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Determinants of Laplacians on random hyperbolic surfaces

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Abstract

For sequences (Xj) of random closed hyperbolic surfaces with volume Vol(Xj) tending to infinity, we prove that there exists a universal constant E > 0 such that for all ϵ > 0, the regularized determinant of the Laplacian satisfies

$${{\log \det ({\Delta _{{X_j}}})} \over {{\rm{Vol}}({X_j})}} \in [E -\epsilon ,E +\epsilon]$$

with high probability as j → +⋡. This result holds for various models of random surfaces, including the Weil–Petersson model.

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References

  1. N. Anantharaman and L. Monk, Friedman–Ramanujan functions in random hyperbolic geometry and application to spectral gaps, arXiv:2304.02678 [math.SP].

  2. N. Bergeron, W. Lück and R. Sauer, The asymptotic growth of twisted torsion, unpublished notes.

  3. N. Bergeron and A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013), 391–447.

    Article  MathSciNet  Google Scholar 

  4. B. Bollobás, Random Graphs, Cambridge University Press, Cambridge, 2001.

    Book  Google Scholar 

  5. J. Bolte and F. Steiner, Determinants of Laplace-like operators on Riemann surfaces, Comm. Math. Phys. 130 (1990), 581–597.

    Article  MathSciNet  Google Scholar 

  6. D. Borthwick, C. Judge and P. Perry, Determinants of Laplacians and isopolar metrics on surfaces of infinite area, Duke Math. J. 118 (2003), 61–102.

    Article  MathSciNet  Google Scholar 

  7. R. Brooks, The spectral geometry of a tower of coverings, J. Differential Geom. 23 (1986), 97–107.

    Article  MathSciNet  Google Scholar 

  8. R. Brooks and E. Makover, Belyi surfaces. in Entire Functions in Modern Analysis (Tel-Aviv, 1997), Bar-Ilan University, Ramat Gan, 2001, pp. 37–46.

    Google Scholar 

  9. R. Brooks and E. Makover, Random construction of Riemann surfaces, J. Differential Geom. 68 (2004), 121–157.

    Article  MathSciNet  Google Scholar 

  10. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser, Boston, MA, 2010.

    Book  Google Scholar 

  11. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Orlando, FL, 1984.

    Google Scholar 

  12. J. Cheeger, Analytic torsion and the heat equation. Ann. of Math. (2) 109 (1979), 259–322.

    Article  MathSciNet  Google Scholar 

  13. E. D’Hoker and D. H. Phong, On determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 104 (1986), 537–545.

    Article  MathSciNet  Google Scholar 

  14. E. D’Hoker and D. H. Phong, The geometry of string perturbation theory, Rev. Modern Phys. 60 (1988), 917–1065.

    Article  MathSciNet  Google Scholar 

  15. I. Efrat, Determinants of Laplacians on surfaces of finite volume, Comm. Math. Phys. 119 (1988), 434–451.

    Article  MathSciNet  Google Scholar 

  16. K. Fedosova and A. Pohl, Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy, Selecta Math. (N.S.) 26 (2020), Article no. 9.

  17. D. A. Hejhal, The Selberg Trace Formula for PSL(2, R).Vol. I, Springer, Berlin-New York, 1976.

    Book  Google Scholar 

  18. D. Jakobson and F. Naud, On the critical line of convex co-compact hyperbolic surfaces, Geom. Funct. Anal. 22 (2012), 352–368.

    Article  MathSciNet  Google Scholar 

  19. M. G. Katz, M. Schaps and U. Vishne, Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups, J. Differential Geom. 76 (2007), 399–422.

    Article  MathSciNet  Google Scholar 

  20. M. W. Liebeck and A. Shalev, Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks, J. Algebra 276 (2004), 552–601.

    Article  MathSciNet  Google Scholar 

  21. M. Magee, F. Naud and D. Puder, A random cover of a compact hyperbolic surface has relative spectral gap \({3 \over {16}} - \epsilon\), Geom. Funct. Anal. 32 (2022), 595–661.

    Article  MathSciNet  Google Scholar 

  22. M. Magee and D. Puder, The asymptotic statistics of random covering surfaces, Forum Math. Pi 11 (2023), Article no. e15

  23. D. Mangoubi, Conformal extension of metrics of negative curvature, J. Anal. Math. 91 (2003), 193–209.

    Article  MathSciNet  Google Scholar 

  24. M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), 179–222.

    Article  MathSciNet  Google Scholar 

  25. M. Mirzakhani, Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus, J. Differential Geom. 94 (2013), 267–300.

    Article  MathSciNet  Google Scholar 

  26. M. Mirzakhani and B. Petri, Lengths of closed geodesics on random surfaces of large genus, Comment. Math. Helv. 94 (2019), 869–889.

    Article  MathSciNet  Google Scholar 

  27. L. Monk, Benjamini–Schramm convergence and spectra of random hyperbolic surfaces of high genus, Anal. PDE 15 (2022), 727–752.

    Article  MathSciNet  Google Scholar 

  28. W. Muller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233–305.

    Article  MathSciNet  Google Scholar 

  29. F. Naud, Random covers of compact surfaces and smooth linear spectral statistics, arXiv:2209.07941 [math.SP].

  30. B. Osgood, R. Phillips and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.

    Article  MathSciNet  Google Scholar 

  31. J. P. Otal and E. Rosas, Pour toute surface hyperbolique de genre g, λ2g−2 > 1/4, Duke Math. J. 150 (2009), 101–115.

    Article  MathSciNet  Google Scholar 

  32. B. Petri, Random regular graphs and the systole of a random surface, J. Topol. 10 (2017), 211–267.

    Article  MathSciNet  Google Scholar 

  33. B. Petri and A. Walker, Graphs of large girth and surfaces of large systole, Math. Res. Lett. 25 (2018), 1937–1956.

    Article  MathSciNet  Google Scholar 

  34. M. Pollicott and A. C. Rocha, A remarkable formula for the determinant of the Laplacian, Invent. Math. 130 (1997), 399–414.

    Article  MathSciNet  Google Scholar 

  35. A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), 207–210.

    Article  MathSciNet  Google Scholar 

  36. P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113–120.

    Article  MathSciNet  Google Scholar 

  37. P. Sarnak and X. X. Xue, Bounds for multiplicities of automorphic representations, Duke Math. J. 64 (1991), 207–227.

    Article  MathSciNet  Google Scholar 

  38. A. Strohmaier, Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces, in Geometric and Computational Spectral Theory, American Mathematical Society, Providence, RI, 2017, pp. 177–205.

    Chapter  Google Scholar 

  39. T. Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology 28 (1989), 125–132.

    Article  MathSciNet  Google Scholar 

  40. A. B. Venkov, Spectral Theory of Automorphic Functions and its Applications, Kluwer, Dordrecht, 1990.

    Book  Google Scholar 

  41. S. A. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987), 283–315.

    Article  MathSciNet  Google Scholar 

  42. A. Wright and M. Lipnowski, Towards optimal spectral gap in large genus, arXiv:2103.07496 [math.GT].

  43. Y. Wu and Y. Xue, Random hyperbolic surfaces of large genus have first eigenvalues greater than \({3 \over {16}} - \epsilon\), Geom. Funct. Anal. 32 (2022), 340–410.

    Article  MathSciNet  Google Scholar 

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Acknowledgement

It is a pleasure to thank my neighbor Bram Petri for several discussions around this work. Thanks to Yuhao Xue and an anonymous referee for pointing out an improvement of Theorem 3.1. I also thank ZeevRudnick for his reading and comments. Finally, Yunhui Wu and Yuxin He have recently shown to me that Theorem 5.1 can also be refined in the Weil–Petterson case; see in §5 for details.

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  1. Institut Mathématique de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France

    Frédéric Naud

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  1. Frédéric Naud
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Correspondence to Frédéric Naud.

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Dedicated to Peter Sarnak on the occasion of his 70th birthday

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Naud, F. Determinants of Laplacians on random hyperbolic surfaces. JAMA 151, 265–291 (2023). https://doi.org/10.1007/s11854-023-0334-8

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  • DOI: https://doi.org/10.1007/s11854-023-0334-8

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