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Random hyperbolic surfaces hit near-optimal spectral gaps

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2026-07-08 09:23 UTC pith:ISUQU6ZO

load-bearing objection Well-executed Bourbaki survey of three major spectral gap results; proof sketches are coherent and the exposition is strong. Deserves a serious referee.

arxiv 2607.06331 v1 pith:ISUQU6ZO submitted 2026-07-07 math.SP math.DGmath.PR

Bass notes of random hyperbolic surfaces of large genus

Bram Petri This is my paper
classification math.SP math.DGmath.PR
keywords randomsurfaceshyperbolicgapsspectralsurveyanantharaman-monkbass
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This survey presents three recent results showing that random hyperbolic surfaces of large genus have spectral gaps approaching 1/4 — the bottom of the spectrum of the Laplacian on the hyperbolic plane — with high probability. The bass note (the smallest nonzero eigenvalue of the Laplacian) of a hyperbolic surface measures how well-connected the surface is, and 1/4 is the natural ceiling: it is the spectral gap of the hyperbolic plane itself, and no sequence of closed surfaces of growing area can do better in the limit. Buser conjectured in 1984 that sequences of closed surfaces with bass notes tending to 1/4 exist; random constructions are currently the only known way to produce them. The survey covers three lines of work. Hide and Magee proved that random finite-degree covers of a non-compact hyperbolic surface have no new spectrum below 1/4 - ε with high probability, using the Bordenave–Collins strong convergence theorem for random permutation representations of free groups as a key input. Anantharaman and Monk proved the same conclusion for surfaces chosen according to the Weil–Petersson measure on moduli space, using an analogue of Friedman's trace method from graph theory, with the Selberg trace formula replacing the adjacency-matrix trace and a new class of Friedman–Ramanujan functions replacing Friedman's Ramanujan functions. Hide, Macera, and Thomas strengthened the Weil–Petersson result to a polynomial error bound: λ₁ > 1/4 - g^{-c} for some constant c > 0.

Core claim

The central mechanism connecting all three results is an analogy between random regular graphs and random hyperbolic surfaces. For d-regular graphs, the Alon–Boppana theorem says the second eigenvalue of the adjacency matrix is at least 2√(d-1) in the limit, and Friedman's theorem (later strengthened by Bordenave–Collins) says random d-regular graphs achieve this bound with high probability. For hyperbolic surfaces, the analogous bound is 1/4 (the bottom of the spectrum of the hyperbolic plane), and the survey's three results say that random surfaces achieve near-optimal spectral gaps with high probability. The graph-to-surface dictionary maps adjacency matrix traces to Selberg trace formula

What carries the argument

The Bordenave–Collins strong convergence theorem (Theorem 4.4) is the load-bearing input for the Hide–Magee result: it states that random permutation representations of free groups converge strongly to the left regular representation. For the Anantharaman–Monk result, the key machinery is the Selberg trace formula combined with the Mirzakhani integration formula and a novel class of Friedman–Ramanujan functions that control the contribution of closed geodesics of all topological types. The Hide–Macera–Thomas improvement uses an effective asymptotic expansion of expected traces in powers of 1/g, with factorial growth control on the error constants.

Load-bearing premise

The Hide–Magee result depends on the Bordenave–Collins strong convergence theorem, which asserts that random permutation representations of free groups converge in operator norm to the left regular representation. If this convergence were to fail or hold more narrowly than stated, the parametrix construction away from cusps would not control the new spectrum, and the near-optimal gap for random covers would not follow.

What would settle it

If one could exhibit an explicit sequence of surfaces of growing genus with bass notes bounded away from 1/4, or if the strong convergence theorem were found to have narrower applicability, the claim that randomness is the route to near-optimal gaps would be weakened.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the polynomial error bound can be pushed to c = 2/3 - ε, one expects the rescaled quantity g^{2/3}(λ₁(X_g) - 1/4) to converge to a Tracy–Widom distribution, mirroring the graph-theoretic result of Huang–McKenzie–Yau for random regular graphs.
  • Sufficiently precise control of the distribution of λ₁ around 1/4 could prove that P(λ₁(X_g) > 1/4) > 0 for large g, yielding surfaces whose spectral gap strictly exceeds that of the hyperbolic plane — currently unknown for surfaces.
  • The strong convergence approach via Bordenave–Collins has been extended to other lattices and groups, suggesting that near-optimal spectral gaps for random covers may hold in broader geometric settings beyond hyperbolic surfaces.
  • The analogy between graph expanders and hyperbolic surface spectral gaps could transfer further tools: quantum ergodicity results, eigenvalue statistics matching random matrix theory, and L^p norm bounds on eigenfunctions are all beginning to be established for these random surfaces.

Where Pith is reading between the lines

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

  • The fact that no explicit (non-random) construction of surfaces with near-optimal spectral gaps is known, despite the random existence proofs, suggests a possible gap between probabilistic and constructive methods in spectral geometry — analogous to the situation for Ramanujan graphs before the work of Marcus–Spielman–Srivastava.
  • The convergence of eigenvalue statistics on random surfaces to GOE/GUE distributions (as shown in several cited works) suggests that random hyperbolic surfaces may serve as a natural geometric model for quantum chaos, with the spectral gap results providing the necessary a priori control.
  • The role of tangles in both Friedman's graph proof and Anantharaman–Monk's surface proof — small substructures that appear with vanishing but not-fast-enough probability — suggests a universal obstruction in random spectral geometry that any proof strategy must address.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. This is a Séminaire Bourbaki survey by Bram Petri on the spectral gap of random hyperbolic surfaces of large genus, covering three bodies of work: Hide–Magee (near-optimal spectral gaps for random covers of non-compact surfaces), Anantharaman–Monk (near-optimal spectral gaps for Weil–Petersson random surfaces), and Hide–Macera–Thomas (polynomial error rates for the Weil–Petersson model). The first four sections provide background on hyperbolic surface geometry, spectral theory, the bass note, Benjamini–Schramm convergence, regular graph expander theory, and the three main models of random surfaces. The final three sections sketch the proofs of the three main results. The survey is expository and makes no new mathematical claims.

Significance. The survey provides a valuable and timely service to the community by synthesizing three major and technically demanding results into a single coherent narrative. The proof sketches are well-structured: the Hide–Magee parametrix decomposition (Section 6) correctly identifies the cusp/interior split, the Hilbert–Schmidt decomposition of the interior error operator, and the role of Bordenave–Collins strong convergence; the Anantharaman–Monk sketch (Section 7) correctly identifies the three obstacles (trivial eigenvalue, logarithmic-length geodesics, tangles) and their resolutions via Friedman–Ramanujan functions and Möbius inversion; the Hide–Macera–Thomas sketch (Section 8) correctly presents the asymptotic expansion and the polynomial method deduction. The background sections on graph expanders (Section 4) and random surface models (Section 5) are well-motivated and place the main results in proper context. The exposition is grounded in established external inputs (Bordenave–Collins, Mirzakhani–Zograf, Selberg trace formula) and introduces no circular reasoning.

minor comments (6)
  1. Introduction, first paragraph: 'near optimalspectral gaps' and 'userandom surfaces' — missing spaces after 'optimal' and 'use'.
  2. Section 1.2.2, p. 6: 'the fact the the automorphism group' — repeated 'the'.
  3. Section 3, p. 19: 'Plancerel measure' should read 'Plancherel measure'.
  4. Section 6.5, p. 42: 'Proppsition 6.2' should read 'Proposition 6.2'.
  5. Section 8.1, p. 49: 'for allg large enough' — missing space before 'g'.
  6. Section 8.3, p. 52: the transition from equation (10) to the final probability bound is somewhat compressed; a brief sentence clarifying how the support condition on f and the bound on w^(m) combine would help the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

The referee recommends minor revision with a positive assessment. The report contains no major comments, only praise for the survey's structure, accuracy of proof sketches, and contextualization. We thank the referee and note that we will conduct a careful proofreading pass to address any typographical issues before the final version.

Circularity Check

0 steps flagged

No circularity found — expository survey with independent load-bearing inputs

full rationale

This is a Séminaire Bourbaki survey presenting three results (Hide–Magee Theorem 6.1, Anantharaman–Monk Theorem 7.1, Hide–Macera–Thomas Theorem 8.1) on spectral gaps of random hyperbolic surfaces. The proof sketches rely on inputs that are independent of the target results: the Bordenave–Collins strong convergence theorem (Theorem 4.4, by different authors), the Selberg trace formula (classical), Mirzakhani's integration formula (Theorem 5.1, independent), Mirzakhani–Zograf volume asymptotics (independent), and Pisier's linearization trick (2018, independent). The author does cite his own joint work (e.g., Budzinski–Curien–Petri on diameters and Cheeger constants, Mirzakhani–Petri on length spectra, Fortier Bourque–Petri on linear programming bounds, Hide–Petri on arithmetic bass notes), but these appear only in peripheral contextual roles — not as load-bearing steps in any of the three main proof sketches. No definition is circularly tied to a result it claims to derive; no parameter is fitted and then presented as a prediction; no uniqueness theorem from the author's prior work is invoked to force a conclusion. The survey is self-contained against external benchmarks in the sense that every load-bearing theorem is attributed to independent, peer-reviewed sources. This is a clean expository text with no circular structure.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The survey introduces no new parameters, axioms, or entities. All results, methods, and assumptions are attributed to external sources. The Friedman–Ramanujan functions (Definition 7.2) are due to Anantharaman–Monk, not introduced here. The tangle condition (Section 7.4) is also from Anantharaman–Monk's work.

axioms (4)
  • domain assumption Bordenave–Collins strong convergence theorem (Theorem 4.4): random permutation representations of free groups converge strongly in probability to the left regular representation.
    Invoked in Section 6.4 as the key input for the Hide–Magee parametrix construction. This is a result from Bordenave–Collins (2019), external to this survey.
  • domain assumption Mirzakhani integration formula (Theorem 5.1) and Mirzakhani–Zograf volume asymptotics.
    Used throughout Section 7 as the foundation for computing expectations in the Anantharaman–Monk trace method. External results from Mirzakhani (2007a, 2013) and Mirzakhani–Zograf (2015).
  • standard math Selberg trace formula for closed hyperbolic surfaces (Theorem 1.6).
    Classical result, used in Sections 7–8 to connect spectral data to geometric data (lengths of closed geodesics).
  • standard math Cheeger–Buser inequality (Theorem 2.1) relating the bass note to the Cheeger constant.
    Classical result used in Section 2.1 to interpret the spectral gap as a measure of connectivity and in Section 5.2 to derive Mirzakhani's spectral gap bound.

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read the original abstract

This is a S\'eminaire Bourbaki survey on spectral gaps of random hyperbolic surfaces. The first part of the text is a brief survey on the geometry and spectra of random hyperbolic surfaces. After this, we discuss the results by Hide-Magee, Anantharaman-Monk and Hide-Macera-Thomas on near optimal spectral gaps for random surfaces.

Figures

Figures reproduced from arXiv: 2607.06331 by Bram Petri.

Figure 1
Figure 1. Figure 1: A regular octagon and the side pairings that turn it into the Bolza surface Of course, the Bolza surface also admits a more explicit description. Namely, it’s the hyperbolic surface one obtains by gluing together the pairs of opposite sides of a regular hyperbolic octagon (so its sides are geodesic segments that all have the same length and its interior angles are all equal) with interior angles π 4 , as i… view at source ↗
Figure 2
Figure 2. Figure 2: Building the thrice punctured sphere To build it explicitly, we can take the ideal quadrilateral in H2 with ideal vertices −1, 0, 1 and ∞ and glue the sides together according to the pattern drawn in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Four topological types of pants We start with a building block (see [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performing a twist An Euler characteristic computation implies that a pants decomposition of an ori￾entable surface of genus g with n punctures and b boundary components contains 3g + n + b curves. This does not count the curves that form the potential bound￾ary components. In the deformation spaces of hyperbolic surfaces that we will consider here, we will always assume the boundary lengths to be fixed. T… view at source ↗
Figure 5
Figure 5. Figure 5: A surface with small Cheeger constant Its connection to the bass note is given by the following theorem due to Cheeger (1970) and Buser (1982), which in the case of hyperbolic surfaces reads: Theorem 2.1 (Cheeger–Buser inequality). — Let X be a hyperbolic surface of finite area. Then h(X) 2 4 ≤ λ1(X) ≤ 2 h(X) + 10 h(X) 2 . This inequality also implies that, whenever (g, n) ∈ { / (0, 3),(1, 1)}, then inf X∈… view at source ↗
Figure 6
Figure 6. Figure 6: 1 0 −1 ∞ [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The density of the Plancherel measure. It follows from the work of Abert, Bergeron, Biringer, Gelander, Nikolov, Raimbault, and Samet (2017) that if (Xn) is a uniformly discrete sequence of hyperbolic surfaces, then for all [a, b] ⊂ R≥0: #{k; λk(Xn) ∈ [a, b]} area(Xn) n→∞ −→ 1 4π Z b a 1[ 1 4 ,∞) (y) tanh  π s y − 1 4   dy. The measure 1 4π 1[ 1 4 ,∞) (y) tanh  π q y − 1 4  dy is the spectral density… view at source ↗
Figure 8
Figure 8. Figure 8: The 4-regular graph on [5] corresponding to the permutations (1 2 3 4)(5) and (1 4 2)(3 5) from S5. We’ve drawn arrows to indicate what the permutations do, but the graph is thought of as undirected. Other well known models are the uniform model (picking a uniformly random iso￾morphism class of d-regular graphs on n vertices) and the configuration model (taking n vertices with d half-edges sticking out and… view at source ↗
Figure 9
Figure 9. Figure 9: Gluing ideals triangles into a surface without boundary. As observed by Brooks and Makover (2001a), this model can also be described as a random cover model. To see this, we recall that as abstract groups PSL(2, Z) ≃ (Z/2Z) ∗ (Z/3Z) = ⟨τ, σ| τ 2 = σ 3 = 1⟩ where “∗” denotes a free product. As such, we can build a random cover of the mod￾ular curve PSL(2, Z)\H2 using a random homomorphism to a symmetric gro… view at source ↗
Figure 10
Figure 10. Figure 10: Gluing ideal squares together according to the permutations (1 2 3)(4)(5) and (1 5 4 2 3) in S5 yields a 5-fold cover of the thrice punctured sphere. random cover of a nice enough space X with a finitely generated fundamental group Γ = π1(X, x) can be built by choosing a homomorphism φn ∈ Hom(Γ, Sn) uniformly at random and setting Xn = Γ \ (Xf × [n]), where Xf denotes the universal cover of X and where th… view at source ↗
Figure 11
Figure 11. Figure 11: Three curves with three self-intersections each, on a surface of genus three. The leftmost two have the same local topological type, even if they sit in the surface differently. The rightmost curve fills a one-holed torus instead of a pair of pants and thus has a different local topological type. To group geodesics together, Anantharaman and Monk define the local topological type of a geodesic. This is th… view at source ↗

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Works this paper leans on

47 extracted references · 47 canonical work pages · 14 internal anchors

  1. [1]

    On the growth ofL2-invariants for sequences of lattices in Lie groups

    M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, and Iddo Samet (2017). “On the growth ofL2-invariants for sequences of lattices in Lie groups”, Ann. of Math. (2)185(3), pp. 711–790. N. Alon (1986). “Eigenvalues and expanders”, in: vol

    work page 2017

  2. [2]

    A high-genus asymptotic expansion of Weil- Petersson volume polynomials

    Theory of computing (Singer Island, Fla., 1984), pp. 83–96. N. Anantharaman and L. Monk (2022). “A high-genus asymptotic expansion of Weil- Petersson volume polynomials”,J. Math. Phys.63(4), Paper No. 043502,

    work page 1984

  3. [3]

    A Moebius inversion formula to discard tangled hyperbolic surfaces

    (2024a). “A Moebius inversion formula to discard tangled hyperbolic surfaces”. arXiv: 2401.01601 [math.GT]. (2024b). “Spectral gap of random hyperbolic surfaces”. arXiv:2403 . 12576 [math.GT]. (2025). “Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps II”. arXiv:2502.12268 [math.MG]. (2026a). “Friedman-Ramanujan f...

    work page arXiv 2025

  4. [4]

    On the spectral stability of finite coverings

    R. Arratia, A. D. Barbour, and S. Tavaré (2003).Logarithmic combinatorial structures: a probabilistic approach. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, pp. xii+363. W. Ballmann, S. Mondal, and P. Polymerakis (2025). “On the spectral stability of finite coverings”. arXiv:2507.17466 [math.DG]. S. Barazer, A. Giacchetto, a...

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  5. [5]

    Disques extrémaux et surfaces modulaires

    C. Bavard (1996). “Disques extrémaux et surfaces modulaires”,Ann. Fac. Sci. Toulouse Math. (6)5(2), pp. 191–202. M. Belolipetsky, T. Gelander, A. Lubotzky, and A. Shalev (2010). “Counting arithmetic lattices and surfaces”,Ann. of Math. (2)172(3), pp. 2197–2221. G. V. Bely˘ ı (1979). “Galois extensions of a maximal cyclotomic field”,Izv. Akad. Nauk SSSR Se...

    work page 1996

  6. [6]

    Recurrence of distributional limits of finite planar graphs

    1255–54 I. Benjamini and O. Schramm (2001). “Recurrence of distributional limits of finite planar graphs”,Electron. J. Probab.6, no. 23,

    work page 2001

  7. [7]

    Spaces of degenerating Riemann surfaces

    N. Bergeron (2016).The spectrum of hyperbolic surfaces. Universitext. Appendix C by Valentin Blomer and Farrell Brumley, Translated from the 2011 French original by Brumley. Springer, Cham; EDP Sciences, Les Ulis, pp. xiii+370. L. Bers (1974). “Spaces of degenerating Riemann surfaces”, in:Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryl...

    work page 2016

  8. [8]

    Lifts, discrepancy and nearly optimal spectral gap

    Y. Bilu and N. Linial (2006). “Lifts, discrepancy and nearly optimal spectral gap”, Combinatorica 26(5), pp. 495–519. B. Bollobás (1981). “Random graphs”, in:Combinatorics (Swansea, 1981). Vol

    work page 2006

  9. [9]

    Bootstrapping closed hyperbolic surfaces

    London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge-New York, pp. 80–102. J. Bonifacio (2022). “Bootstrapping closed hyperbolic surfaces”,J. High Energy Phys. (3), Paper No. 093,

    work page 2022

  10. [10]

    Twist-minimal trace formulas and the Selberg eigenvalue conjecture

    A. R. Booker, M. Lee, and A. Strömbergsson (2020). “Twist-minimal trace formulas and the Selberg eigenvalue conjecture”,J. Lond. Math. Soc. (2)102(3), pp. 1067–1134. A. R. Booker and A. Strömbergsson (2007). “Numerical computations with the trace formula and the Selberg eigenvalue conjecture”,J. Reine Angew. Math.607, pp. 113–

    work page 2020

  11. [11]

    A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts

    C. Bordenave (2020). “A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts”,Ann. Sci. Éc. Norm. Supér. (4)53(6), pp. 1393–1439. C. Bordenave and B. Collins (2019). “Eigenvalues of random lifts and polynomials of random permutation matrices”,Ann. of Math. (2)190(3), pp. 811–875. D. Borthwick (2016). Spectral theory of infin...

    work page 2020

  12. [12]

    Birkhäuser/Springer, [Cham], pp

    Progress in Mathematics. Birkhäuser/Springer, [Cham], pp. xiii+463. (2020). Spectral theory—basic concepts and applications. Vol

    work page 2020

  13. [13]

    Benjamini-Schramm limits of high genus translation surfaces: research announcement

    Graduate Texts in Mathematics. Springer, Cham, pp. x+338. L. Bowen, K. Rafi, and H. Vallejos (2025). “Benjamini-Schramm limits of high genus translation surfaces: research announcement”. arXiv:2501.03474 [math.GT]. E. Breuillard (2014). “Expander graphs, property(τ) and approximate groups”, in: Geometric group theory. Vol

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  14. [14]

    On the second eigenvalue of random regular graphs

    IAS/Park City Math. Ser. Amer. Math. Soc., Prov- idence, RI, pp. 325–377. A. Broder and E. Shamir (1987). “On the second eigenvalue of random regular graphs”. In: The 28th Annual Symposium on Foundations of Computer Science, pp. 286–294. R. Brooks (1985). “The bottom of the spectrum of a Riemannian covering”,J. Reine Angew. Math.357, pp. 101–114. (1986). ...

    work page 1987

  15. [15]

    Platonic surfaces

    Lecture Notes in Math. Springer, Berlin, pp. 14–32. 1255–55 (1999). “Platonic surfaces”,Comment. Math. Helv.74(1), pp. 156–170. R. Brooks and E. Makover (2001a). “Belyi surfaces”, in:Entire functions in modern analysis (Tel-Aviv, 1997). Vol

    work page 1999

  16. [16]

    Random punctured hyperbolic surfaces & the Brownian sphere

    Israel Math. Conf. Proc. Bar-Ilan Univ., Ramat Gan, pp. 37–46. (2001b). “Riemann surfaces with large first eigenvalue”,J. Anal. Math. 83, pp. 243–258. (2004). “Random construction of Riemann surfaces”,J. Differential Geom.68 (1), pp. 121–157. R. Brooks and A. Zuk (2002). “On the asymptotic isoperimetric constants for Riemann surfaces and graphs”,J. Differ...

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  17. [17]

    On the minimal diameter of closed hyperbolic surfaces

    T. Budzinski, N. Curien, and B. Petri (2021a). “On the minimal diameter of closed hyperbolic surfaces”,Duke Math. J.170(2), pp. 365–377. (2021b). “The diameter of random Bely˘ ı surfaces”,Algebr. Geom. Topol.21(6), pp. 2929–2957. (2025). “On Cheeger constants of hyperbolic surfaces”,Invent. Math.242(2), pp. 511–530. M. Burger (1986). “Grandes valeurs prop...

    work page 2025

  18. [18]

    Spectral gap for random Schottky surfaces

    Lecture Notes in Math. Springer, Berlin, pp. 54–63. 1255–56 P. Buser and P. Sarnak (1994). “On the period matrix of a Riemann surface of large genus”,Invent. Math. 117(1). With an appendix by J. H. Conway and N. J. A. Sloane, pp. 27–56. I. Calderón, M. Magee, and F. Naud (2024). “Spectral gap for random Schottky sur- faces”. arXiv:2407.21506 [math.SP]. I....

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  19. [19]

    A new approach to strong convergence II. The classical ensembles

    Pure and Applied Mathematics. Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk. Academic Press, Inc., Orlando, FL, pp. xiv+362. J. Cheeger (1970). “A lower bound for the smallest eigenvalue of the Laplacian”, in: Problems in analysis (Papers dedicated to Salomon Bochner, 1969). Princeton Univ. Press, Princeton, N. J., pp. 195–199. C...

    work page internal anchor Pith review Pith/arXiv arXiv 1970

  20. [20]

    Linear programming bounds for hyperbolic surfaces

    Princeton Mathematical Series. Princeton University Press, Princeton, NJ, pp. xiv+472. M. Fortier Bourque and B. Petri (2026). “Linear programming bounds for hyperbolic surfaces”. arXiv:2302.02540 [math.GT]. J. Friedman (1993). “Some geometric aspects of graphs and their eigenfunctions”,Duke Math. J.69(3), pp. 487–525. (2008). “A proof of Alon’s second ei...

    work page arXiv 2026

  21. [21]

    Short geodesic loops andLp norms of eigenfunctions on large genus random surfaces

    Generalized Functions. Translated from the Rus- sian by K. A. Hirsch, Reprint of the 1969 edition. Academic Press, Inc., Boston, MA, pp. xviii+426. 1255–57 C. Gilmore, E. Le Masson, T. Sahlsten, and J. Thomas (2021). “Short geodesic loops andLp norms of eigenfunctions on large genus random surfaces”,Geom. Funct. Anal. 31(1), pp. 62–110. W. M. Goldman (198...

    work page 1969

  22. [22]

    Non-simple systoles on random hyperbolic surfaces for large genus

    L. Guth, H. Parlier, and R. Young (2011). “Pants decompositions of random surfaces”, Geom. Funct. Anal.21(5), pp. 1069–1090. Y. He, Y. Shen, Y. Wu, and Y. Xue (2025). “Non-simple systoles on random hyperbolic surfaces for large genus”. arXiv:2308.16447 [math.GT]. Y. He and Y. Wu (2025). “Averages of determinants of Laplacians over moduli spaces for large ...

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  23. [23]

    Uniform spectral gaps for random hyperbolic surfaces with not many cusps

    Y. He, Y. Wu, and Y. Xue (2026). “Uniform spectral gaps for random hyperbolic surfaces with not many cusps”. arXiv:2602.08352 [math.DG]. D. A. Hejhal (1976).The Selberg trace formula forPSL(2, R). Vol. I. Vol. Vol

    work page arXiv 2026

  24. [24]

    Springer-Verlag, Berlin-New York, pp

    Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, pp. vi+516. (1983). The Selberg trace formula forPSL(2, R). Vol

    work page 1983

  25. [25]

    Effective lower bounds for spectra of random covers and random unitary bundles

    Lecture Notes in Mathematics. Springer-Verlag, Berlin, pp. viii+806. W. Hide (2023). “Spectral gap for Weil-Petersson random surfaces with cusps”,Int. Math. Res. Not.(20), pp. 17411–17460. (2024).“Effectivelowerboundsforspectraofrandomcoversandrandomunitary bundles”. arXiv:2305.04584 [math.SP]. W. Hide, D. Macera, and J. Thomas (2025a). “Spectral gap with...

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  26. [26]

    Limit points of uniform arithmetic bass notes

    W. Hide and B. Petri (2024). “Limit points of uniform arithmetic bass notes”. arXiv: 2412.15111 [math.SP]. W. Hide and J. Thomas (2025a). “Large-n asymptotics for Weil-Petersson volumes of moduli spaces of bordered hyperbolic surfaces”,Comm. Math. Phys.406(9), Paper No. 203,

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  27. [27]

    Ramanujan Property and Edge Universality of Random Regular Graphs

    (2025b). “Short geodesics and small eigenvalues on random hyperbolic punc- tured spheres”,Comment. Math. Helv.100(3), pp. 463–506. S. Hoory, N. Linial, and A. Wigderson (2006). “Expander graphs and their applica- tions”,Bull. Amer. Math. Soc. (N.S.)43(4), pp. 439–561. 1255–58 J. Huang, T. McKenzie, and H.-T. Yau (2025). “Ramanujan Property and Edge Uni- v...

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  28. [28]

    PWN, Warsaw, pp

    Banach Center Publ. PWN, Warsaw, pp. 217–306. Y.ImayoshiandM.Taniguchi(1992). An introduction to Teichmüller spaces.Translated and revised from the Japanese by the authors. Springer-Verlag, Tokyo, pp. xiv+279. H. Iwaniec (2002).Spectral methods of automorphic forms. Second. Vol

    work page 1992

  29. [29]

    Unicellular maps vs. hyperbolic surfaces in large genus: simple closed curves

    Graduate Studies in Mathematics. American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, pp. xii+220. S. Janson and B. Louf (2023). “Unicellular maps vs. hyperbolic surfaces in large genus: simple closed curves”,Ann. Probab.51(3), pp. 899–929. G. Jones and D. Singerman (1996). “Bely˘ ı functions, hypermaps and Galois grou...

    work page arXiv 2023

  30. [30]

    Automorphic spectra and the conformal bootstrap

    Cours Spécialisés [Specialized Courses]. Société Mathématique de France, Paris, pp. x+276. P. Kravchuk, D. Mazáč, and S. Pal (2024). “Automorphic spectra and the conformal bootstrap”,Commun. Am. Math. Soc.4, pp. 1–63. J.-F. Le Gall (2013). “Uniqueness and universality of the Brownian map”,Ann. Probab. 41(4), pp. 2880–2960. E. Le Masson and T. Sahlsten (20...

    work page 2024

  31. [31]

    Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks

    Mathematical Sciences Research Insti- tute Publications. The beauty of Klein’s quartic curve. Cambridge University Press, Cambridge, pp. x+331. M. W. Liebeck and A. Shalev (2004). “Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks”,J. Algebra276(2), pp. 552–

    work page 2004

  32. [32]

    Random surfaces with large systoles

    1255–59 M. Lipnowski and A. Wright (2024). “Towards optimal spectral gaps in large genus”, Ann. Probab.52(2), pp. 545–575. M. Liu (2022). “Length statistics of random multicurves on closed hyperbolic surfaces”, Groups Geom. Dyn.16(2), pp. 437–459. M. Liu and B. Petri (2024). “Random surfaces with large systoles”. arXiv:2312.11428 [math.GT]. L. Louder and ...

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  33. [33]

    Ramanujan graphs

    A. Lubotzky, R. Phillips, and P. Sarnak (1988). “Ramanujan graphs”,Combinatorica 8(3), pp. 261–277. A. Lubotzky and D. Segal (2003).Subgroup growth. Vol

    work page 1988

  34. [34]

    Birkhäuser Verlag, Basel, pp

    Progress in Mathematics. Birkhäuser Verlag, Basel, pp. xxii+453. N. A. M. Lulov (1996).Random walks on the symmetric group generated by conjugacy classes. Thesis (Ph.D.)–Harvard University. ProQuest LLC, Ann Arbor, MI, p

    work page 1996

  35. [35]

    Maclachlan and A

    C. Maclachlan and A. W. Reid (2003). The arithmetic of hyperbolic 3-manifolds. Vol

    work page 2003

  36. [36]

    The limit points of the bass notes of arithmetic hyperbolic surfaces

    Graduate Texts in Mathematics. Springer-Verlag, New York, pp. xiv+463. M. Magee (2020). “Letter to Bram Petri”. Available at:https://www.mmagee.net/ diameter.pdf. (2024). “The limit points of the bass notes of arithmetic hyperbolic surfaces”. arXiv: 2403.00928 [math.NT]. (2025). “Strong convergence of unitary and permutation representations of dis- crete ...

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  37. [37]

    The asymptotic statistics of random covering surfaces

    (2023). “The asymptotic statistics of random covering surfaces”,Forum Math. Pi 11, Paper No. e15,

    work page 2023

  38. [38]

    Strong convergence of uniformly random permutation representations of surface groups

    M. Magee, D. Puder, and R. van Handel (2025). “Strong convergence of uniformly ran- dom permutation representations of surface groups”. arXiv:2504.08988 [math.GT]. Y. Maoz (2025). “Asymptotic independence for random permutations from surface groups”,Geom. Dedicata219(3), Paper No. 43,

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  39. [39]

    Lengths of saddle connections on random translation surfaces of large genus

    (2026). “Smooth linear eigenvalue statistics on random covers of compact hy- perbolic surfaces—a central limit theorem and almost sure RMT statistics”,Israel Journal of Mathematics, pp. 1–50. A. W. Marcus, D. A. Spielman, and N. Srivastava (2015). “Interlacing families I: Bi- partite Ramanujan graphs of all degrees”,Ann. of Math. (2)182(1), pp. 307–325. 1...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  40. [40]

    Determinants of Laplacians on random hyperbolic surfaces

    F. Naud (2023). “Determinants of Laplacians on random hyperbolic surfaces”,J. Anal. Math. 151(1), pp. 265–291. (2026). “Random covers of compact surfaces and smooth linear spectral statis- tics”,Ann. Henri Poincaré27(1), pp. 347–373. A. Nica (1994). “On the number of cycles of given length of a free word in several random permutations”,Random Structures A...

    work page 2023

  41. [41]

    Pour toute surface hyperbolique de genreg, λ2g−2 > 1/4

    J.-P. Otal and E. Rosas (2009). “Pour toute surface hyperbolique de genreg, λ2g−2 > 1/4”,Duke Math. J.150(1), pp. 101–115. H. Parlier (2024). “A shorter note on shorter pants”,Bull. Lond. Math. Soc.56(4), pp. 1483–1487. H. Parlier, Y. Wu, and Y. Xue (2022). “The simple separating systole for hyperbolic surfaces of large genus”,J. Inst. Math. Jussieu21(6),...

    work page 2009

  42. [42]

    Local statistics of random permutations from free products

    Lecture Notes in Math- ematics. Lectures from the 32nd Summer School on Probability Theory held in Saint- Flour, July 7–24, 2002, With a foreword by Jean Picard. Springer-Verlag, Berlin, pp. x+256. D. Puder and T. Zimhoni (2024). “Local statistics of random permutations from free products”,Int. Math. Res. Not.(5), pp. 4242–4300. A. Putman and B. Wieland (...

    work page 2002

  43. [43]

    Géométrie et topologie des variétés hyperboliques de grand vol- ume

    J. Raimbault (2014). “Géométrie et topologie des variétés hyperboliques de grand vol- ume”,Séminaire de théorie spectrale et géométrie31, pp. 163–195. 1255–62 M. Ratner (1987). “The rate of mixingfor geodesic and horocycle flows”,Ergodic Theory Dynam. Systems7(2), pp. 267–288. W. Roelcke (1953/55). “Über die Wellengleichung bei Grenzkreisgruppen erster Ar...

    work page 2014

  44. [44]

    Random harmonic maps into spheres

    (2026). “Nearly optimal spectral gaps for random Belyi surfaces”. arXiv:2511. 02517 [math.SP]. C. L. Siegel (1945). “A mean value theorem in geometry of numbers”,Ann. of Math. (2) 46, pp. 340–347. A. Song (2025). “Random harmonic maps into spheres”. arXiv:2402.10287 [math.DG]. A. Strohmaier and V. Uski (2013). “An algorithm for the computation of eigenval...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  45. [45]

    Delocalisation of eigenfunctions on large genus random surfaces

    J. Thomas (2022). “Delocalisation of eigenfunctions on large genus random surfaces”, Israel J. Math.250(1), pp. 53–83. R. van Handel (2026). “The strong convergence phenomenon”. arXiv:2507 . 00346 [math.PR]. D. Witte Morris (2015).Introduction to arithmetic groups. Deductive Press, [place of publication not identified], pp. xii+475. 1255–63 S. A. Wolpert ...

    work page 2022

  46. [46]

    The asymptotic connectivity of labelled regular graphs

    CBMS Regional Conference Series in Mathematics. Conference Board of the Math- ematical Sciences, Washington, DC; by the American Mathematical Society, Provi- dence, RI, pp. viii+118. N. C. Wormald (1981). “The asymptotic connectivity of labelled regular graphs”,J. Combin. Theory Ser. B31(2), pp. 156–167. (1999). “Models of random regular graphs”, in:Surve...

    work page 1981

  47. [47]

    A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

    London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, pp. 239–298. A. Wright (2020). “A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces”,Bull. Amer. Math. Soc. (N.S.)57(3), pp. 359–408. S. Wright (2025). “Random covers of surfaces”. arXiv:2511.22731 [math.GT]. Y. Wu and Y. Xue (2022). “Random hyperbolic surfaces of larg...

    work page arXiv 2020

This paper was first reviewed by glm-5.2 on July 8, 2026.