited by Thomas H. Seligman and Hidetoshi Nishioka, Lecture Notes in Physics Vol. 263 (Springer-Verlag, Berlin, 1986), p. 18. 21

    work page 1986

  8. [8]

    M.; Keating, J

    Bui, H. M.; Keating, J. P.; Smith, D. J. On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class. J. Lond. Math. Soc. (2) 94 (2016), no. 1, 161–185. 6, 21

    work page 2016

  9. [9]

    V. A. Bykovskii, Density theorems and the mean value of arithmetic functions in short intervals (Russian), Zap. Nauchn. Semin. POMI 212 (1994), 56–70; translation in J. Math. Sci. (N. Y.) 83 (1997), no. 6, 720–730. 4 CLOSED GEODESICS IN SHORT INTERVALS 23

    work page 1994

  10. [10]

    Prime geodesic theorem

    Cai, Yingchun. Prime geodesic theorem. J. Th´ eor. Nombres Bordeaux 14 (2002), no. 1, 59–72. 4

    work page 2002

  11. [11]

    Goldston, D. A. Notes on pair correlation of zeros and prime numbers. London Math. Soc. Lecture Note Ser., 322 Cambridge University Press, Cambridge, 2005, 79–110. 18

    work page 2005

  12. [12]

    D. A. Goldston, and H. L. Montgomery,Pair correlation of zeros and primes in short intervals. Analytic number theory and Diophantine problems (Stillwater, OK, 1984), 183–203, Progr. Math., 70, Birkh¨ auser Boston, Boston, MA, 1987. 3, 6, 7, 21

    work page 1984

  13. [13]

    New large value estimates for Dirichlet poly- nomials

    Larry Guth and James Maynard. New large value estimates for Dirichlet poly- nomials. Annals of Math, Volume 203 (2026), Issue 2, 623–675. 3

    work page 2026

  14. [14]

    Variance of arith- metic sums and L-functions inF q[t]

    Hall, Chris; Keating, Jonathan P.; Roditty-Gershon, Edva. Variance of arith- metic sums and L-functions inF q[t]. Algebra Number Theory 13 (2019), no. 1, 19–92. 3

    work page 2019

  15. [15]

    The Selberg trace formula for PSL(2,R)

    Hejhal, Dennis A. The Selberg trace formula for PSL(2,R) . Vol. I Lecture Notes in Math., Vol. 548 Springer-Verlag, Berlin-New York, 1976, vi+516 pp. 4

    work page 1976

  16. [16]

    Zur analytischen Theorie hyperbolischen Raumformen und Be- wegungsgruppen

    Huber, Heinz. Zur analytischen Theorie hyperbolischen Raumformen und Be- wegungsgruppen. Math. Ann. 138 (1959), 1–26. 4

    work page 1959

  17. [17]

    arXiv:2507.20653 [math.NT] 18, 20

    Yujiao Jiang, On Hypothesis H of Rudnick and Sarnak. arXiv:2507.20653 [math.NT] 18, 20

    work page arXiv

  18. [18]

    Iwaniec, Prime geodesic theorem, J

    H. Iwaniec, Prime geodesic theorem, J. Reine Angew. Math.,349, 136–159 (1984). 4

    work page 1984

  19. [19]

    The variance of the number of prime polynomials in short intervals and in residue classes

    Keating, Jonathan P.; Rudnick, Ze´ ev. The variance of the number of prime polynomials in short intervals and in residue classes. Int. Math. Res. Not. IMRN 2014, no. 1, 259–288 3

    work page 2014

  20. [20]

    and Ye, Y

    Liu, J., Wang, Y. and Ye, Y. A proof of Selberg’s orthogonality for automorphic L-functions. manuscripta math. 118, 135–149 (2005). 19

    work page 2005

  21. [21]

    Perron’s formula and the prime number theorem for automorphic L-functions

    Liu, Jianya; Ye, Yangbo. Perron’s formula and the prime number theorem for automorphic L-functions. Pure Appl. Math. Q. 3 (2007), no. 2, Special Issue: In honor of Leon Simon. Part 1, 481–497. 5

    work page 2007

  22. [22]

    The variance of arithmetic mea- sures associated to closed geodesics on the modular surface, Jour

    Wenzhi Luo, Zeev Rudnick and Peter Sarnak. The variance of arithmetic mea- sures associated to closed geodesics on the modular surface, Jour. of Modern Dynamics, volume 3 no. 2, 2009, 271–309 . 7

    work page 2009

  23. [23]

    Luo, W., Sarnak, P.: Number variance for arithmetic hyperbolic surfaces. Commun. Math. Phys. 161(2), 419–432 (1994). 4

    work page 1994

  24. [24]

    Hautes ´Etudes Sci

    Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on PSL 2(Z)\H2, Inst. Hautes ´Etudes Sci. Publ. Math. No. 81 (1995), 207–237. 4

    work page 1995

  25. [25]

    Primes in short intervals Michigan Math

    Maier, Helmut. Primes in short intervals Michigan Math. J. 32 (1985), no. 2, 221–225. 3

    work page 1985

  26. [26]

    The moduli space of twisted Laplacians and random matrix theory, International Mathematics Research Notices, rnae239 7

    Marklof J., Monk L. The moduli space of twisted Laplacians and random matrix theory, International Mathematics Research Notices, rnae239 7

    work page

  27. [27]

    and Petri, B

    Mirzakhani, M. and Petri, B. Lengths of closed geodesics on random surfaces of large genus. Comment. Math. Helv. 94 (2019), no. 4, 869–889. 7, 8, 9, 10, 12, 13

    work page 2019

  28. [28]

    Montgomery, H. L. The pair correlation of zeros of the zeta function. Proc. Sympos. Pure Math., Vol. XXIV American Mathematical Society, Providence, RI, 1973, pp. 181–193. 18, 19 24 ZE ´EV RUDNICK

    work page 1973

  29. [29]

    and Vaughan, R.C

    Montgomery, H.L. and Vaughan, R.C. (1974), Hilbert’s Inequal- ity. Journal of the London Mathematical Society, s2-8: 73–82. https://doi.org/10.1112/jlms/s2-8.1.73 19

    work page doi:10.1112/jlms/s2-8.1.73 1974

  30. [30]

    X. Nie, Y. Wu and Y. Xue, Large genus asymptotics for lengths of separating closed geodesics on random surfaces, J. Top. 16 (2023), 106—-175. 8

    work page 2023

  31. [31]

    Pollicott M., Sharp R.: Correlations for pairs of closed geodesics. Invent. Math. 163, 1–24 (2006) 7

    work page 2006

  32. [32]

    GOE statistics on the moduli space of surfaces of large genus

    Rudnick, Z. GOE statistics on the moduli space of surfaces of large genus. Geom. Funct. Anal. 33, 1581–1607 (2023). 7, 21

    work page 2023

  33. [33]

    Rudnick and P

    Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory. Duke Math. J. 81, 269–322 (1996). 16

    work page 1996

  34. [34]

    Journal d’Analyse Mathe- matique 151, 293–302 (2023)

    Zeev Rudnick and Igor Wigman On the Central Limit Theorem for linear eigen- value statistics on random surfaces of large genus. Journal d’Analyse Mathe- matique 151, 293–302 (2023). 7

    work page 2023

  35. [35]

    Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus

    Zeev Rudnick and Igor Wigman. Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus. Ann. Henri Poincare 26 (2025), no. 6, 2279–2291. 7

    work page 2025

  36. [36]

    Selberg, On the normal density of primes in small intervals, and the differ- ence between consecutive primes

    A. Selberg, On the normal density of primes in small intervals, and the differ- ence between consecutive primes. Arch. Math. Naturvid 47(6) (1943), 87–105. 3

    work page 1943

  37. [37]

    Selberg, Some notes from Selberg’s 1954-55 IAS lectures on the trace for- mula Institute for Advanced Study (unpublished handwritten notes), page 19, December 7, 1954

    A. Selberg, Some notes from Selberg’s 1954-55 IAS lectures on the trace for- mula Institute for Advanced Study (unpublished handwritten notes), page 19, December 7, 1954. 4 https://publications.ias.edu/sites/default/files/DOCstfA_0.pdf

    work page 1954

  38. [38]

    Weak subconvexity without a Ramanujan hypothesis

    Soundararajan, Kannan and Thorner, Jesse. Weak subconvexity without a Ramanujan hypothesis. With an appendix by Farrell Brumley. Duke Math. J. 168 (2019), no. 7, 1231–1268. 20

    work page 2019

  39. [39]

    The prime geodesic theorem

    Soundararajan, K.; Young, Matthew P. The prime geodesic theorem. J. Reine Angew. Math. 676 (2013), 105–120. 4

    work page 2013

  40. [40]

    Yunhui Wu, Yuhao Xue, Prime geodesic theorem and closed geodesics for large genus. J. Eur. Math. Soc. (2025). 4 School of Mathematical Sciences, Tel A viv University, Tel A viv 69978, Israel Email address:rudnick@tauex.tau.ac.il

    work page 2025