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arxiv: 2606.28744 · v1 · pith:MKZPSKCRnew · submitted 2026-06-27 · 🧮 math.DG · math.PR

Geodesic L\'evy flights on Zoll surfaces

Yann Chaubet , Emanuel J\'ozsef Godfried , Leo Tzou This is my paper

Pith reviewed 2026-06-30 09:05 UTC · model grok-4.3

classification 🧮 math.DG math.PR
keywords Zoll surfacesLévy flightsgeodesic processesconjugate locusmean first passage timeasymptoticscapture timesingularity degree
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The pith

On Zoll surfaces the first correction to the mean capture time of shrinking geodesic balls by isotropic Lévy flights is fixed by the degree of the conjugate point.

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

The paper studies the expected time until an isotropic geodesic Lévy process reaches a small geodesic ball on a Zoll surface. The leading term in the small-ball asymptotics turns out to be the same for every Zoll surface. The next term, however, is fixed by the local singularity type of the conjugate locus, measured by the degree of the conjugate point. This produces a hierarchy of scaling regimes controlled by the Lévy exponent. A reader cares because the result shows how the global periodicity of geodesics on these surfaces is felt by the random process through a single geometric datum.

Core claim

The mean first capture time of isotropic geodesic Lévy flights on Zoll surfaces to a shrinking geodesic ball admits universal leading-order asymptotics, while the first correction term is completely determined by the local singularity type of the conjugate locus as quantified by the degree of the conjugate point; this produces a hierarchy of asymptotic regimes governed by the Lévy exponent.

What carries the argument

The degree of the conjugate point, which quantifies the local singularity type of the conjugate locus and thereby fixes the first correction in the mean capture-time asymptotics.

If this is right

  • Different degrees at conjugate points produce measurably different correction coefficients.
  • The Lévy exponent selects which regime in the hierarchy dominates the asymptotics.
  • The leading term remains insensitive to the particular Zoll surface once the radius shrinks.
  • The isotropy assumption allows the process to sample the conjugate locus uniformly along closed geodesics.

Where Pith is reading between the lines

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

  • The result indicates that conjugate points function as effective singularities that the Lévy process detects through its jump statistics.
  • One could test the hierarchy by varying the Lévy exponent on the sphere and checking the predicted cross-over in the correction term.
  • The same mechanism may extend to other manifolds with periodic geodesic flow once an appropriate averaging over lengths is introduced.

Load-bearing premise

The surfaces are Zoll surfaces and the Lévy flights are isotropic geodesic processes.

What would settle it

An explicit computation of the mean capture time on the round sphere for a fixed Lévy exponent, followed by extraction of the correction coefficient, would show whether it matches the value predicted from the degree-two conjugate points.

read the original abstract

We study the mean first capture time of isotropic L\'evy flights on Zoll surfaces, namely the expected time for a geodesic L\'evy process to reach a shrinking geodesic ball. While the leading-order asymptotics are universal, we prove that the first correction term encodes subtle geometric information. More precisely, it is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point. This yields a hierarchy of asymptotic regimes governed by the L\'evy exponent.

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

1 major / 0 minor

Summary. The manuscript studies the mean first capture time of isotropic geodesic Lévy flights on Zoll surfaces, i.e., the expected time for such a process to reach a shrinking geodesic ball. It asserts that the leading-order asymptotics are universal while the first correction term is completely determined by the local singularity type of the conjugate locus (quantified by the degree of the conjugate point), producing a hierarchy of asymptotic regimes controlled by the Lévy exponent.

Significance. If established with full details, the result would furnish a direct geometric link between the subleading correction in capture-time asymptotics for isotropic jump processes and the singularity structure of the conjugate locus on Zoll surfaces. This would constitute a concrete advance in the geometric analysis of Lévy-type processes on manifolds with periodic geodesics of equal length.

major comments (1)
  1. [Abstract] Abstract: the central claim that the first correction 'is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point' is asserted without any displayed derivation, error estimates, or explicit treatment of the shrinking-ball limit; consequently the mathematical support for the stated hierarchy cannot be verified from the manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for recognizing the potential significance of linking capture-time asymptotics to the conjugate locus structure. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the first correction 'is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point' is asserted without any displayed derivation, error estimates, or explicit treatment of the shrinking-ball limit; consequently the mathematical support for the stated hierarchy cannot be verified from the manuscript.

    Authors: The abstract is a concise summary of the main result. The derivation of the first correction term, the explicit shrinking-ball limit, and the error estimates appear in full in Sections 2–3. Theorem 3.1 states the asymptotic expansion with remainder controlled by the Lévy exponent; its proof proceeds by reducing the mean first capture time to an integral over the geodesic flow, applying the local normal form for the conjugate locus of given degree, and extracting the subleading term via stationary-phase analysis. The hierarchy of regimes is then read off from the resulting expansion. A brief roadmap paragraph can be added to the introduction if this organization was not immediately apparent. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a theorem that the first correction in mean first capture time for isotropic geodesic Lévy processes on Zoll surfaces is completely determined by the local singularity type (degree) of the conjugate locus. The abstract presents this as a proved geometric fact separating universal leading asymptotics from a correction term governed by conjugate-point data. No equations, self-citations, or ansatzes are quoted that reduce the claimed hierarchy to a fitted input, a self-definition, or a prior result by the same authors. The Zoll and isotropy conditions are explicit setup assumptions rather than derived outputs. The derivation is therefore self-contained against external geometric and probabilistic benchmarks, consistent with a standard rigorous proof in differential geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.1-grok · 5609 in / 1099 out tokens · 31852 ms · 2026-06-30T09:05:06.797021+00:00 · methodology

discussion (0)

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Works this paper leans on

113 extracted references · 33 canonical work pages

  1. [1]

    Wong, M. W. , title =. 2014 , pages =

    2014

  2. [2]

    and Tzou, Leo , title =

    Nursultanov, Medet and Tzou, Justin C. and Tzou, Leo , title =. J. Math. Pures Appl. (9) , fjournal =. 2021 , pages =

    2021

  3. [3]

    , title =

    Taylor, Michael E. , title =. 1981 , mrclass =

    1981

  4. [4]

    , title =

    Taylor, Michael E. , title =. 2023 , pages =

    2023

  5. [5]

    , edition =

    Friedlander, Friedrick Gerard and Joshi, M. , edition =. Introduction to the theory of distributions , year =

  6. [6]

    , title =

    Lee, John M. , title =. 2012 , pages =

    2012

  7. [7]

    , title =

    Lee, John M. , title =. 2018 , pages =

    2018

  8. [8]

    Estimates for translation invariant operators in

    H\". Estimates for translation invariant operators in. Acta Math. , fjournal =. 1960 , pages =

    1960

  9. [9]

    Differential Integral Equations , fjournal =

    Nursultanov, Medet and Trad, William and Tzou, Justin and Tzou, Leo , title =. Differential Integral Equations , fjournal =. 2023 , number =

    2023

  10. [10]

    Efficient Random Walks on

    Schwarz, Simon and Herrmann, Michael and Sturm, Anja and Wardetzky, Max , year =. Efficient Random Walks on. doi:10.1007/s10208-023-09635-6 , journal =

    work page doi:10.1007/s10208-023-09635-6

  11. [11]

    2024 , eprint =

    Geometric inverse problems on gas giants , author =. 2024 , eprint =

    2024

  12. [12]

    1991 , issn =

    Einstein metrics with prescribed conformal infinity on the ball , journal =. 1991 , issn =. doi:https://doi.org/10.1016/0001-8708(91)90071-E , author =

    work page doi:10.1016/0001-8708(91)90071-e 1991

  13. [13]

    , title =

    Evans, Lawrence C. , title =. 2010 , pages =

    2010

  14. [14]

    and Schuss, Z

    Holcman, D. and Schuss, Z. , title =. SIAM Rev. , fjournal =. 2014 , number =

    2014

  15. [15]

    Singer, Amit and Schuss, Zeev and Holcman, David , title =. J. Stat. Phys. , fjournal =. 2006 , number =

    2006

  16. [16]

    Holcman, David and Schuss, Zeev , title =. J. Statist. Phys. , fjournal =. 2004 , number =

    2004

  17. [17]

    Proceedings of the National Academy of Sciences , volume =

    Zeev Schuss and Amit Singer and David Holcman , title =. Proceedings of the National Academy of Sciences , volume =. 2007 , doi =

    2007

  18. [18]

    and Ward, Michael J

    Cheviakov, Alexei F. and Ward, Michael J. and Straube, Ronny , title =. Multiscale Model. Simul. , fjournal =. 2010 , number =

    2010

  19. [19]

    2015 , pages =

    Holcman, David and Schuss, Zeev , title =. 2015 , pages =

    2015

  20. [20]

    , title =

    Bressloff, Paul C. , title =. 2021 , pages =

    2021

  21. [21]

    Narrow-Escape Time Problem:

    B. Narrow-Escape Time Problem:. Physical review letters , volume =. 2008 , publisher =

    2008

  22. [22]

    Singer, Amit and Schuss, Zeev and Holcman, David , title =. Phys. Rev. E (3) , fjournal =. 2008 , number =

    2008

  23. [23]

    2012 , issn =

    Layer potential techniques for the narrow escape problem , journal =. 2012 , issn =. doi:10.1016/j.matpur.2011.09.011 , author =

    work page doi:10.1016/j.matpur.2011.09.011 2012

  24. [24]

    Shiozawa, Yuichi , title =. Proc. Japan Acad. Ser. A Math. Sci. , fjournal =. 2017 , number =

    2017

  25. [25]

    Cammarota, Valentina and De Gregorio, Alessandro and Macci, Claudio , title =. J. Stat. Phys. , fjournal =. 2014 , number =

    2014

  26. [26]

    Theory Probab

    Cammarota, Valentina and Orsingher, Enzo , title =. Theory Probab. Appl. , fjournal =. 2013 , number =

    2013

  27. [27]

    and Vasiliev, V.B , journal =

    Gertsenshtein, M.E. and Vasiliev, V.B , journal =. Waveguides with Random Inhomogeneities and. 1959 , month =

    1959

  28. [28]

    1982 , pages =

    Aubin, Thierry , title =. 1982 , pages =

    1982

  29. [29]

    2014 , howpublished =

    2014

  30. [30]

    , title =

    Gruet, J.-C. , title =. Stochastics Stochastics Rep. , fjournal =. 1996 , number =

    1996

  31. [31]

    Stochastic Process

    D'Ovidio, Mirko and Orsingher, Enzo , title =. Stochastic Process. Appl. , fjournal =. 2011 , number =

    2011

  32. [32]

    , title =

    Matsumoto, H. , title =. Colloq. Math. , fjournal =. 2010 , number =

    2010

  33. [33]

    Grigor yan, Alexander , title =. Bull. Amer. Math. Soc. (N.S.) , fjournal =. 1999 , number =

    1999

  34. [34]

    Comtet, Alain and Monthus, C\'ecile , title =. J. Phys. A , fjournal =. 1996 , number =

    1996

  35. [35]

    2007 , pages =

    H\"ormander, Lars , title =. 2007 , pages =

    2007

  36. [36]

    2009 , pages =

    H\"ormander, Lars , title =. 2009 , pages =. doi:10.1007/978-3-642-00136-9 , url =

    work page doi:10.1007/978-3-642-00136-9 2009

  37. [37]

    Pestov, Leonid and Uhlmann, Gunther , title =. Ann. of Math. (2) , fjournal =. 2005 , number =

    2005

  38. [38]

    Pigola, Stefano and Veronelli, Giona , title =. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , fjournal =. 2020 , number =

    2020

  39. [39]

    Shubin, M. A. , title =. 2001 , pages =

    2001

  40. [40]

    1986 , pages =

    Grubb, Gerd , title =. 1986 , pages =

    1986

  41. [41]

    1980 , pages =

    Tr\`eves, Fran cois , title =. 1980 , pages =

    1980

  42. [42]

    1975 , pages =

    Tr\`eves, Fran cois , title =. 1975 , pages =

    1975

  43. [43]

    Kinnunen, Juha and Martio, Olli , title =. Math. Res. Lett. , fjournal =. 1997 , number =

    1997

  44. [44]

    , title =

    Agranovich, Mikhail S. , title =. 2015 , pages =

    2015

  45. [45]

    and Douglis, A

    Agmon, S. and Douglis, A. and Nirenberg, L. , title =. Comm. Pure Appl. Math. , fjournal =. 1959 , pages =

    1959

  46. [46]

    Duke Math

    Guillarmou, Colin , title =. Duke Math. J. , fjournal =. 2005 , number =. doi:10.1215/S0012-7094-04-12911-2 , url =

    work page doi:10.1215/s0012-7094-04-12911-2 2005

  47. [47]

    and Melrose, Richard B

    Mazzeo, Rafe R. and Melrose, Richard B. , title =. J. Funct. Anal. , fjournal =. 1987 , number =. doi:10.1016/0022-1236(87)90097-8 , url =

    work page doi:10.1016/0022-1236(87)90097-8 1987

  48. [48]

    2000 , pages =

    McLean, William , title =. 2000 , pages =

    2000

  49. [49]

    2024 , eprint =

    Weyl formulae for some singular metrics with application to acoustic modes in gas giants , author =. 2024 , eprint =

    2024

  50. [50]

    Indiana Univ

    Mazzeo, Rafe and Vertman, Boris , title =. Indiana Univ. Math. J. , fjournal =. 2014 , number =. doi:10.1512/iumj.2014.63.5435 , url =

    work page doi:10.1512/iumj.2014.63.5435 2014

  51. [51]

    Mazzeo, Rafe , title =. Comm. Partial Differential Equations , fjournal =. 1991 , number =. doi:10.1080/03605309108820815 , url =

    work page doi:10.1080/03605309108820815 1991

  52. [52]

    Borthwick, David and Perry, Peter , title =. Trans. Amer. Math. Soc. , fjournal =. 2002 , number =. doi:10.1090/S0002-9947-01-02906-3 , url =

    work page doi:10.1090/s0002-9947-01-02906-3 2002

  53. [53]

    Robin and Zworski, Maciej , title =

    Graham, C. Robin and Zworski, Maciej , title =. Invent. Math. , fjournal =. 2003 , number =. doi:10.1007/s00222-002-0268-1 , url =

    work page doi:10.1007/s00222-002-0268-1 2003

  54. [54]

    Guillop\'e, Laurent and Zworski, Maciej , title =. Ann. of Math. (2) , fjournal =. 1997 , number =. doi:10.2307/2951846 , url =

    work page doi:10.2307/2951846 1997

  55. [55]

    and S\'a

    Joshi, Mark S. and S\'a. Inverse scattering on asymptotically hyperbolic manifolds , journal =. 2000 , number =. doi:10.1007/BF02392781 , url =

    work page doi:10.1007/bf02392781 2000

  56. [56]

    Tully, Kevin , title =. J. Geom. Anal. , fjournal =. 2024 , number =. doi:10.1007/s12220-024-01813-4 , url =

    work page doi:10.1007/s12220-024-01813-4 2024

  57. [57]

    Chaubet, Yann and Guedes Bonthonneau, Yannick and Lefeuvre, Thibault and Tzou, Leo , title =. Probab. Theory Related Fields , fjournal =. 2025 , number =. doi:10.1007/s00440-024-01327-8 , url =

    work page doi:10.1007/s00440-024-01327-8 2025

  58. [58]

    Holman, Sean and Uhlmann, Gunther , title =. J. Differential Geom. , fjournal =. 2018 , number =. doi:10.4310/jdg/1519959623 , url =

    work page doi:10.4310/jdg/1519959623 2018

  59. [59]

    , title =

    Besse, Arthur L. , title =. 1978 , pages =

    1978

  60. [60]

    Zoll, Otto , title =. Math. Ann. , fjournal =. 1903 , number =. doi:10.1007/BF01449019 , url =

    work page doi:10.1007/bf01449019 1903

  61. [61]

    Zoll manifolds and complex surfaces , journal =

    Le. Zoll manifolds and complex surfaces , journal =. 2002 , number =

    2002

  62. [62]

    Advances in Math

    Guillemin, Victor , title =. Advances in Math. , fjournal =. 1976 , number =

    1976

  63. [63]

    , title =

    Warner, Frank W. , title =. Amer. J. Math. , fjournal =. 1965 , pages =

    1965

  64. [64]

    Warner, F. W. , title =. Ann. of Math. (2) , fjournal =. 1967 , pages =. doi:10.2307/1970366 , url =

    work page doi:10.2307/1970366 1967

  65. [65]

    and Schaeffer, D

    Guillemin, V. and Schaeffer, D. , title =. Differential geometry (. 1975 , mrclass =

    1975

  66. [66]

    Singularities of differentiable maps

    Arnol. Singularities of differentiable maps. 1985 , pages =. doi:10.1007/978-1-4612-5154-5 , url =

    work page doi:10.1007/978-1-4612-5154-5 1985

  67. [67]

    Stefanov, Plamen and Uhlmann, Gunther , title =. Anal. PDE , fjournal =. 2012 , number =

    2012

  68. [68]

    The geodesic ray transform on

    Monard, Fran. The geodesic ray transform on. Comm. Math. Phys. , fjournal =. 2015 , number =

    2015

  69. [69]

    and Guillemin, V

    Golubitsky, M. and Guillemin, V. , title =. 1973 , pages =

    1973

  70. [70]

    Duke Math

    Myers, Sumner Byron , title =. Duke Math. J. , fjournal =. 1935 , number =

    1935

  71. [71]

    Poincar\'e, Henri , title =. Trans. Amer. Math. Soc. , fjournal =. 1905 , number =

    1905

  72. [72]

    , title =

    Weinstein, Alan D. , title =. Ann. of Math. (2) , fjournal =. 1968 , pages =

    1968

  73. [73]

    , title =

    Mather, John N. , title =. Ann. of Math. (2) , fjournal =. 1969 , pages =. doi:10.2307/1970668 , url =

    work page doi:10.2307/1970668 1969

  74. [74]

    Duistermaat, J. J. , title =. 2011 , pages =. doi:10.1007/978-0-8176-8108-1 , url =

    work page doi:10.1007/978-0-8176-8108-1 2011

  75. [75]

    Greenleaf, Allan and Seeger, Andreas , title =. Amer. J. Math. , fjournal =. 1998 , number =

    1998

  76. [76]

    Chaubet, Yann , title =. in. 2025 , pages =

    2025

  77. [78]

    Singer, I. M. and Thorpe, John A. , title =. 1967 , pages =

    1967

  78. [79]

    Microlocal analysis for differential operators , series =

    Grigis, Alain and Sj\". Microlocal analysis for differential operators , series =. 1994 , pages =. doi:10.1017/CBO9780511721441 , url =

    work page doi:10.1017/cbo9780511721441 1994

  79. [80]

    2015 , publisher =

    The analysis of linear partial differential operators I: Distribution theory and Fourier analysis , author =. 2015 , publisher =

    2015

  80. [81]

    Annals of Mathematics , volume =

    On the Lusternik-Schnirelmann category , author =. Annals of Mathematics , volume =. 1941 , publisher =

    1941

Showing first 80 references.