pith. sign in

arxiv: 2604.21582 · v1 · submitted 2026-04-23 · 🧮 math.SP · math-ph· math.DG· math.DS· math.MP

Quantum Mixing for Schr\"odinger eigenfunctions in Benjamini-Schramm limit

Kai Hippi , F\'elix Lequen , S{\o}ren Mikkelsen , Tuomas Sahlsten , Henrik Uebersch\"ar This is my paper

Pith reviewed 2026-05-08 12:41 UTC · model grok-4.3

classification 🧮 math.SP math-phmath.DGmath.DSmath.MP
keywords quantum mixingSchrödinger eigenfunctionsBenjamini-Schramm convergencehyperbolic surfacesspectral gapgeodesic flowDuhamel formulaquantum ergodicity
0
0 comments X

The pith

Eigenfunctions of Schrödinger operators on Benjamini-Schramm converging hyperbolic surfaces exhibit quantum mixing in large spectral windows.

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

The paper proves that eigenfunctions of the Schrödinger operator minus the Laplacian plus a potential on sequences of compact hyperbolic surfaces display quantum mixing when the surfaces Benjamini-Schramm converge to the hyperbolic plane. This holds in any sufficiently large spectral window provided the sequence has a uniform spectral gap and the geodesic flow mixes exponentially. The potential on the surfaces is induced from a fixed potential on the plane in suitable integrability classes. A reader would care because this gives a way to transfer mixing properties from the infinite hyperbolic plane to approximating finite surfaces, covering cases like arithmetic covers, random surfaces, and approximations to many-body quantum systems.

Core claim

Let −Δ_H + V be the Schrödinger operator on the hyperbolic plane H where V belongs to L^p intersect L^infty for some p>0. If (X_n) is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to H, we prove quantum mixing for the eigenfunctions of −Δ_{X_n} + V_n in any sufficiently large spectral window I, where V_n is the induced potential on X_n. The proof relies on the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on the unit tangent bundle of X_n.

What carries the argument

The Duhamel formula applied to the hyperbolic wave equation combined with the exponential mixing property of the geodesic flow on T^1 X_n, which transfers mixing from the flow to the quantum evolution of eigenfunctions.

Load-bearing premise

The surfaces in the sequence must maintain a uniform spectral gap and their geodesic flows must mix exponentially fast.

What would settle it

A counterexample would be a sequence of hyperbolic surfaces satisfying the Benjamini-Schramm convergence and uniform gap but where the matrix elements of the eigenfunctions in a large spectral window fail to decay according to the mixing rate predicted by the geodesic flow.

read the original abstract

Let $-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger operator on $\mathbb{H}$ where $V \in L^p(\mathbb{H}) \cap L^\infty(\mathbb{H})$ for some $p > 0$. If $(X_n)$ is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to $\mathbb{H}$, we prove quantum mixing for the eigenfunctions of $-\Delta_{X_n}+V_n$ in any sufficiently large spectral window $I$, where $V_n$ is the potential on $X_n$ induced by $V$. These apply to large degree lifts of a potential on a base surface such as congruence covers of arithmetic surfaces, with high probability to random hyperbolic surfaces in the Weil-Petersson model of large genus, and to Hartree one-particle operators arising in thermodynamic limit of many-body Bose gas on hyperbolic surfaces. The proof uses the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on $T^1 X_n$.

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

0 major / 3 minor

Summary. The manuscript proves quantum mixing for the eigenfunctions of the Schrödinger operators −Δ_{X_n} + V_n, where (X_n) is a uniformly discrete sequence of compact hyperbolic surfaces with uniform spectral gap that Benjamini-Schramm converges to the hyperbolic plane H, and V_n is the potential induced by a fixed V ∈ L^p(H) ∩ L^∞(H) for p > 0. The result holds in any sufficiently large spectral window I. The proof combines the Duhamel formula for the hyperbolic wave equation with exponential mixing of the geodesic flow on T¹X_n. Applications are given to large-degree lifts (including congruence covers), random surfaces in the Weil-Petersson model, and Hartree operators arising from many-body Bose gases.

Significance. If the central claim holds, the result supplies a modular extension of quantum ergodicity/mixing to perturbed operators on BS-converging sequences. It directly applies to arithmetic surfaces, high-probability random hyperbolic surfaces, and thermodynamic limits of many-body systems, thereby linking spectral theory, hyperbolic dynamics, and statistical mechanics. The reliance on established exponential mixing (from the uniform spectral gap) and BS convergence rather than new dynamical estimates makes the argument reusable across models.

minor comments (3)
  1. The abstract and introduction should explicitly define the induced potential V_n on X_n (e.g., via the covering map or pull-back) and state the precise regularity it inherits from V, since this is used to control error terms in the Duhamel expansion.
  2. Clarify the dependence of the 'sufficiently large' spectral window I on the uniform spectral gap, the L^p norm of V, and the BS convergence rate; a quantitative statement would strengthen the applications to random surfaces.
  3. In the statement of the main theorem, specify whether the quantum mixing is in the sense of matrix coefficients against continuous test functions or in a weaker averaged sense, and indicate the topology on the space of measures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external standard tools on stated assumptions

full rationale

The paper states its assumptions explicitly (uniform spectral gap, BS convergence to H, exponential mixing of the geodesic flow on T^1 X_n, and V in L^p ∩ L^∞ inducing V_n) and invokes the Duhamel formula for the hyperbolic wave equation as a standard analytic tool rather than deriving it internally. The central step transfers the assumed mixing to the Schrödinger eigenfunctions via BS convergence and error control uniform in n; no step reduces by construction to a fitted parameter, self-definition, or self-citation chain. The argument is therefore self-contained against its listed external inputs and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the uniform spectral gap assumption for the sequence, the Benjamini-Schramm convergence to H, and the exponential mixing property of the geodesic flow, all treated as given inputs from prior work.

axioms (2)
  • domain assumption The sequence of surfaces has a uniform spectral gap
    Stated as a hypothesis on (X_n) to control the spectrum in the limit.
  • domain assumption The geodesic flow on T^1 X_n has exponential mixing
    Invoked directly in the proof via the Duhamel formula.

pith-pipeline@v0.9.0 · 5520 in / 1242 out tokens · 43462 ms · 2026-05-08T12:41:26.197381+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

105 extracted references · 105 canonical work pages · 1 internal anchor

  1. [1]

    Abert, N

    M. Abert, N. Bergeron, and E. Le Masson. Eigenfunctions and random waves in the Benjamini-Schramm limit,

    work page

  2. [2]

    Aizenman and S

    M. Aizenman and S. Warzel. Extended states in a lifshitz tail regime for random schrödinger operators on trees. Phys. Rev. Lett., 106:136804, Mar 2011

    work page 2011

  3. [3]

    Anantharaman

    N. Anantharaman. Entropy and the localization of eigenfunctions.Ann. of Math. (2), 168(2):435–475, 2008

    work page 2008

  4. [4]

    Anantharaman.Quantum ergodicity and delocalization of Schrödinger eigenfunctions

    N. Anantharaman.Quantum ergodicity and delocalization of Schrödinger eigenfunctions. Zurich Lectures in Ad- vanced Mathematics. European Mathematical Society (EMS), Zürich, [2022]©2022

    work page 2022

  5. [5]

    Anantharaman and E

    N. Anantharaman and E. Le Masson. Quantum ergodicity on large regular graphs.Duke Math. J., 164(4):723–765, 2015

    work page 2015

  6. [6]

    Friedman-Ramanujan functions in random hyper- bolic geometry and application to spectral gaps II

    N. Anantharaman and L. Monk. Friedman–Ramanujan functions in random hyperbolic geometry and application to spectral gaps, 2023. arXiv:2304.02678

    work page arXiv 2023

  7. [7]

    Anantharaman and L

    N. Anantharaman and L. Monk. A Möbius inversion formula to discard tangled hyperbolic surfaces, 2024. arXiv:2401.01601

    work page arXiv 2024

  8. [8]

    Spectral gap of random hyperbolic surfaces

    N. Anantharaman and L. Monk. Spectral gap of random hyperbolic surfaces, 2024. arXiv:2403.12576

    work page arXiv 2024

  9. [9]

    Counting curves, and the stable length of currents

    N. Anantharaman and L. Monk. Friedman–Ramanujan functions in random hyperbolic geometry and application to spectral gaps II, 2025. arXiv:2502.12268

    work page arXiv 2025

  10. [10]

    Anantharaman and S

    N. Anantharaman and S. Nonnenmacher. Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold.Ann. Inst. Fourier (Grenoble), 57(7):2465–2523, 2007. Festival Yves Colin de Verdière

    work page 2007

  11. [11]

    Anantharaman and M

    N. Anantharaman and M. Sabri. Quantum ergodicity for the Anderson model on regular graphs.J. Math. Phys., 58(9):091901, 10, 2017

    work page 2017

  12. [12]

    Anantharaman and M

    N. Anantharaman and M. Sabri. Quantum ergodicity on graphs: from spectral to spatial delocalization.Ann. of Math. (2), 189(3):753–835, 2019

    work page 2019

  13. [13]

    Anantharaman and M

    N. Anantharaman and M. Sabri. Recent results of quantum ergodicity on graphs and further investigation.Ann. Fac. Sci. Toulouse Math. (6), 28(3):559–592, 2019

    work page 2019

  14. [14]

    P. W. Anderson. Absence of diffusion in certain random lattices.Phys. Rev., 109:1492–1505, Mar 1958

    work page 1958

  15. [15]

    Anker and V

    J.-P. Anker and V. Pierfelice. Nonlinear Schrödinger equation on real hyperbolic spaces.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 26(5):1853–1869, 2009

    work page 2009

  16. [16]

    Anker and V

    J.-P. Anker and V. Pierfelice. Wave and Klein-Gordon equations on hyperbolic spaces.Anal. PDE, 7(4):953–995, 2014

    work page 2014

  17. [17]

    Anker, V

    J.-P. Anker, V. Pierfelice, and M. Vallarino. The wave equation on hyperbolic spaces.J. Differential Equations, 252(10):5613–5661, 2012

    work page 2012

  18. [18]

    Anker, V

    J.-P. Anker, V. Pierfelice, and M. Vallarino. The wave equation on Damek-Ricci spaces.Ann. Mat. Pura Appl. (4), 194(3):731–758, 2015

    work page 2015

  19. [19]

    Bauerschmidt, J

    R. Bauerschmidt, J. Huang, and H.-T. Yau. Local Kesten-McKay law for random regular graphs.Comm. Math. Phys., 369(2):523–636, 2019

    work page 2019

  20. [20]

    Bauerschmidt, A

    R. Bauerschmidt, A. Knowles, and H.-T. Yau. Local semicircle law for random regular graphs.Comm. Pure Appl. Math., 70(10):1898–1960, 2017

    work page 1960

  21. [21]

    M. Beceanu. Dispersive estimates for Schrödinger’s and wave equations on Riemannian manifolds, 2025. arXiv:2501.06957

    work page arXiv 2025

  22. [22]

    Bordenave and B

    C. Bordenave and B. Collins. Eigenvalues of random lifts and polynomials of random permutation matrices.Ann. of Math. (2), 190(3):811–875, 2019. 34 KAI HIPPI, FÉLIX LEQUEN, SØREN MIKKELSEN, TUOMAS SAHLSTEN, AND HENRIK UEBERSCHÄR

    work page 2019

  23. [23]

    Bordenave, C

    C. Bordenave, C. Letrouit, and M. Sabri. Quantum mixing on large Schreier graphs, 2026. arXiv:2601.14182

    work page arXiv 2026

  24. [24]

    Bourgain and C

    J. Bourgain and C. E. Kenig. On localization in the continuous Anderson-Bernoulli model in higher dimension. Invent. Math., 161(2):389–426, 2005

    work page 2005

  25. [25]

    Breteaux

    S. Breteaux. A geometric derivation of the linear Boltzmann equation for a particle interacting with a Gaussian random field, using a Fock space approach.Ann. Inst. Fourier (Grenoble), 64(3):1031–1076, 2014

    work page 2014

  26. [26]

    Brooks, E

    S. Brooks, E. Le Masson, and E. Lindenstrauss. Quantum ergodicity and averaging operators on the sphere, 2016

    work page 2016

  27. [27]

    Brooks and E

    S. Brooks and E. Lindenstrauss. Non-localization of eigenfunctions on large regular graphs.Israel J. Math., 193(1):1–14, 2013

    work page 2013

  28. [28]

    Brumley, S

    F. Brumley, S. Marshall, J. Matz, and C. Peterson. Quantum ergodicity in the Benjamini–Schramm limit for locally symmetric spaces, 2026. arXiv:2604.01075

    work page arXiv 2026

  29. [29]

    Brumley and J

    F. Brumley and J. Matz. Quantum ergodicity for compact quotients ofSLd(R)/SO(d)in the Benjamini-Schramm limit.J. Inst. Math. Jussieu, 22(5):2075–2115, 2023

    work page 2075

  30. [30]

    C.-F. Chen, J. Garza-Vargas, J. A. Tropp, and R. van Handel. A new approach to strong convergence.Ann. of Math. (2), 203(2):555–602, 2026

    work page 2026

  31. [31]

    X. Chen. Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds III: Global-in-time Strichartz estimates without loss.Ann. Inst. H. Poincaré C Anal. Non Linéaire, 35(3):803–829, 2018

    work page 2018

  32. [32]

    Cipolloni, L

    G. Cipolloni, L. Erdős, and D. Schröder. Eigenstate thermalization hypothesis for Wigner matrices.Comm. Math. Phys., 388(2):1005–1048, 2021

    work page 2021

  33. [33]

    Colin de Verdière

    Y. Colin de Verdière. Pseudo-laplaciens. I.Ann. Inst. Fourier (Grenoble), 32(3):xiii, 275–286, 1982

    work page 1982

  34. [34]

    Colin de Verdière

    Y. Colin de Verdière. Ergodicité et fonctions propres du laplacien.Comm. Math. Phys., 102(3):497–502, 1985

    work page 1985

  35. [35]

    J. M. Deutsch. Eigenstate thermalization hypothesis.Rep. Prog. Phys., 81(8):082001, 2018

    work page 2018

  36. [36]

    S. Dyatlov. Around quantum ergodicity.Ann. Math. Qué., 46(1):11–26, 2022

    work page 2022

  37. [37]

    Dyatlov and L

    S. Dyatlov and L. Jin. Semiclassical measures on hyperbolic surfaces have full support.Acta Math., 220(2):297–339, 2018

    work page 2018

  38. [38]

    Dyatlov and J

    S. Dyatlov and J. Zahl. Spectral gaps, additive energy, and a fractal uncertainty principle.Geom. Funct. Anal., 26(4):1011–1094, 2016

    work page 2016

  39. [39]

    Eng and L

    D. Eng and L. Erdős. The linear Boltzmann equation as the low density limit of a random Schrödinger equation. Rev. Math. Phys., 17(6):669–743, 2005

    work page 2005

  40. [40]

    Erdős, A

    L. Erdős, A. Knowles, H.-T. Yau, and J. Yin. Spectral statistics of Erdős-Rényi graphs I: Local semicircle law. Ann. Probab., 41(3B):2279–2375, 2013

    work page 2013

  41. [41]

    Erdős, B

    L. Erdős, B. Schlein, and H.-T. Yau. Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys., 287(2):641–655, 2009

    work page 2009

  42. [42]

    M. B. Erdoğan, M. Goldberg, and W. Schlag. Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials inR3.J. Eur. Math. Soc. (JEMS), 10(2):507–531, 2008

    work page 2008

  43. [43]

    Erdős, M

    L. Erdős, M. Salmhofer, and H.-T. Yau. Quantum diffusion for the Anderson model in the scaling limit.Ann. Henri Poincaré, 8(4):621–685, 2007

    work page 2007

  44. [44]

    Erdős and H.-T

    L. Erdős and H.-T. Yau. Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Comm. Pure Appl. Math., 53(6):667–735, 2000

    work page 2000

  45. [45]

    Rep., 103(1-4):9–25, 1984.Common trends in particle and condensed matter physics (Les Houches, 1983)

    J.FröhlichandT.Spencer.ArigorousapproachtoAndersonlocalization.Phys. Rep., 103(1-4):9–25, 1984.Common trends in particle and condensed matter physics (Les Houches, 1983)

    work page 1984

  46. [46]

    Gérard and É

    P. Gérard and É. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem.Duke Math. J., 71(2):559–607, 1993

    work page 1993

  47. [47]

    Germinet, P

    F. Germinet, P. Hislop, and A Klein. On localization for the Schrödinger operator with a Poisson random potential. C. R. Math. Acad. Sci. Paris, 341(8):525–528, 2005

    work page 2005

  48. [48]

    Gilmore, E

    C. Gilmore, E. Le Masson, T. Sahlsten, and J. Thomas. Short geodesic loops andLp norms of eigenfunctions on large genus random surfaces.Geom. Funct. Anal., 31(1):62–110, 2021

    work page 2021

  49. [49]

    Goldberg

    M. Goldberg. Strichartz estimates for the Schrödinger equation with time-periodicLn/2 potentials.J. Funct. Anal., 256(3):718–746, 2009

    work page 2009

  50. [50]

    I. Ja. Gol’dše˘id, S. A. Molčanov, and L. A. Pastur. A random homogeneous Schrödinger operator has a pure point spectrum.Funkcional. Anal. i Priložen., 11(1):1–10, 96, 1977

    work page 1977

  51. [51]

    A. Hassani. Wave equation on Riemannian symmetric spaces.J. Math. Phys., 52(4):043514, 15, 2011

    work page 2011

  52. [52]

    Hezari and G

    H. Hezari and G. Rivière. Quantitative equidistribution properties of toral eigenfunctions.J. Spectr. Theory, 7(2):471–485, 2017

    work page 2017

  53. [53]

    W. Hide, D. Macera, and J. Thomas. Spectral gap with polynomial rate for random covering surfaces, 2025. arXiv:2505.08479. QUANTUM MIXING FOR SCHRÖDINGER EIGENFUNCTIONS IN BENJAMINI-SCHRAMM LIMIT 35

    work page arXiv 2025

  54. [54]

    W. Hide, D. Macera, and J. Thomas. Spectral gap with polynomial rate for Weil-Petersson random surfaces, 2025. arXiv:2508.14874

    work page arXiv 2025

  55. [55]

    Hide and M

    W. Hide and M. Magee. Near optimal spectral gaps for hyperbolic surfaces.Ann. of Math. (2), 198(2):791–824, 2023

    work page 2023

  56. [56]

    K. Hippi. Quantum mixing and benjamini-schramm convergence of hyperbolic surfaces, 2025. arXiv:2512.15504

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  57. [57]

    A. D. Ionescu. Fourier integral operators on noncompact symmetric spaces of real rank one.J. Funct. Anal., 174(2):274–300, 2000

    work page 2000

  58. [58]

    Journé, A

    J.-L. Journé, A. Soffer, and C. D. Sogge. Decay estimates for Schrödinger operators.Comm. Pure Appl. Math., 44(5):573–604, 1991

    work page 1991

  59. [59]

    M. G. Katz, M. Schaps, and U. Vishne. Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups.J. Differential Geom., 76(3):399–422, 2007

    work page 2007

  60. [60]

    J. P. Keating, N. Linden, and H. J. Wells. Spectra and eigenstates of spin chain Hamiltonians.Comm. Math. Phys., 338(1):81–102, 2015

    work page 2015

  61. [61]

    J. P. Keating and H. Ueberschär. Multifractal eigenfunctions for a singular quantum billiard.Comm. Math. Phys., 389(1):543–569, 2022

    work page 2022

  62. [62]

    Kim and Z

    E. Kim and Z. Tao. Eigenvalue rigidity of hyperbolic surfaces in the random cover model, 2026. arXiv:2603.01127

    work page arXiv 2026

  63. [63]

    A. Klein. Extended states in the Anderson model on the Bethe lattice.Adv. Math., 133(1):163–184, 1998

    work page 1998

  64. [64]

    Kurlberg and H

    P. Kurlberg and H. Ueberschär. Quantum ergodicity for point scatterers on arithmetic tori.Geom. Funct. Anal., 24(5):1565–1590, 2014

    work page 2014

  65. [65]

    Kurlberg and H

    P. Kurlberg and H. Ueberschär. Superscars in the šeba billiard.J. Eur. Math. Soc. (JEMS), 19(10):2947–2964, 2017

    work page 2017

  66. [66]

    V. F. Lazutkin.KAM theory and semiclassical approximations to eigenfunctions, volume 24 ofErgebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin,

    work page

  67. [67]

    With an addendum by A. I. Shnirelman

    work page

  68. [68]

    Le Masson and T

    E. Le Masson and T. Sahlsten. Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces. Duke Math. J., 166(18):3425–3460, 2017

    work page 2017

  69. [69]

    Le Masson and T

    E. Le Masson and T. Sahlsten. Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus. Math. Ann., 389(1):845–898, 2024

    work page 2024

  70. [70]

    Lemm and O

    M. Lemm and O. Siebert. Bose-Einstein condensation on hyperbolic spaces.J. Math. Phys., 63(8):Paper No. 081903, 18, 2022

    work page 2022

  71. [71]

    E.Lindenstrauss.Invariantmeasuresandarithmeticquantumuniqueergodicity.Ann. of Math. (2), 163(1):165–219, 2006

    work page 2006

  72. [72]

    Magee, F

    M. Magee, F. Naud, and D. Puder. A random cover of a compact hyperbolic surface has relative spectral gap 3 16 −ε.Geom. Funct. Anal., 32(3):595–661, 2022

    work page 2022

  73. [73]

    Magee, D

    M. Magee, D. Puder, and R. van Handel. Strong convergence of uniformly random permutation representations of surface groups, 2025. arXiv:2504.08988

    work page arXiv 2025

  74. [74]

    Magee, J

    M. Magee, J. Thomas, and Y. Zhao. Quantum unique ergodicity for Cayley graphs of quasirandom groups.Comm. Math. Phys., 402(3):3021–3044, 2023

    work page 2023

  75. [75]

    J. Marklof. Selberg’s trace formula: an introduction. InHyperbolic geometry and applications in quantum chaos and cosmology, volume 397 ofLondon Math. Soc. Lecture Note Ser., pages 83–119. Cambridge Univ. Press, Cambridge, 2012

    work page 2012

  76. [76]

    C. Matheus. Some quantitative versions of Ratner’s mixing estimates.Bull. Braz. Math. Soc. (N.S.), 44(3):469–488, 2013

    work page 2013

  77. [77]

    McKenzie

    T. McKenzie. The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph.C. R. Math. Acad. Sci. Paris, 360:399–408, 2022

    work page 2022

  78. [78]

    Mikkelsen

    S. Mikkelsen. Schrödinger evolution in a low-density random potential: annealed convergence to the linear boltz- mann equation for general semiclassical wigner measures, 2023. arXiv:2303.05176

    work page arXiv 2023

  79. [79]

    Mirzakhani

    M. Mirzakhani. Growth of Weil-Petersson volumes and random hyperbolic surfaces of large genus.J. Differential Geom., 94(2):267–300, 2013

    work page 2013

  80. [80]

    Monk.Geometry and Spectrum of Typical Hyperbolic Surfaces

    L. Monk.Geometry and Spectrum of Typical Hyperbolic Surfaces. PhD thesis, Université de Strasbourg, Strasbourg, France, 2021. Available athttps://lauramonk.github.io/thesis.pdf

    work page 2021

Showing first 80 references.