Pith. sign in

REVIEW 3 major objections 5 minor 57 references

Lattice random walks and quantum A-period conjecture

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper derives closed-form signed-area counts for square and triangular lattice walks and conjectures that their generating function equals the quantum A-period of the associated toric Calabi-Yau threefold.

desk verdict New anisotropic walk-counting formulas worth taking seriously, but the key derivation step is unproved and the headline conjecture is a known relation. read the letter →

arxiv 2412.21128 v2 pith:MVBZPXJX submitted 2024-12-30 math-ph cond-mat.stat-mechhep-thmath.COmath.MPquant-ph

Li Gan This is my paper · ORCID
classification math-phcond-mat.stat-mechhep-thmath.COmath.MPquant-ph MSC 82B4181T30
keywords signedareaenumerationlatticerandomwalksHofstadtermodelexclusionstatisticsKreftcoefficientsquantumA-periodtoricCalabi-YauthreefoldlocalB3geometry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper gives closed-form formulas for the number of closed random walks on square and triangular lattices that have a given length, signed area, and number of moves in each direction. It obtains the formulas by writing the walk count as the trace of a power of an anisotropic Hofstadter-like Hamiltonian, and it connects those traces to exclusion-statistics cluster coefficients. The paper's main conjecture is that the same trace generating function is the quantum A-period of a toric Calabi-Yau threefold: square walks correspond to local $\mathbb{F}_0$, triangular walks to local $\mathcal{B}_3$. If true, this would tie a concrete combinatorial counting problem to topological string theory and mirror symmetry.

What carries the argument

The load-bearing object is the anisotropic Hofstadter-like Hamiltonian $H_{\mathrm{tri}}$ (with the square-lattice case $H_{\mathrm{sq}}$ obtained by setting two hopping amplitudes to zero) and its $q\times q$ clock-and-shift matrix representation. The argument runs through the secular determinant $\det(1 - z H_{1,2})$, whose Kreft coefficients $Z_n$ are interpreted as $n$-body partition functions for exclusion particles: $g=2$ exclusons for square walks, and a mixture of $g=1$ fermions and $g=2$ bound pairs for triangular walks. The cluster coefficients $b_n$ obtained from $\log Z(z)$ then express $\operatorname{Tr} H^N$, giving the closed-form counts $C_N(A)$. On the geometry side, the mirror curve of the Calabi-Yau is promoted to operators with $[x,y]=i\hbar$, and the quantum A-period is extracted by a residue computation; the conjecture identifies that period with the trace generating function.

What would settle it

Compute the coefficient of $z^{13}$ (or any order beyond $z^{12}$) on both sides of (11) for local $\mathcal{B}_3$: evaluate the residue formula (10) to that order and compare it with $-\frac{1}{13}\sum_A C_{13}(A) Q^A$ using (8). A mismatch at the first unchecked order would disprove the conjecture. Alternatively, for a fixed rational flux $Q = e^{2\pi i p/q}$, evaluate $\operatorname{Tr} H^N$ and $\frac{1}{q}\operatorname{tr} H_{1,2}^N$ for some $N \ge q$; the paper's own statement says these differ for $N \ge q$, so a finite-$q$ version of (11) would fail visibly unless the limit is taken with care.

Watch

Extended reading notes

Core claim

The author claims that the signed-area enumeration of closed random walks on the square and triangular lattices is captured exactly by the anisotropic Hofstadter trace, and that the generating function of these counts equals the quantum A-period of the corresponding toric geometry. Concretely, equation (11) states $t = -\log(z) - \sum_{N\ge 1} z^N \frac{1}{N} \sum_A C_N(A) Q^A$ with $Q = e^{i\hbar}$, where $C_N(A)$ is given in closed form by (6) for the square lattice and (8) for the triangular lattice, and $t$ is the quantum A-period of local $\mathbb{F}_0$ or local $\mathcal{B}_3$. The equality is checked against known expansions up to order $z^{12}$. In the square case the derivative of the period is also written as a complete elliptic integral, recovering and generalizing the known Hofstadter/quantum-geometry result, and a strong-weak coupling energy relation is noted.

Load-bearing premise

The load-bearing premise is that a trace identity proved only for $N<q$ continues to hold for every term in an infinite series after taking $q$ to infinity; the paper gives no convergence or limit-interchange argument for that step.

Editorial extensions

If this is right

  • The closed forms (6) and (8) upgrade earlier isotropic walk-counting results to counts that record the number of moves in each direction, so they can be used to probe anisotropic lattice models.
  • The walk enumeration is re-expressed as exclusion statistics: square walks are $g=2$ exclusons and triangular walks are a mixture of $g=1$ and $g=2$; this gives a physical interpretation of the Kreft coefficients.
  • If the quantum A-period conjecture (11) is correct, the signed-area generating function for lattice walks is a direct calculational path to quantum periods of local $\mathbb{F}_0$ and local $\mathcal{B}_3$, and conversely topological string techniques give new information about walk areas.
  • The square-lattice case yields an elliptic-integral formula for the derivative of the quantum A-period and recovers the known strong-weak coupling energy relation for local $\mathbb{F}_0$ (up to a normalization factor), connecting the Hofstadter spectrum to quantum geometry.
  • The same framework suggests analogous correspondences for other planar lattices (honeycomb, Lieb, king's, kagome) and their associated toric Calabi-Yau threefolds, as well as possible extensions to three-dimensional walks and Calabi-Yau fourfolds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if (11) holds to all orders, then the coefficients of the quantum A-period, which are related to enumerative invariants on the Calabi-Yau side, are themselves lattice-walk counts, so open Gromov-Witten-type data could in principle be computed by a purely combinatorial trace expansion.
  • The paper does not address the convergence question; a natural test is whether the $q\to\infty$ trace identity can be justified termwise, since each fixed $N$ only uses the proof for $N<q$. One could try to prove (11) by showing both sides satisfy the same difference equation, the quantized Picard-Fuchs equation mentioned in the conclusion.
  • The strong-weak coupling relation (13) may be a shadow of a duality in the exclusion-statistics picture: the same Kreft coefficients reappear under the exchange of $q$ and $p$, suggesting an S-duality-type symmetry between the walk counts at rational flux $p/q$ and $q/p$.
  • The enumeration method likely extends to other lattices whose Hofstadter spectra have known Kreft coefficients; if the correspondence with toric geometries survives, it would give a dictionary between planar walk combinatorics and mirror symmetry that the paper only sketches.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies signed-area enumeration of closed random walks on square and triangular lattices with anisotropic hopping weights, and connects these counts to cluster coefficients of exclusion statistics and to the quantum A-period of toric Calabi-Yau threefolds. The main technical results are closed-form expressions for the anisotropic generating functions C_N(A) for square walks (Eq. (6)) and for a special family of triangular walks (Eq. (8)), obtained from traces of anisotropic Hofstadter-like Hamiltonians. The paper then proposes a conjecture (Eq. (11)) relating these enumeration counts to the quantum A-period of local F0 and local B3 geometries, with verification up to z^12.

Significance. If the enumeration formulas are correct, they generalize known isotropic lattice-walk area results by tracking step counts in each direction, and they connect classical random-walk combinatorics to exclusion statistics and topological strings. The paper also contains useful ancillary material: explicit low-degree traces, numerical tables, recurrence relations, and a suggested interpretation of Kreft coefficients in terms of exclusion particles. The conjecture (11) is clearly stated and the paper honestly labels it as a conjecture; however, its novelty is tempered by the fact that the quantum A-period is defined through the same log-determinant/trace identity used in the derivation, so (11) is more a restatement of the known Hofstadter/Calabi-Yau correspondence than an independent prediction.

major comments (3)
  1. [Section 2.2, Eq. (6)] The derivation of Eq. (6) is incomplete at the step labeled 'Replacing all binomials of the form binom(2l,k) with sum_m binom(l,m) binom(l,k-m)(bb')^m(cc')^{l-m}'. This replacement is not an immediate algebraic identity for the anisotropic spectral function s_k = (1-Q^{-k})(cc' - bb'Q^k): expanding s_k^l gives coefficients of Q^{kr} of the form (-1)^r sum_i binom(l,i) binom(l,r+i)(cc')^{l-r-i}(bb')^{r+i}, which is structurally different from the expression inserted into (6). Since Eq. (6) is the paper's main explicit enumeration result for square lattice walks, this unproved step is load-bearing; a proof or a precise reference establishing the identity is required.
  2. [Section 2.3, Eq. (8)] Equation (8) is derived only under the restriction a' = ab^2/c'^2 and b' = cc'/b, as explicitly stated in the text before Eq. (8). The abstract and conclusion, however, claim closed-form expressions for triangular lattice walks without this restriction. The paper itself says the general case 'can be treated in a similar approach' but the resulting expression 'is cumbersome and will not be presented.' The general claim in the abstract is therefore unsupported; the formula as stated applies to a codimension-two subfamily of hopping parameters.
  3. [Section 3, Eq. (11)] The conjecture (11) is not an independent relation: the paper derives t = (1/q) log det(1/z - H_{1,2}) + O(z^q) from Eq. (7), and the log-determinant identity log det = -sum_N z^N/N tr H^N is used earlier in Section 2.2. Thus (11) follows from the definition of the quantum A-period and the trace representation, up to the unresolved interchange of the infinite sum with the q -> infinity limit. The verification up to z^12 is evidence of consistency but not of a new structural relation. The authors should either state clearly that the conjecture is an interpretive reformulation of known equivalences or provide an independent test that goes beyond the log-det identity.
minor comments (5)
  1. [Introduction] The Introduction contains duplicated paragraphs, which should be removed.
  2. [Section 2.2] The identity TrH_sq^N = (1/q) tr H_2^N for N < q is stated without proof or reference; since the cluster expansion in Eq. (5) depends on it, a derivation or a citation to the relevant trace theorem would improve the paper.
  3. [Section 2.2, Eq. (5)] The notation in the composition sum is unclear: the bounds and the meaning of 'composition of N/2' should be stated explicitly, and the equation should be checked for a simple example such as N=4 to show how the terms combine.
  4. [Section 3, Eq. (11)] In Eq. (11), the variable A on the right-hand side is half-integer for triangular lattice walks while the quantum A-period expansion (10) is in integer powers of z; the convention for the unit area should be stated in the equation itself.
  5. [Appendix B] The tables would be more useful if they included the anisotropic cases used in Eq. (8), not only the isotropic values.

Circularity Check

1 steps flagged · score 6.0 of 10

Quantum A-period conjecture (11) restates the trace-log identity: C_N is defined by the trace, so the CY/lattice link is not an independent prediction.

  1. renaming known result [Section 3, Eq. (11) and the sentences following it]
    "We observe that (10) is related to the signed area enumeration of triangular lattice walks, namely t = − log(z) − P∞ N=1 zN 1/N TrH N tri = − log(z) − P∞ N=1 zN 1/N P A CN (A)QA ... From (7), we obtain t = (1/q) log det(1/z − H1,2) + O(zq)."

    By (4), TrH_tri^N = Σ_A C_N(A) Q^A by definition, so the right-hand side of (11) is just the trace series of the same Hamiltonian whose mirror curve defines the quantum A-period t. The paper then rewrites this series as (1/q) log det(1/z − H_{1,2}), which is the standard log-det = tr-log identity. Hence (11) is not an independent bridge between lattice-walk combinatorics and Calabi-Yau geometry: it restates the known spectral-determinant representation of t in terms of the trace coefficients C_N(A). The conjecture therefore reduces to equation (4) plus the trace-log identity, rather than being a new prediction forced by the enumeration data.

full rationale

The enumeration formulas (6) and (8) are not themselves circular: they are obtained from the Kreft/cluster expansion and are tested against known isotropic and total-count limits. The main circularity is concentrated in the quantum A-period conjecture (11). Equation (4) defines C_N(A) as the coefficient of Q^A in TrH_tri^N, so the right-hand side of (11) is literally the trace series of H_tri; the left-hand side t is the quantum A-period of the same Hamiltonian, and the paper's own 'From (7), we obtain t=(1/q)log det...' invokes the trace-log identity. Thus the claimed CY/lattice correspondence is a renaming/restatement of the known relation between quantum periods and spectral determinants, not an independent derivation. The unproved 'binomial replacement' in the passage leading to (6) is a serious derivation gap but is not a circularity; the N<q identity being extended to all N is a limit-interchange gap, also not circular. Self-citations [30,46] provide auxiliary combinatorial ingredients but are not used to force the conjecture. Because the counting content has independent value while the central conjecture reduces by construction, the circularity is partial.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation does not fit any free parameters to data; the hopping amplitudes are symbolic. The main mathematical assumptions are the noncommutative commutation relation, the ad hoc phase choice to remove spurious terms, imported trigonometric sum identities, and the known quantum A-period and Hofstadter-CY correspondences. No new particles, forces, or entities are introduced.

assumptions (5)
  • domain assumption Commutation relation v u = Q u v for lattice hopping operators (Eq. 1).
    This is the fundamental algebraic model that maps signed area to a noncommutative phase. It is standard in the Hofstadter model literature and is not derived in the paper.
  • ad hoc to paper The equivalence between H_sq and H_2 via the choice e^{ikx} = -b/c' Q^{-1/2}, ky = 0, and neglecting g_q.
    This specific choice removes the spurious Wilson loop term in the secular determinant. It is an ad hoc normalization that is not derived from first principles, though it is said to follow the procedure in Ref. [30].
  • domain assumption The trigonometric sum identity for the isotropic spectral function S_k from Refs. [26, 29].
    The anisotropic sums in (6) and (8) are computed by importing the isotropic trigonometric sum result and replacing binomial coefficients. This imported identity is a known result, but its extension to the anisotropic case is not fully proven.
  • domain assumption The quantum A-period residue formula (10) obtained following Refs. [53, 34].
    The conjecture (11) is built on this formula for the quantum A-period of local B3. The formula is taken from established WKB/residue literature and is not rederived.
  • domain assumption The correspondence between the Hofstadter model on square/triangular lattices and local F0 / local B3 geometries from Refs. [33, 34, 35, 36, 37].
    This correspondence grounds the association of walk enumeration to specific toric Calabi-Yau threefolds. The paper relies on this external result for the physical interpretation of the conjecture.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lattice random walks and quantum A-period conjecture." pith.science (2026). https://pith.science/paper/MVBZPXJX

@misc{pith2026241221128,
  author       = {Pith},
  title        = {Pith review of: Lattice random walks and quantum A-period conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MVBZPXJX}},
  note         = {Machine review of arXiv:2412.21128}
}
abstract

We derive explicit closed-form expressions for the generating function $C_N(A)$, which enumerates classical closed random walks on square and triangular lattices with $N$ steps and a signed area $A$, characterized by the number of moves in each hopping direction. This enumeration problem is mapped to the trace of powers of anisotropic Hofstadter-like Hamiltonian and is connected to the cluster coefficients of exclusion particles: exclusion strength parameter $g = 2$ for square lattice walks, and a mixture of $g = 1$ and $g = 2$ for triangular lattice walks. By leveraging the intrinsic link between the Hofstadter model and high energy physics, we propose a conjecture connecting the above signed area enumeration $C_N(A)$ in statistical mechanics to the quantum A-period of associated toric Calabi-Yau threefold in topological string theory: square lattice walks correspond to local $\mathbb{F}_0$ geometry, while triangular lattice walks are associated with local $\mathcal{B}_3$.

Figures

Figures reproduced from arXiv: 2412.21128 by the authors.

Figure 1
Figure 1. A closed random walk of length 18 on the square (left), triangular (middle), and honeycomb lattice (right), starting and ending at the bullet point, with signed area −1, −14 and 2, respectively. The region inside the walk, i.e., winding sector, is colored green if its area is positive, otherwise it is colored red. Additionally, recent studies [33, 34, 35, 36, 37] in high energy physics have revealed a profound conne… view at source ↗
Figure 2
Figure 2. , left). Considering the connection between the deformed triangular lattice (see [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 55 canonical work pages

  1. [34]

    Calabi–Yau geometry and electrons on 2d lattices,

    Y. Hatsuda, Y. Sugimoto, and Z. Xu, “Calabi–Yau geometry and electrons on 2d lattices,” Phys. Rev. D 95, 086004 (2017)

    work page 2017

  2. [53]

    Topological strings from quantum mechan- ics,

    A. Grassi, Y. Hatsuda, and M. Mari˜ no, “Topological strings from quantum mechan- ics,” Ann. Henri Poincar´ e17, 3177 (2016)

    work page 2016

  3. [1]

    Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,

    D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Phys. Rev. B 14, 2239 (1976)

    work page 1976

  4. [2]

    A tale of two fractals: The Hofstadter butterfly and the integral Apol- lonian gaskets,

    I. I. Satija, “A tale of two fractals: The Hofstadter butterfly and the integral Apol- lonian gaskets,” Eur. Phys. J. Special Topics 225, 2533 (2016)

    work page 2016

  5. [3]

    Nests and chains of Hofstadter butterflies,

    I. I. Satija, M. Wilkinson, “Nests and chains of Hofstadter butterflies,” J. Phys. A: Math. Theor. 53, 085703 (2020)

    work page 2020

  6. [4]

    New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,

    K. v. Klitzing, G. Dorda, M. Pepper, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance,” Phys. Rev. Lett. 45, 494 (1980)

    work page 1980

  7. [5]

    Quantized Hall Conductance in a Two-Dimensional Periodic Potential,

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall Conductance in a Two-Dimensional Periodic Potential,” Phys. Rev. Lett. 9, 405 (1982)

    work page 1982

  8. [6]

    Topological invariant and the quantization of the Hall conductance,

    M. Kohmoto, “Topological invariant and the quantization of the Hall conductance,” Ann. Phys. 160, 343 (1985)

    work page 1985

Show all 57 references
  1. [7]

    Spectroscopic signatures of localiza- tion with interacting photons in superconducting qubits,

    P. Roushan, C. Neill, J. Tangpanitanon, et al., “Spectroscopic signatures of localiza- tion with interacting photons in superconducting qubits,” Science 358, 1175 (2017)

    2017

  2. [8]

    Magnetic subband structure of electrons in hexag- onal lattices,

    F. H. Claro and G. H. Wannier, “Magnetic subband structure of electrons in hexag- onal lattices,” Phys. Rev. B 19, 6068 (1979)

    1979

  3. [9]

    Energy spectrum and the quantum Hall effect on the square lattice with next-nearest-neighbor hopping,

    Y. Hatsugai and M. Kohmoto, “Energy spectrum and the quantum Hall effect on the square lattice with next-nearest-neighbor hopping,” Phys. Rev. B 42, 8282 (1990). 18

    1990

  4. [10]

    Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods,

    J. Bellissard, C. Kreft, and R. Seiler, “Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods,” J. Phys. A: Math. Gen. 24, 2329 (1991)

    1991

  5. [11]

    A study of the ambiguity in the solutions to the Diophantine equation for Chern numbers,

    J. E. Avron, O. Kenneth, and G. Yehoshua, “A study of the ambiguity in the solutions to the Diophantine equation for Chern numbers,” J. Phys. A: Math. Theor. 47, 185202 (2014)

    2014

  6. [12]

    Landau level spectrum of Bloch electrons in a honeycomb lattice,

    R. Rammal, “Landau level spectrum of Bloch electrons in a honeycomb lattice,” J. Phys. France 46, 1345 (1985)

    1985

  7. [13]

    Models of the Hofstadter-type,

    C. Kreft and R. Seiler, “Models of the Hofstadter-type,” J. Math. Phys. 37, 5207 (1996)

    1996

  8. [14]

    The colored Hofstadter butterfly for the honeycomb lattice,

    A. Agazzi, J.-P. Eckmann, and G. M. Graf, “The colored Hofstadter butterfly for the honeycomb lattice,” J. Stat. Phys. 156, 417 (2014)

    2014

  9. [15]

    Landau levels in the case of two degenerate coupled bands: Kagom´ e lattice tight-binding spectrum,

    Y. Xiao, V. Pelletier, P. M. Chaikin, and D. A. Huse, “Landau levels in the case of two degenerate coupled bands: Kagom´ e lattice tight-binding spectrum,” Phys. Rev. B 67, 104505 (2003)

    2003

  10. [16]

    Light-induced Hofstadter’s butterfly spectrum of ultracold atoms on the two-dimensional kagom´ e lattice,

    J.-M. Hou, “Light-induced Hofstadter’s butterfly spectrum of ultracold atoms on the two-dimensional kagom´ e lattice,” Chinese Phys. Lett.26, 123701 (2009)

    2009

  11. [17]

    Quantized Hall effect in three dimensions,

    G. Montambaux and M. Kohmoto, “Quantized Hall effect in three dimensions,” Phys. Rev. B 41, 11417 (1990)

    1990

  12. [18]

    Generalized flux states on 3-dimensional lattice,

    Y. Hasegawa, “Generalized flux states on 3-dimensional lattice,” J. Phys. Soc. Jpn. 59, 4384 (1990)

    1990

  13. [19]

    Hofstadter butterfly and integer quantum Hall effect in three dimensions,

    M. Koshino, H. Aoki, K. Kuroki, S. Kagoshima, and T. Osada, “Hofstadter butterfly and integer quantum Hall effect in three dimensions,” Phys. Rev. Lett. 86, 1062 (2001)

    2001

  14. [20]

    Integer quantum Hall effect in isotropic three-dimensional crystals,

    M. Koshino and H. Aoki, “Integer quantum Hall effect in isotropic three-dimensional crystals,” Phys. Rev. B 67, 195336 (2003)

    2003

  15. [21]

    Hofstadter-type spectral diagrams for the Bloch electron in three dimensions,

    J. Br¨ uning, V. V. Demidov, and V. A. Geyler, “Hofstadter-type spectral diagrams for the Bloch electron in three dimensions,” Phys. Rev. B 69, 033202 (2004)

    2004

  16. [22]

    Linking topological features of the Hofstadter model to optical diffraction figures,

    F. Di Colandrea, A. D’Errico, M. Maffei, H. M. Price, M. Lewenstein, L. Marrucci, F. Cardano, A. Dauphin, and P. Massignan, “Linking topological features of the Hofstadter model to optical diffraction figures,” New J. Phys. 24, 013028 (2022)

    2022

  17. [23]

    Universality of Hofstadter Butterflies on Hyperbolic Lattices,

    A. Stegmaier, L. K. Upreti, R. Thomale, and I. Boettcher, “Universality of Hofstadter Butterflies on Hyperbolic Lattices,” Phys. Rev. Lett. 128, 166402 (2022)

    2022

  18. [24]

    Signed area enumeration for lattice paths,

    S. Ouvry and A. P. Polychronakos, “Signed area enumeration for lattice paths,” S´ emin. Lothar. Combin.87B, 4 (2023). 19

    2023

  19. [25]

    Exact random walk distri- butions using noncommutative geometry,

    J. Bellissard, C. J. Camacho, A. Barelli, and F. Claro, “Exact random walk distri- butions using noncommutative geometry,” J. Phys. A: Math. Gen. 30, L707 (1997)

    1997

  20. [26]

    The algebraic area of closed lattice random walks,

    S. Ouvry and S. Wu, “The algebraic area of closed lattice random walks,” J. Phys. A: Math. Theor. 52, 255201 (2019)

    2019

  21. [27]

    Exclusion statistics and lattice random walks,

    S. Ouvry and A. P. Polychronakos, “Exclusion statistics and lattice random walks,” Nucl. Phys. B 948, 114731 (2019)

    2019

  22. [28]

    Algebraic area of Lattice Random Walks and Exclusion Statistics,

    L. Gan, “Algebraic area of Lattice Random Walks and Exclusion Statistics,” Uni- versit´ e Paris-Saclay, 2023 (PhD thesis)

    2023

  23. [29]

    Lattice walk area combinatorics, some remark- able trigonometric sums and Ap´ ery-like numbers,

    S. Ouvry and A. P. Polychronakos, “Lattice walk area combinatorics, some remark- able trigonometric sums and Ap´ ery-like numbers,” Nucl. Phys. B960, 115174 (2020)

    2020

  24. [30]

    Algebraic area enumeration of random walks on the honeycomb lattice,

    L. Gan, S. Ouvry, and A. P. Polychronakos, “Algebraic area enumeration of random walks on the honeycomb lattice,” Phys. Rev. E 105, 014112 (2022)

    2022

  25. [31]

    ‘Fractional statistics’ in arbitrary dimensions: A generalization of the Pauli principle,

    F. D. M. Haldane, “‘Fractional statistics’ in arbitrary dimensions: A generalization of the Pauli principle,” Phys. Rev. Lett. 67, 937 (1991)

    1991

  26. [32]

    Statistical Distribution for Generalized Ideal Gas of Fractional-Statistics Particles,

    Y.-S. Wu, “Statistical Distribution for Generalized Ideal Gas of Fractional-Statistics Particles,” Phys. Rev. Lett. 73, 922 (1994); Erratum Phys. Rev. Lett. 74, 3906 (1995)

    1994

  27. [33]

    Hofstadter’s butterfly in quantum ge- ometry,

    Y. Hatsuda, H. Katsura, and Y. Tachikawa, “Hofstadter’s butterfly in quantum ge- ometry,” New J. Phys. 18, 103023 (2016)

    2016

  28. [35]

    Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves,

    Z. Duan, J. Gu, Y. Hatsuda, and T. Sulejmanpasic, “Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves,” J. High En- erg. Phys. 2019, 79 (2019)

    2019

  29. [36]

    Recent developments in topological string theory,

    M.-X. Huang, “Recent developments in topological string theory,” Sci. China Phys. Mech. Astron. 62, 990001 (2019)

    2019

  30. [37]

    Bloch electrons on honeycomb lattice and toric Calabi- Yau geometry,

    Y. Hatsuda and Y. Sugimoto, “Bloch electrons on honeycomb lattice and toric Calabi- Yau geometry,” J. High Energ. Phys. 2020, 26 (2020)

    2020

  31. [38]

    Chern-Simons theory and topological strings,

    M. Mari˜ no, “Chern-Simons theory and topological strings,” Rev. Mod. Phys.77, 675 (2005)

    2005

  32. [39]

    Spectral theory and mirror symmetry,

    M. Mari˜ no, “Spectral theory and mirror symmetry,” Proc. Symp. Pure Math. 98, 259 (2018)

    2018

  33. [40]

    Lectures on mirror symmetry and topological string theory,

    M. Alim, “Lectures on mirror symmetry and topological string theory,” arXiv:1207.0496 (2012). 20

    2012 arXiv

  34. [41]

    Instantons, topological strings, and enumerative geometry,

    R. J. Szabo, “Instantons, topological strings, and enumerative geometry,” Adv. math. phys. 2010, 107857 (2010)

    2010

  35. [42]

    Topological Strings and Integrable Hierarchies,

    M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mari˜ no, and C. Vafa, “Topological Strings and Integrable Hierarchies,” Commun. Math. Phys. 261, 451 (2006)

    2006

  36. [43]

    Les Houches lectures on matrix models and topological strings,

    M. Mari˜ no, “Les Houches lectures on matrix models and topological strings,” arXiv:hep-th/0410165 (2005)

    2005 arXiv

  37. [44]

    Spectral equations for the modular oscillator,

    R. M. Kashaev and S. M. Sergeev, “Spectral equations for the modular oscillator,” Rev. Math. Phys. 30, 1840009 (2018)

    2018

  38. [45]

    Explicit computation of the discriminant for the Harper equation with rational flux,

    C. Kreft, “Explicit computation of the discriminant for the Harper equation with rational flux,” SFB 288 Preprint No. 89, TU-Berlin, 1993 (unpublished)

    1993

  39. [46]

    Combinatorics of generalized Dyck and Motzkin paths,

    L. Gan, S. Ouvry, and A. P. Polychronakos, “Combinatorics of generalized Dyck and Motzkin paths,” Phys. Rev. E 106, 044123 (2022)

    2022

  40. [47]

    Hofstadter point spectrum trace and the almost Mathieu operator,

    S. Ouvry, S. Wagner, and S. Wu, “Hofstadter point spectrum trace and the almost Mathieu operator,” J. Phys. A: Math. Theor. 59, 073504 (2018)

    2018

  41. [48]

    Maximal and minimal polyiamonds,

    W. C. Yang and R. R. Meyer, “Maximal and minimal polyiamonds,” Math Prog Technical Reports, University of Wisconsin–Madison (2002)

    2002

  42. [49]

    WKB method and quantum periods beyond genus one,

    F. Fischbach, A. Klemm and C. Nega, “WKB method and quantum periods beyond genus one,” J. Phys. A: Math. Theor. 52, 075402 (2019)

    2019

  43. [50]

    On the resurgent structure of quantum periods,

    J. Gu, M. Mari˜ no, “On the resurgent structure of quantum periods,” SciPost Phys. 15, 035 (2023)

    2023

  44. [51]

    Quantum geometry of refined topological strings,

    M. Aganagic, M. C. N. Cheng, R. Dijkgraaf, D. Krefl, and C. Vafa, “Quantum geometry of refined topological strings,” J. High Energ. Phys. 2012, 19 (2012)

    2012

  45. [52]

    Topological strings and quantum spectral problems,

    M.-x. Huang and X.-f. Wang, “Topological strings and quantum spectral problems,” J. High Energ. Phys. 2014, 150 (2014)

    2014

  46. [54]

    On the algebraic area of lattice walks and the Hofstadter model,

    S. Ouvry, S. Wagner, and S. Wu, “On the algebraic area of lattice walks and the Hofstadter model,” J. Phys. A: Math. Theor. 49, 495205 (2016)

    2016

  47. [55]

    Hofstadter-Toda spectral duality and quantum groups,

    P. Marra, V. Proietti, and X. Sheng, “Hofstadter-Toda spectral duality and quantum groups,” J. Math. Phys. 65, 071903 (2024)

    2024

  48. [56]

    Exact quantization conditions for the relativistic Toda lattice,

    Y. Hatsuda and M. Mari˜ no, “Exact quantization conditions for the relativistic Toda lattice,” J. High Energ. Phys. 2016, 133 (2016)

    2016

  49. [57]

    Algebraic area of cubic lattice walks,

    L. Gan, “Algebraic area of cubic lattice walks,” Phys. Rev. E 108, 054104 (2023). 21

    2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.