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A p-adic operator spectrum balances exponential energies and degeneracies to reproduce the string's Hardy-Ramanujan entropy scaling, with log-periodic modulations.

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2026-08-03 12:15 UTC pith:NMM4EOH4

load-bearing objection A genuinely new p-adic spectrum with string-like entropy, but the general-conductor eigenvalue formula needs a proof before the main claim is settled. the 3 major comments →

arxiv 2601.03738 v2 pith:NMM4EOH4 submitted 2026-01-07 hep-th

A glimpse into the Ultrametric spectrum

An Huang , Christian B. Jepsen This is my paper
classification hep-th MSC 11S8011P8281T3005C50
keywords p-adic numbersVladimirov derivativeultrametric spectrumHardy-Ramanujan scalingp-adic string theorylog-periodic oscillationstree normal modesNeumann-to-Dirichlet operator
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 paper asks whether an ultrametric, p-adic analog of the string can reproduce the entropy-energy relation of the ordinary non-relativistic string, where the microstate count grows as exp(√E). Its central result is that the eigenvalue spectrum of the Vladimirov derivative on the p-adic unit circle has exponentially growing energies and exponentially growing degeneracies, and these two effects balance so that a saddle-point evaluation of the thermal partition function gives a coarse-grained degeneracy of the form E^(−a) exp(c√E). This recovers the string's Hardy-Ramanujan scaling, but modulated by log-periodic oscillations rather than smooth behavior. The authors also show that more naive tree-graph replacements fail to achieve the same scaling, so the p-adic continuum construction is what makes the string-like counting possible.

Core claim

The paper's central discovery is that the Vladimirov derivative acting on the p-adic unit circle U_p has a spectrum consisting of the trivial character with eigenvalue 0, p−2 characters of conductor one with eigenvalue ε(p−1)/p, and, for n>1, (p−1)^2 p^(n−2) characters of conductor n with eigenvalue E_n = ε[(p+1)/p^2 p^n − 2/p]. Promoting each level to a quantum harmonic oscillator and evaluating the partition function at high temperature gives, after dropping log-periodic fluctuations, a coarse-grained degeneracy Ω(E) ∝ E^(−a) exp[(p−1)π√(2E/ε)/(√(3(p+1) log p))] with a = 5/4 − (p−1)^2/[p(p+1) log p], so log Ω ∼ √E. The same eigenfunctions and multiplicities, and the same leading entropy en

What carries the argument

The key objects are the multiplicative characters of the p-adic units, classified by their conductor n (the number of p-adic digits on which the character depends), and the eigenvalue integral λ_π = ∫_{U_p} dz (1−π(z))/|z−1|_p^2 for the Vladimirov derivative, a bi-local derivative operator on p-adic numbers. Because characters depend on only finitely many digits, the integral collapses to a finite sum over residue classes; the paper evaluates these sums explicitly for conductors 1, 2, and 3 and asserts the general-n result. The thermodynamic argument then turns on the infinite product partition function and a saddle-point evaluation in which the exponential spacing p^n of energies and the ex

Load-bearing premise

The load-bearing premise is that the eigenvalue integral for multiplicative characters of arbitrary conductor n collapses to λ=(p+1)p^(n−2)−2/p by the same method as the three cases actually shown (n=1,2,3), since the whole spectrum—and with it the entropy law—stands on that single unproven general-n formula; the coarse-grained envelope additionally rests on a numerically checked but not analytically proven saddle-point formula.

What would settle it

Enumerate all multiplicative characters of conductor n=4 for p=3 (there are (p−1)^2 p^(n−2)=36 of them), evaluate the defining eigenvalue integral directly as a finite sum over residue classes mod p^n, and check whether every eigenvalue equals (p+1)p^(n−2)−2/p = 36 − 2/3; any deviation, or any dependence on the character label j, falsifies the spectrum and hence the thermodynamics.

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

If this is right

  • The ultrametric spectrum yields a concrete infinite tower of microstates with string-like entropy, opening a path toward p-adic string theories that contain more than a single tachyon.
  • The balance condition—energy spacing p^n with degeneracy (p−1)^2 p^(n−2)—explains why the Neumann-tree spectrum, whose levels grow only as p^(n/2), instead produces S ∼ E^(2/3).
  • Log-periodic oscillations in log temperature are small for p=2 but grow with p, so thermodynamic quantities derived from this spectrum carry modulations that must be controlled in any large-p application.
  • The Neumann-to-Dirichlet variant shares the same eigenfunctions and the same √E entropy envelope but has a tachyonic ground state, so subleading thermodynamics differ while the leading entropy law remains robust.
  • Periodic boundary conditions on trees are severely constrained: vertex-transitive identifications exist only for finitely many sizes, so the p-adic continuum spectrum is not a limit of a periodic tree graph.

Where Pith is reading between the lines

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

  • If the general-conductor eigenvalue formula survives closer scrutiny, the saddle-point mechanism is generic: any spectrum with E_n ∝ p^n and ρ_n ∝ p^n should produce a √E entropy envelope, so the same counting should appear for other p-adic operators or unramified extensions.
  • The log-periodic modulation means the exact microstate count is a rough, oscillating function; a natural next step would be to seek a Rademacher-style convergent expansion for the ultrametric partition function, analogous to the refinement of Hardy-Ramanujan for ordinary partitions.
  • The opposite regime—exponentially decreasing spacings, as in q-deformed strings—is the mirror image of this construction; whether an analytic continuation links the two families is a natural open question suggested by the formulas.
  • A sharper numerical test of the claimed balance would be to compute exact partition multiplicities for p=5 at moderate energies and check both the exponential coefficient and the period log p of the oscillations; the paper's numerical checks are shown for p=3.

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

3 major / 5 minor

Summary. The paper studies deformations of the non-relativistic string spectrum motivated by p-adic geometry. After showing that normal modes of (p+1)-regular tree graphs with Neumann, Dirichlet, or periodic boundary conditions do not yield Hardy–Ramanujan microstate scaling (with the periodic case obstructed by Moore-graph nonexistence), the authors compute the spectrum of the Vladimirov derivative acting on functions on the p-adic unit circle U_p. They claim (eq. 66) that the eigenvalues are E_n = ε[(p+1)/p^2 p^n − 2/p] with degeneracies ρ(1)=p−2 and ρ(n>0)=(p−1)^2 p^{n−2}. Associating a quantum harmonic oscillator to each eigenvalue, they evaluate the thermal partition function in a high-temperature saddle-point approximation and obtain a coarse-grained degeneracy (eq. 79) whose logarithm scales as √E, i.e. string-like Hardy–Ramanujan behavior, but with log-periodic fluctuations. Appendix B identifies the same eigenfunctions and degeneracies for a Neumann-to-Dirichlet operator on the Bruhat–Tits tree, with eigenvalues that differ only by an overall shift and normalization.

Significance. If the central derivation is correct, the paper gives a concrete p-adic operator spectrum whose microstate count reproduces the non-relativistic string entropy–energy relation, despite exponentially growing energy levels, and it does so without fitting parameters: the thermodynamic constants A, B, C, c_n are computed in closed form, the zero-point cancellation (67) is explicitly checked, and the log-periodic modulation is a consequence of the calculation rather than an input. The exact degeneracy counting in Section IV is independently checked by the character count, and the numerical comparison in Figure 7 supports the envelope formula. The extension of the saddle-point method to exponentially spaced, exponentially degenerate spectra is a useful technical contribution. However, the general-conductor eigenvalue formula on which the whole spectrum rests is asserted rather than proved, and the derivation of the envelope from the saddle point is partly by analogy with the string case; these gaps must be closed before the central claim can be considered established.

major comments (3)
  1. [§IV, eqs. (64)–(66)] The eigenvalue formula λ = (p+1)p^{n−2} − 2/p for characters of conductor n is the single load-bearing input of the paper: it determines the spectrum (66), the zero-point cancellation (67), and all thermodynamic constants in Appendix A. The text works out n = 1, 2, 3 (eqs. 59, 61, 63) and then states that the general sum (64) reduces to (65) “by the same method.” No general-n evaluation of the character sums is supplied, and the typesetting of (64) is partly garbled, making the claimed collapse difficult to verify. The reader is left with a proof gap, not an identified error, but the gap is load-bearing. A direct derivation—or a precise reference to a standard p-adic harmonic-analysis result—is required before the spectrum can be treated as settled.
  2. [§V, eqs. (76)–(79)] The paper states explicitly that E(β0) in (78) could not be inverted analytically, and then obtains the envelope (79) by reusing the string saddle-point formula (19) with modified values of A, B, C after “dropping the log-periodic fluctuations.” This transfer is not justified in detail: it is not shown that the saddle point of the averaged entropy has the same form once the oscillatory terms in (76) are discarded, nor that the prefactor and the exponent in (79) are the correct envelope rather than an ansatz. Figure 7 is suggestive, but a numerical check is not a proof. The central Hardy–Ramanujan claim rests on this envelope, so the averaging procedure needs to be made precise.
  3. [§IV, eq. (67) and §V] The zero-point energy divergence is regulated by analytic continuation in p / geometric summation, and the authors find that the total zero-point contribution vanishes exactly. This is a particular regulator choice; a zeta-function or cutoff regulator would in general give a different (possibly divergent or finite) ground-state shift. Since the partition function (68) starts without any zero-point term, the low-temperature and subleading thermodynamics of the model depend on this convention. The authors should discuss the physical status of this regulator and, if possible, show that the leading √E scaling is independent of it.
minor comments (5)
  1. [§IV, eq. (64)] The displayed formula for the general-conductor sum is difficult to parse: the exponents, denominators, and summation limits are not typeset cleanly. Please rewrite it with clearly grouped factors and define all summation ranges.
  2. [§I, eq. (3)] The sample spectra E^{(2)}, E^{(3)}, E^{(5)} are presented without stating the chosen normalization of ε. For example, the p=3 values correspond to ε=3/2. Please state the normalization explicitly.
  3. [Appendix A, eq. (A19)] The text says A, B, C, c_n, etc. are “p-independent constants,” but A, B, and C as given in (A20)–(A22) depend explicitly on p. This should be corrected to “p-dependent constants” or similar.
  4. [Figure 7] The three columns of plots are described only in the caption as differing in plot region. Please label the panels and add axis legends, especially for the lower-row plots where the three curves nearly coincide.
  5. [References] Several entries are incomplete or lack journal/volume/page data (e.g. [45], [58], [61]). The arXiv identifiers are present for most items, but the bibliographic details should be completed for publication.

Circularity Check

0 steps flagged

No significant circularity: the Vladimirov spectrum is computed from a direct eigenvalue integral, and the thermodynamic constants are derived rather than fitted; the main weakness is an asserted induction for general conductor n, which is a proof gap rather than a circular reduction.

full rationale

The central derivation is self-contained. The spectrum (66) comes from evaluating the Vladimirov-derivative eigenvalue integral (56) for multiplicative characters; the degeneracies are counts of characters of each conductor. The free-energy expansion (A19) and all constants A, B, C, c_n in (A20)-(A25) are obtained by Poisson resummation and polylogarithm expansions, not by matching to the target entropy. The envelope (79) reuses the general saddle-point formula (19) with these derived constants, so the Hardy-Ramanujan exponent follows from the computed spectrum rather than being an input. Figure 7 compares exact degeneracies with the derived asymptotics, not fitting them. The self-citations ([66], [79], [17]) are pointers to related work and are not load-bearing for the spectrum or thermodynamics. The genuine weakness is that (65) is asserted 'by the same method' after only n=1,2,3 are shown, leaving the full spectrum and all subsequent thermodynamics dependent on an unproved induction; Appendix B's Neumann-to-Dirichlet calculation gives different eigenvalues and does not close this gap. These are proof gaps and correctness risks, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 6 axioms · 0 invented entities

The model is economical: one dimensionful scale ε (a unit choice, not fitted), a standard eigenfunction fact (multiplicative characters), and two conventions (operator normalization, zero-point regulator). The genuinely load-bearing unproven input is the general-conductor eigenvalue evaluation (64)→(65). No new physical entities are postulated: the 'ultrametric string spectrum' is a name for the derived operator eigenvalues, and the authors explicitly defer the dynamical completion (oscillator algebra, Virasoro, Lorentzian realization) to future work (Section VI).

free parameters (1)
  • ε (overall energy scale)
    Dimensionful normalization for E_n in (66) — the analogue of ℏω₀. Not fitted to data; cancels from the entropy–energy scaling law.
axioms (6)
  • standard math Multiplicative characters of U_p are eigenfunctions of the modified Vladimirov derivative and form a complete basis
    Section IV, eqs. (53)–(56): Dπ = λ_π π is verified for characters; completeness is standard harmonic analysis on U_p.
  • ad hoc to paper The general-conductor eigenvalue sum (64) evaluates to λ = (p+1)p^{n−2} − 2/p (eq. 65)
    Shown for n = 1, 2, 3 only (eqs. 59, 61, 63; Figs. 5–6); asserted for general n 'by the same method'. The spectrum (66) and Section V rest on it.
  • ad hoc to paper Divergent zero-point energy is regulated by analytic continuation/geometric summation in p, making Σ ρ(n)E_n = 0 (eq. 67)
    Section V; 'arguably the most natural regulator'. A different regulator shifts the ground state (the NtD case (B6) yields a tachyonic shift) but not the asymptotic entropy.
  • domain assumption The coarse-grained envelope, obtained by dropping the log-periodic Fourier modes of the free energy, is the right object for the claim that Hardy–Ramanujan scaling is 'realized'
    Section V: the paper admits the exact degeneracy admits no strict √E law; the claim is thereby by construction about the envelope. Numerical support in Fig. 7.
  • standard math Poisson resummation and polylogarithm asymptotics used in Appendix A are valid on the relevant integration/expansion domains
    Appendix A, eqs. (A8)–(A11); standard analytic tools; the branch-cut cancellation (A16)–(A17) is intricate but tracked explicitly.
  • standard math Singleton's classification of Moore graphs bounds periodic-tree constructions
    Section III C, cited as Singleton (1966); used to conclude no large-L periodic trees exist for p>1.

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read the original abstract

The non-relativistic string spectrum is built from integer-spaced energy quanta in such a way that the high-temperature asymptotics, via the Hardy-Ramanujan formula for integer partitions, reduces to standard two-dimensional thermodynamics. Here we explore deformed realizations of this behavior motivated by $p$-adic string theory and Lorentzian versions thereof with a non-trivial spectrum. We study the microstate scaling that results on associating quantum harmonic oscillators to the normal modes of tree-graphs rather than string graphs and observe that Hardy-Ramanujan scaling is not realized. But by computing the eigenvalues of the derivative operator on the $p$-adic circle and by determining the eigenspectrum of the Neumann-to-Dirichlet operator, we uncover a spectrum of exponentially growing energies but with exponentially growing degeneracies balanced in such a way that Hardy-Ramanujan scaling is realized, but modulated with log-periodic fluctuations.

Figures

Figures reproduced from arXiv: 2601.03738 by An Huang, Christian B. Jepsen.

Figure 1
Figure 1. Figure 1: FIG. 1. The fractal tree for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plot of the values of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plot of the values of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. By identifying boundary vertices of a finite tree, one [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Computation of the eigenvalue for the multiplicative character of conductor two for [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Computation of the eigenvalue for the multiplicative character of conductor three for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots of the exact ultrametric degeneracies along with the averaged and unaveraged asymptotic values for [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Harmonic extension [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

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Forward citations

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Reference graph

Works this paper leans on

98 extracted references · 57 linked inside Pith · cited by 1 Pith paper

  1. [1]

    the precise determination of the eigenvalue spectrum of the Vladimirov derivative when acting on thep-adic unitsU p, as given in equation (66), and also realized as the spectrum of thep-adic Neumann-to-Dirichlet opera- tor computed in Appendix B, and

  2. [2]

    An important qualification to the second point, however, is that the leading scaling is modulated by a periodic function in logE[72]

    a saddle-point evaluation of the partition function of the quantum theory built from the infinite set of oscilla- tors with energy quanta matching this spectrum, reveal- ing that, as in string theory, Hardy-Ramanujan scaling is realized, by which we mean that, at large total energyE, the total degeneracy scales ase # √ E as shown in equation (79). An impo...

  3. [3]

    Kumar and W

    V. Kumar and W. Taylor, String Universality in Six Dimensions, Adv. Theor. Math. Phys.15, 325 (2011), arXiv:0906.0987 [hep-th]

    Pith/arXiv arXiv 2011

  4. [4]

    Belin, C

    A. Belin, C. A. Keller, and A. Maloney, Per- mutation Orbifolds in the large N Limit, An- nales Henri Poincar´ e 10.1007/s00023-016-0529-y (2015), arXiv:1509.01256 [hep-th]

    Pith/arXiv arXiv 2015

  5. [5]

    Caron-Huot, Z

    S. Caron-Huot, Z. Komargodski, A. Sever, and A. Zhi- boedov, Strings from Massive Higher Spins: The Asymp- totic Uniqueness of the Veneziano Amplitude, JHEP10, 026, arXiv:1607.04253 [hep-th]

    Pith/arXiv arXiv

  6. [6]

    Sever and A

    A. Sever and A. Zhiboedov, On Fine Structure of Strings: The Universal Correction to the Veneziano Amplitude, JHEP06, 054

  7. [7]

    S.-J. Lee, W. Lerche, and T. Weigand, Emergent strings from infinite distance limits, JHEP02, 190, arXiv:1910.01135 [hep-th]

    Pith/arXiv arXiv 1910

  8. [8]

    Huang, J.-Y

    Y.-t. Huang, J.-Y. Liu, L. Rodina, and Y. Wang, Carving out the Space of Open-String S-matrix, JHEP04, 195, arXiv:2008.02293 [hep-th]

    Pith/arXiv arXiv 2008

  9. [9]

    Kaplan and S

    J. Kaplan and S. Kundu, Closed Strings and Weak Gravity from Higher-Spin Causality, JHEP02, 145, arXiv:2008.05477 [hep-th]

    Pith/arXiv arXiv 2008

  10. [10]

    Guerrieri, J

    A. Guerrieri, J. Penedones, and P. Vieira, Where Is String Theory in the Space of Scattering Amplitudes?, Phys. Rev. Lett.127, 081601 (2021), arXiv:2102.02847 [hep- th]

    Pith/arXiv arXiv 2021

  11. [11]

    Arkani-Hamed, C

    N. Arkani-Hamed, C. Cheung, C. Figueiredo, and G. N. Remmen, Multiparticle Factorization and the Rigidity of String Theory, Phys. Rev. Lett.132, 091601 (2024), arXiv:2312.07652 [hep-th]

    Pith/arXiv arXiv 2024

  12. [12]

    Cheung, A

    C. Cheung, A. Hillman, and G. N. Remmen, Bootstrap Principle for the Spectrum and Scattering of Strings, Phys. Rev. Lett.133, 251601 (2024), arXiv:2406.02665 [hep-th]

    Pith/arXiv arXiv 2024

  13. [13]

    Cheung, A

    C. Cheung, A. Hillman, and G. N. Remmen, Uniqueness criteria for the Virasoro-Shapiro amplitude, Phys. Rev. D111, 086034 (2025), arXiv:2408.03362 [hep-th]

    Pith/arXiv arXiv 2025

  14. [14]

    Cheung, G

    C. Cheung, G. N. Remmen, F. Sciotti, and M. Tarquini, Strings from Almost Nothing, arXiv (2025), 2508.09246 [hep-th]

    Pith/arXiv arXiv 2025

  15. [15]

    Volovich, p-adic string, Classical and Quantum Gravity 4, L83 (1987)

    I. Volovich, p-adic string, Classical and Quantum Gravity 4, L83 (1987)

    1987

  16. [16]

    Grossman, p-adic strings, the weil conjectures and anomalies, Physics Letters B197, 101 (1987)

    B. Grossman, p-adic strings, the weil conjectures and anomalies, Physics Letters B197, 101 (1987)

    1987

  17. [17]

    P. G. O. Freund and M. Olson, Nonarchimedean Strings, Phys. Lett. B199, 186 (1987)

    1987

  18. [18]

    A. V. Zabrodin, Nonarchimedean Strings and Bruhat-tits Trees, Commun. Math. Phys.123, 463 (1989)

    1989

  19. [19]

    Huang, B

    A. Huang, B. Stoica, and S.-T. Yau, General relativity fromp-adic strings, Adv. Theor. Math. Phys.26, 1203 (2022), arXiv:1901.02013 [hep-th]

    Pith/arXiv arXiv 2022

  20. [20]

    Coon, Uniqueness of the veneziano representation, Physics Letters B29, 669 (1969)

    D. Coon, Uniqueness of the veneziano representation, Physics Letters B29, 669 (1969)

    1969

  21. [21]

    Figueroa and P

    F. Figueroa and P. Tourkine, Unitarity and Low Energy Expansion of the Coon Amplitude, Phys. Rev. Lett.129, 121602 (2022), arXiv:2201.12331 [hep-th]

    Pith/arXiv arXiv 2022

  22. [22]

    Geiser and L

    N. Geiser and L. W. Lindwasser, Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon, JHEP12, 112, arXiv:2207.08855 [hep-th]

    Pith/arXiv arXiv

  23. [23]

    Chakravarty, P

    J. Chakravarty, P. Maity, and A. Mishra, On the pos- itivity of Coon amplitude in D = 4, JHEP10, 043, arXiv:2208.02735 [hep-th]

    Pith/arXiv arXiv

  24. [24]

    Cheung and G

    C. Cheung and G. N. Remmen, Veneziano variations: how unique are string amplitudes?, JHEP01, 122, arXiv:2210.12163 [hep-th]

    Pith/arXiv arXiv

  25. [25]

    Geiser and L

    N. Geiser and L. W. Lindwasser, Generalized Veneziano and Virasoro amplitudes, JHEP04, 031, arXiv:2210.14920 [hep-th]

    Pith/arXiv arXiv

  26. [26]

    Maldacena and G

    J. Maldacena and G. N. Remmen, Accumulation- point amplitudes in string theory, JHEP08, 152, arXiv:2207.06426 [hep-th]

    Pith/arXiv arXiv

  27. [27]

    Bhardwaj, S

    R. Bhardwaj, S. De, M. Spradlin, and A. Volovich, On unitarity of the Coon amplitude, JHEP08, 082, arXiv:2212.00764 [hep-th]

    Pith/arXiv arXiv

  28. [28]

    Cheung and G

    C. Cheung and G. N. Remmen, Stringy dynamics from an amplitudes bootstrap, Phys. Rev. D108, 026011 (2023), arXiv:2302.12263 [hep-th]

    Pith/arXiv arXiv 2023

  29. [29]

    C. B. Jepsen, Cutting the Coon amplitude, JHEP06, 114, arXiv:2303.02149 [hep-th]

    Pith/arXiv arXiv

  30. [30]

    Li and H.-Y

    Y. Li and H.-Y. Sun, Towardsα ′-finiteness:q-deformed open string amplitude, arXiv (2023), 2307.13117 [hep- th]

    Pith/arXiv arXiv 2023

  31. [31]

    Geiser, The Baker-Coon-Romans N-point amplitude 20 and an exact field theory limit of the Coon amplitude, JHEP10, 010, arXiv:2311.04130 [hep-th]

    N. Geiser, The Baker-Coon-Romans N-point amplitude 20 and an exact field theory limit of the Coon amplitude, JHEP10, 010, arXiv:2311.04130 [hep-th]

    Pith/arXiv arXiv

  32. [32]

    Almheiri and F

    A. Almheiri and F. K. Popov, Holography on the quan- tum disk, JHEP06, 070, arXiv:2401.05575 [hep-th]

    Pith/arXiv arXiv

  33. [33]

    Eckner, F

    C. Eckner, F. Figueroa, and P. Tourkine, Regge boot- strap: From linear to nonlinear trajectories, Phys. Rev. D111, 126005 (2025), arXiv:2401.08736 [hep-th]

    Pith/arXiv arXiv 2025

  34. [34]

    K. C. Rigatos and B. Wang, Coon unitarity via partial waves or: how I learned to stop worrying and love the harmonic numbers, Phys. Rev. D110, 126024 (2024), arXiv:2401.13031 [hep-th]

    Pith/arXiv arXiv 2024

  35. [35]

    Wang, Positivity of the hypergeometric Coon ampli- tude, JHEP04, 143, arXiv:2403.00906 [hep-th]

    B. Wang, Positivity of the hypergeometric Coon ampli- tude, JHEP04, 143, arXiv:2403.00906 [hep-th]

    Pith/arXiv arXiv

  36. [36]

    Belaey, T

    A. Belaey, T. G. Mertens, and J. Papalini, Probing the singularity at the holographic screen viaq-holography, arXiv (2025), 2507.13873 [hep-th]

    Pith/arXiv arXiv 2025

  37. [37]

    F. J. Dyson, Existence of a phase transition in a one- dimensional Ising ferromagnet, Commun. Math. Phys. 12, 91 (1969)

    1969

  38. [38]

    Melzer, Nonarchimedean conformal field theories, Int

    E. Melzer, Nonarchimedean conformal field theories, Int. J. Mod. Phys. A4, 4877 (1989)

    1989

  39. [39]

    S. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk,p-adic AdS/CFT, Commun. Math. Phys. 352, 1019 (2017), arXiv:1605.01061 [hep-th]

    Pith/arXiv arXiv 2017

  40. [40]

    Heydeman, M

    M. Heydeman, M. Marcolli, I. Saberi, and B. Stoica, Ten- sor networks,p-adic fields, and algebraic curves: arith- metic and the AdS 3/CFT2 correspondence, Adv. Theor. Math. Phys.22, 93 (2018), arXiv:1605.07639 [hep-th]

    Pith/arXiv arXiv 2018

  41. [41]

    Periwal, E

    A. Periwal, E. S. Cooper, P. Kunkel, J. F. Wienand, E. J. Davis, and M. Schleier-Smith, Programmable in- teractions and emergent geometry in an array of atom clouds, Nature600, 630 (2021), [Erratum: Nature 603, E29 (2022)], arXiv:2106.04070 [quant-ph]

    Pith/arXiv arXiv 2021

  42. [42]

    S. S. Gubser, C. Jepsen, Z. Ji, and B. Trundy, Contin- uum limits of sparse coupling patterns, Phys. Rev. D98, 045009 (2018), arXiv:1805.07637 [hep-th]

    Pith/arXiv arXiv 2018

  43. [43]

    Bentsen, T

    G. Bentsen, T. Hashizume, A. S. Buyskikh, E. J. Davis, A. J. Daley, S. S. Gubser, and M. Schleier-Smith, Treelike interactions and fast scrambling with cold atoms, Phys. Rev. Lett.123, 130601 (2019), arXiv:1905.11430 [quant- ph]

    Pith/arXiv arXiv 2019

  44. [44]

    H. Yan, C. B. Jepsen, and Y. Oz, p-adic hologra- phy from the hyperbolic fracton model, JHEP08, 096, arXiv:2306.07203 [hep-th]

    Pith/arXiv arXiv

  45. [45]

    See also [80] for recent work on quantum hierarchical models and [81] for recent work on critical phenomena on the Bethe lattice

  46. [46]

    See also [82–85] for alternative approaches top-adic quan- tum mechanics and for related notions of statistical field theory [86, 87]

  47. [47]

    I. R. Klebanov, String theory in two-dimensions, in Spring School on String Theory and Quantum Gravity (to be followed by Workshop)(1991) arXiv:hep-th/9108019

    Pith/arXiv arXiv 1991

  48. [48]

    M. N. Tran, M. V. N. Murthy, and R. K. Bhaduri, On the quantum density of states and partitioning an integer, Annals Phys.311, 204 (2004), arXiv:math-ph/0309020

    Pith/arXiv arXiv 2004

  49. [49]

    equation (10) of [88]

    See e.g. equation (10) of [88]

  50. [50]

    equation (3) of [89]

    See e.g. equation (3) of [89]

  51. [51]

    K. S. Tikhonov and A. D. Mirlin, Fractality of wave func- tions on a cayley tree: Difference between tree and locally treelike graph without boundary, Physical Review B94, 184203 (2016), arXiv:1608.00331 [cond-mat]

    Pith/arXiv arXiv 2016

  52. [52]

    Aryal and S

    D. Aryal and S. Kettemann, Complete solution of the tight binding model on a cayley tree: strongly localised versus extended states, Journal of Physics Communica- tions4, 105010 (2020), arXiv:2001.11814 [cond-mat]

    Pith/arXiv arXiv 2020

  53. [53]

    Ostilli, C

    M. Ostilli, C. G. Bezerra, and G. Viswanathan, Spectrum of the tight-binding model on cayley trees and compari- son with bethe lattices, Physical Review E105, 034123 (2022), arXiv:2106.06879 [cond-mat]

    Pith/arXiv arXiv 2022

  54. [54]

    Mahan, Energy bands of the bethe lattice, Physical Review B63, 155110 (2001)

    G. Mahan, Energy bands of the bethe lattice, Physical Review B63, 155110 (2001)

    2001

  55. [55]

    Solomyak, On the spectrum of the laplacian on regular metric trees, Waves in Random Media14, S155 (2003)

    M. Solomyak, On the spectrum of the laplacian on regular metric trees, Waves in Random Media14, S155 (2003)

    2003

  56. [56]

    M. Solomyak, Laplace and schr¨ odinger operators on regu- lar metric trees: the discrete spectrum case, inFunction Spaces, Differential Operators and Nonlinear Analysis: The Hans Triebel Anniversary Volume(Springer, 2003) pp. 161–181, arXiv:0111023 [math]

    2003

  57. [57]

    A complication present more generally is that a clean separation of oscillations along independent and identical transverse directions does not subsist beyond the special case of chain graphs. In the more general case, the form of the oscillation modes that do not change edge lengths to first order in displacement will depend on the details of the graph’s...

  58. [58]

    However, as the smallest roots is shared by the two equations, the approximation (37) remains valid

    While the equationh ℓ−1(1−λ) = 0 can be shown to imply (36), the converse is not true as (36) has two additional roots. However, as the smallest roots is shared by the two equations, the approximation (37) remains valid

  59. [59]

    To obtain a finite total massM=mN p,L, the individual bead massmmust scale as 1/Np,L

    In the simplest model where we consider the fieldsϕ x as representing small transverse oscillations of a cobweb stretched along a single spatial direction and think of the vertices as beads of massm, with the edges connecting them serving as springs with a uniform spring constantk, we have the relationλ=mω 2/k. To obtain a finite total massM=mN p,L, the i...

  60. [60]

    Ebert, H.-Y

    S. Ebert, H.-Y. Sun, and M.-Y. Zhang, Probing holog- raphy in p-adic CFT, Phys. Rev. D107, 126011 (2023), arXiv:1911.06313 [hep-th]

    Pith/arXiv arXiv 2023

  61. [61]

    On a technical note, the roots for different values ofℓ sometimes overlap so that one must add together the re- spective multiplicities to get the total degeneracy. E.g., forp= 2 andL= 6, the eigenvalueλ= 3 has a to- tal degeneracy of 64, which comes from a multiplicity- oneλ L,n eigenvalue, a multiplicity-3λ L−2,n eigenvalue, a multiplicity-12λ L−4,n eig...

  62. [62]

    But this does not lead to a sensi- ble spectrum either

    One could attempt to obtain a non-trivial spectrum through the introduction by hand of a mass term that 21 effectively replaces the graph Laplacian□in (21) with □−(p+ 1−2 √p). But this does not lead to a sensi- ble spectrum either. For sinceλ L−ℓ,n −(p+ 1− √p) = √p n2π2 L2 +O(1/L 4) with noℓdependence on the right- hand side, this mass shift results in an...

  63. [63]

    Bollob´ as,Modern graph theory, Vol

    B. Bollob´ as,Modern graph theory, Vol. 184 (Springer Sci- ence & Business Media, 1998)

    1998

  64. [64]

    Singleton, On minimal graphs of maximum even girth, Journal of Combinatorial Theory1, 306 (1966)

    R. Singleton, On minimal graphs of maximum even girth, Journal of Combinatorial Theory1, 306 (1966)

    1966

  65. [65]

    C. W. Lam, The search for a finite projective plane of order 10, The American mathematical monthly98, 305 (1991)

    1991

  66. [66]

    In this case the set ofp-adic numbers Qp is replaced with the degreenunramified extension of Qp

    The whole analysis can also be straightforwardly ex- tended to the cases whenpis a prime power:p=p n with n∈Nandp∈P. In this case the set ofp-adic numbers Qp is replaced with the degreenunramified extension of Qp

  67. [67]

    On a technical note, since the boundary of the tree also includes the point at infinite, the boundary space should more precisely be identified with the projective space P1(Qp)

  68. [68]

    Huang, R

    A. Huang, R. Rohrlich, Y. Sun, and E. Whyman, Green’s function on the Tate curve, arXiv:2512.24935 (2026)

    arXiv 2026

  69. [69]

    Derrida, C

    B. Derrida, C. Itzykson, and J. M. Luck, Oscillatory Crit- ical Amplitudes in Hierarchical Models, Commun. Math. Phys.94, 115 (1984)

    1984

  70. [70]

    R. F. S. Andrade, Emergence of log-periodic oscillations in periodic and aperiodic ising models, Brazilian Journal of Physics30, 671 (2000)

    2000

  71. [71]

    Costin and G

    O. Costin and G. Giacomin, Oscillatory critical ampli- tudes in hierarchical models and the tail of the harris random variable, arXiv:1206.1468 (2012)

    Pith/arXiv arXiv 2012

  72. [72]

    Calcagni, Complex dimensions and their observability, Phys

    G. Calcagni, Complex dimensions and their observability, Phys. Rev. D96, 046001 (2017), arXiv:1705.01619 [gr- qc]

    Pith/arXiv arXiv 2017

  73. [73]

    P. D. Bhoyar and P. M. Gade, Emergence of logarithmic- periodic oscillations in contact process with topological disorder, Physical Review E103, 022115 (2021)

    2021

  74. [74]

    While the present work focuses on the entropy and free energy of non-relativistic quantum systems and every- thing is real, in the context of relativistic CFT correla- tors, such log-periodic fluctuations would be indicative of a complex CFT [90, 91]

  75. [75]

    Meinardus, Asymptotische aussagen ¨ uber partitionen, Mathematische Zeitschrift59, 388 (1953)

    G. Meinardus, Asymptotische aussagen ¨ uber partitionen, Mathematische Zeitschrift59, 388 (1953)

    1953

  76. [76]

    Hwang, Limit theorems for the number of sum- mands in integer partitions, Journal of Combinatorial Theory, Series A96, 89 (2001)

    H.-K. Hwang, Limit theorems for the number of sum- mands in integer partitions, Journal of Combinatorial Theory, Series A96, 89 (2001)

    2001

  77. [77]

    B. L. Granovsky, D. Stark, and M. Erlihson, Meinardus’ theorem on weighted partitions: Extensions and a proba- bilistic proof, Advances in Applied Mathematics41, 307 (2008)

    2008

  78. [78]

    B. L. Granovsky and D. Stark, A meinardus theorem with multiple singularities, Communications in Mathe- matical Physics314, 329 (2012)

    2012

  79. [79]

    Latapy, Partitions of an integer into powers, Discrete Mathematics & Theoretical Computer Science (2001)

    M. Latapy, Partitions of an integer into powers, Discrete Mathematics & Theoretical Computer Science (2001)

    2001

  80. [80]

    The founding references are [37] and [38], see also [92] for its tensor-network implementation and [93–97] for more recent work on the subject

Showing first 80 references.