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

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

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

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

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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.

Desk Editor's Note private letter to a colleague

The paper proves quantum mixing for Schrödinger eigenfunctions on BS-converging hyperbolic surfaces with potentials, via Duhamel and geodesic flow mixing from the spectral gap.

read the letter

The paper shows quantum mixing for eigenfunctions of Schrödinger operators on sequences of hyperbolic surfaces that Benjamini-Schramm converge to the plane, provided they have a uniform spectral gap. They extend earlier results on quantum ergodicity to include a potential term and obtain mixing in spectral windows. The argument applies the Duhamel formula to the perturbed hyperbolic wave equation and uses the exponential mixing of the geodesic flow on the unit tangent bundle, which follows from the spectral gap. This setup covers lifts of potentials from base surfaces, random surfaces in the Weil-Petersson model, and Hartree operators from many-body systems. The approach looks solid on the surface. The assumptions on V in L^p cap L^infty seem chosen to control the error terms uniformly across the sequence, and the stress-test finds no internal gaps in transferring the mixing via BS convergence. A minor point to check is whether the spectral window size depends on the potential in a way that stays uniform, but that is likely handled in the details. This is for researchers in spectral geometry and mathematical physics who care about quantum limits on varying or random hyperbolic surfaces. It fills a gap between fixed-surface results and the BS limit with perturbations. The paper deserves a serious referee. The result is new enough and the method is direct enough that it should get a full review rather than a desk reject. I recommend sending it out.

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)

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.
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.
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)

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Works this paper leans on

105 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Abert, N

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

2011
[3]

Anantharaman

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

2008
[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

2022
[5]

Anantharaman and E

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

2015
[6]

Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps.arXiv:2304.02678,

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]

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]

Anantharaman and L

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

work page arXiv 2024
[9]

Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II.arXiv:2502.12268,

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]

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

2007
[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

2017
[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

2019
[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

2019
[14]

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

1958
[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

2009
[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

2014
[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

2012
[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

2015
[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

2019
[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

1960
[21]

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

work page arXiv 2025
[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

2019
[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]

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

2005
[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

2014
[26]

Brooks, E

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

2016
[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

2013
[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]

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

2075
[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

2026
[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

2018
[32]

Cipolloni, L

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

2021
[33]

Colin de Verdière

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

1982
[34]

Colin de Verdière

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

1985
[35]

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

2018
[36]

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

2022
[37]

Dyatlov and L

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

2018
[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

2016
[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

2005
[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

2013
[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

2009
[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

2008
[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

2007
[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

2000
[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)

1984
[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

1993
[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

2005
[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

2021
[49]

Goldberg

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

2009
[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

1977
[51]

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

2011
[52]

Hezari and G

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

2017
[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]

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]

Hide and M

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

2023
[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]

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

2000
[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

1991
[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

2007
[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

2015
[61]

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

2022
[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]

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

1998
[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

2014
[65]

Kurlberg and H

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

2017
[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,
[67]

With an addendum by A. I. Shnirelman
[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

2017
[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

2024
[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

2022
[71]

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

2006
[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

2022
[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]

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

2023
[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

2012
[76]

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

2013
[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

2022
[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]

Mirzakhani

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

2013
[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

2021

Showing first 80 references.