Pith. sign in

REVIEW 2 major objections 4 minor 92 references

Riemann surfaces for KPZ with periodic boundaries

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper claims that the exact finite-volume KPZ height probabilities for flat, wedge, and stationary initial conditions are traces of holomorphic differentials on one infinite-genus Riemann surface, and that two independent derivations…

desk verdict Careful Riemann-surface reformulation of known KPZ formulas with a genuinely new equivalence proof between Prolhac and Baik-Liu; one explicitly-flagged uniqueness conjecture is load-bearing for half of that equivalence. read the letter →

arxiv 1908.08907 v2 pith:KLUE6HNO submitted 2019-08-23 cond-mat.stat-mech math-phmath.MP

Sylvain Prolhac This is my paper · ORCID
classification cond-mat.stat-mechmath-phmath.MP MSC 14H5530F3037K1060K35
keywords KPZequationperiodicboundaryconditionsRiemannsurfaceshalf-integerpolylogarithmsBetheansatzTASEPFredholmdeterminantsKdV
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

The paper is trying to establish that the known exact formulas for one-dimensional KPZ fluctuations in a finite ring—flat, sharp-wedge, and stationary initial conditions, and even multiple-time joint distributions—are not separate calculations but different faces of one geometric object: a holomorphic differential on a Riemann surface associated to half-integer polylogarithms, integrated around an infinite cylinder and summed over all sheets of a covering map. If correct, this unification turns the large sums over Bethe-ansatz particle-hole excitations in the earlier formulas into traces of a single differential, and it supplies the missing proof that the two known derivations of the flat and wedge distributions agree. The same surface already appears in stationary large deviations, so the paper suggests that the full finite-time dynamics can be recovered from static large-deviation data by analytic continuation. The flat-initial-condition probability is also identified as a KdV soliton tau function averaged over the soliton velocity, with the integration variable playing the role of a moduli parameter for a degenerate hyperelliptic surface.

What carries the argument

The Riemann surface $R$ is the natural single-valued domain of the half-integer polylogarithm $\chi_{\emptyset}(\nu)=-\mathrm{Li}_{5/2}(-e^{\nu})/\sqrt{2\pi}$; its sheets are indexed by finite subsets of $\mathbb{Z}+1/2$, which gives it the topology of an infinite-dimensional hypercube. Quotients by translation automorphisms produce $\check{R}$, while quotients by involutions that remove chosen branch points produce $R_{\Delta}$. The key operation is the trace of a holomorphic differential over the covering map to the cylinder $C$, $(\mathrm{tr}_{\rho}\,\omega)(q)=\sum_{p:\rho(p)=q}\omega(p)$; because the trace of a holomorphic differential is holomorphic on $C$, the loop of integration in the probability formula may be moved freely. The paper proves that the exponential building blocks $e^{2I}$, $e^{I+J}$, $e^{2J}$, and $e^{2K}$ are well-defined meromorphic functions on these surfaces once the Vandermonde determinants and powers of $i/4$ are supplied by analytic continuation.

What would settle it

Find one finite set $P\subset\mathbb{Z}+1/2$ and one $s>0$ for which $\chi'_P(\nu)=s$ has two distinct solutions with $\mathrm{Re}\,\nu>0$, or none; that would falsify the conjectured uniqueness behind (122) and require modifying the contour deformation leading to (4). Short of that, truncating the sheet sum in (4) at increasing $|P|$ and checking that the $c<0$ and $c>0$ evaluations agree, or comparing with Monte Carlo data for TASEP on a ring at intermediate times, would test the claimed equivalence numerically.

Watch

Extended reading notes

Core claim

Equations (1), (6), (11), and (13) are claimed to be exactly equivalent to the known formulas of [39]–[42]. The mechanism is that the finite subsets of half-integers labelling the sheets of $\check{R}$ and $R_{\Delta}$ are the same objects as the particle-hole excitation sets in the Bethe-ansatz sums, so what looked like a discrete sum over excitations is actually a trace over a covering map from the Riemann surface to the cylinder $C$. Carrying out the trace reproduces the earlier formulas, and comparing the two choices of fundamental domain shows that the expressions in [39] and [40] agree, a fact that had previously only been checked numerically. The paper also rewrites the multiple-time formula (13) from [42] and analyses the pole structure of the integrand, showing that poles occur exactly when two points coincide on the same Riemann surface.

Load-bearing premise

The load-bearing premise is that for every sheet $P$ and every $s>0$ the equation $\chi'_P(\nu_P(s))=s$ has exactly one solution with $\mathrm{Re}\,\nu_P(s)>0$—a uniqueness property the paper states as conjectured, not proved.

Editorial extensions

If this is right

  • The flat formula (4) and the wedge formula (10) are provably the same whether one starts from the derivation in [39] or the derivation in [40], closing the previous gap.
  • For sharp wedge initial condition, the particle-hole constraints $|P|_+=|H|_-$ and $|P|_-=|H|_+$ emerge automatically from the sheet structure of $R_{\Delta}$, rather than being imposed by hand.
  • The flat probability is an $N=\infty$ KdV soliton tau function, and the integration variable $\nu$ is the common soliton velocity; higher KdV time variables appear as derivatives of $\chi_{\emptyset}$.
  • The stationary-initial-condition probability costs only the factor $-\sqrt{2\pi}e^{-\nu}\partial_u$ applied to the wedge differential, so the same Riemann-surface machinery covers it.
  • In the multiple-time formula, poles in $\nu_{\ell+1}=\nu_{\ell}+2i\pi m$ appear exactly when $\Delta_{\ell+1}=\Delta_{\ell}+m$ and $P_{\ell+1}=P_{\ell}+m$, i.e. when the points coincide on the same Riemann surface.

Reading between the lines

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

  • If the paper's picture holds, the entire finite-time transition probability is an analytic continuation of stationary large-deviation data, making the excited-state spectrum a derived object rather than an input.
  • A direct numerical check at small time that the KdV tau function approaches the Painlevé II scaling solution would test the soliton-gas interpretation of the flat probability.
  • The trace formalism looks transferable to other solvable exclusion processes and to finite asymmetry, with the half-integer polylogarithm replaced by the corresponding special function.
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

2 major / 4 minor

Summary. The paper proposes a unified geometric reformulation of exact finite-volume KPZ fluctuation formulas with periodic boundary conditions. It constructs infinite-genus Riemann surfaces R, ˇR, and R∆ associated with half-integer polylogarithms, defines meromorphic functions χ, χ′, χ′′, e^{2J}, e^{2K}, etc. on these surfaces, and expresses the one-point probabilities for flat, sharp wedge, and stationary initial conditions, as well as the multiple-time joint distribution for sharp wedge initial condition, as integrals of traces of holomorphic differentials over the covering maps to an infinite cylinder. In Section 5, these formulas are shown to be equivalent to the earlier Bethe-ansatz-based expressions of [39]–[42], with the additional claim that the distinct formulas of [39] and [40] agree once their sheet sums are reinterpreted as traces on the same Riemann surface. The paper also discusses connections to stationary large deviations, particle-hole excitations, and KdV/KP soliton tau functions.

Significance. If the claimed equivalences hold, the paper gives a genuinely unifying picture of exact finite-volume KPZ results: complicated sums over particle-hole excitations become traces of holomorphic differentials over sheets of ramified coverings, and the previously separate [39] and [40] derivations appear as different choices of fundamental domain. The analytic-continuation identities are derived in detail in the appendices, and the manipulations from the known formulas are transparent and largely self-contained once the ingredients of [39]–[42] are accepted. The main caveat is that one direction of the equivalence relies on an unproved and explicitly labeled conjecture about the uniqueness of the solutions of χ′_P(ν)=s. This does not affect the reformulation itself, but it does affect the strength of the claimed proof that the [39] and [40] formulas agree.

major comments (2)
  1. [§5.1.1, Eq. (122)] The passage from the generating function (122) to the probability formula (123), and hence to (4) for c>0, requires that for every finite P the equation χ′_P(ν_P(s))=s have a unique solution with Re ν_P(s)>0 when s>0. The manuscript states at (122) that this is “conjectured to be unique” and does not supply a proof or a reference to one. The change of variables s=−iχ′_P(ν) is a bijective substitution only under this uniqueness; if multiple roots exist, the deformed contour {c+iℝ} is not the image of the s-integration contour, and extra contributions can appear. Consequently the claimed equivalence between the [39] expression and (4) for c>0 is not fully established, and the subsequent conclusion that the [39] and [40] flat-initial-condition formulas agree is conditional on this conjecture. This is a load-bearing gap that should be resolved by a proof, a precise citation of a proof, or a clearly stated weakening of the claim.
  2. [§5.2.1, Eq. (131)] The same unproved uniqueness issue appears for sharp wedge initial condition: ν_{P,H}(s) is defined as the solution of χ′_{P,H}(ν)=s and is again stated to be “conjectured to be unique.” The derivation of (133) from (131) and the later identification of (10) with the [39] formula for c>0 depend on this conjecture. The direct derivation from [40] in §5.2.2 gives (10) only for c<0. Since the paper’s stated goal includes showing that the [39] and [40] sharp-wedge expressions agree, this second instance of the same unresolved assumption should also be addressed explicitly. The manuscript would be strengthened either by proving the conjecture or by making the conditional nature of the [39]=[40] statement prominent throughout Section 5.
minor comments (4)
  1. [§2.2, Eq. (4)] The definition of I0 is split across the text and depends on the sign of c, but this is stated only parenthetically in the paragraph after the display. Since (4) is claimed for both c<0 and c>0, a displayed convention for I0 with explicit branch choices would improve clarity.
  2. [§3.8.2] The sentence describing the automorphism T says that the map is a homeomorphism and “hence an automorphism since it is locally holomorphic.” Strictly speaking one should also note that the inverse is holomorphic, or cite the standard fact that a bijective locally biholomorphic map between Riemann surfaces is a biholomorphism.
  3. [§5.1.2] The sentence “The rest of the section is essentially a more detailed version of the derivation of equation (22), run backwards” is vague because equation (22) is presented later in §2.6.3. A forward reference and a one-sentence explanation of the logical structure would help the reader.
  4. [General] The manuscript contains numerous small typographical and spacing issues, such as missing spaces around mathematical expressions in prose (e.g., “R∆”, “χ∆P”, “S0P”). A careful copyedit would noticeably improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the Riemann-surface formulas are explicitly derived from independent Bethe-ansatz results [39,40], with no fitted parameter renamed as a prediction.

full rationale

The paper is a reformulation paper: Eqs. (1), (6), (11), and (13) are introduced as equivalent to known results from [39-42], and Section 5 proves the equivalence by direct manipulation of the earlier formulas. The Riemann surfaces R, R^Δ, and the traced differentials are constructed in Sections 3-4 from the analytic structure of half-integer polylogarithms, independently of the KPZ formulas; the comparison in Section 5 then matches them to [39] and [40] through Cauchy determinants, regularized integrals, and analytic-continuation identities. No parameter is fitted to a data subset and then called a prediction: coefficients such as V_P, Ξ_x^Δ, and W_P are fixed by the requirement that the integrands be single-valued on the relevant Riemann surfaces, and they coincide with the previously known Bethe-ansatz factors only after explicit evaluation. The claimed agreement between [39] and [40] is supported by showing that both reduce to the same integral representation (4) or (10), with two independent routes: [40] supplies a derivation for Re ν < 0, and [39] supplies one for Re ν > 0. Although [39] is the author's own earlier work, it is a peer-reviewed Bethe-ansatz result used as an external benchmark, and the equivalence also uses the non-overlapping work [40]. The main caveat is a correctness risk rather than circularity: at eq. (122), and in the sharp-wedge analogue (131), the change of variables s = -iχ'_P(ν) relies on the statement that χ'_P(ν_P(s)) = s has a unique solution with Re ν_P(s) > 0, which the paper explicitly labels as 'conjectured to be unique'. If that conjecture fails, the c > 0 direction of the equivalence would require additional contour arguments, but this is an unproved assumption inherited from [39], not a reduction of the target identity to itself by construction. The derivation chain does not define any target quantity in terms of itself, nor does it replace proof by a self-citation chain, so no significant circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the only adjustable quantities are contour positions and regularization constants, which do not enter the physical predictions. The paper introduces infinite-genus Riemann surfaces as mathematical constructions, but they are not new physical entities with independent falsifiable handles. The core postulates are the correctness of the cited exact formulas and the uniqueness conjecture for nu_P(s), the latter being the weakest unproved input.

assumptions (4)
  • domain assumption The exact one-point and multi-point formulas of [39-42] for TASEP/KPZ in finite volume are correct.
    The paper uses these formulas as input and derives its Riemann-surface expressions by equivalence to them, as stated in sections 5.1 to 5.3.
  • ad hoc to paper For each sheet P, the equation chi'_P(nu_P(s)) = s has a unique solution with Re nu_P(s) > 0 for s > 0.
    Stated as a conjecture below equation (122); needed to pass from the generating function to the cumulative distribution via Fourier transform and contour deformation. It is not proved in this paper.
  • domain assumption The Bethe ansatz for TASEP on a ring is complete and its KPZ-scaling asymptotics are valid for the relevant initial conditions.
    The input formulas from [39-42] come from large-scale asymptotics of the Bethe ansatz, and the conclusions section says the results are based on complicated asymptotics of Bethe ansatz formulas.
  • standard math Standard results on compact Riemann surfaces, ramified coverings and traces of differentials extend to the infinite-genus limit considered here.
    Section 3 recalls these facts and cites [81-83,85]; section 3.8 extends them to infinite genus by considering finite-sheet gluing and treating the points at infinity as punctures.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Riemann surfaces for KPZ with periodic boundaries." pith.science (2026). https://pith.science/paper/KLUE6HNO

@misc{pith2026190808907,
  author       = {Pith},
  title        = {Pith review of: Riemann surfaces for KPZ with periodic boundaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLUE6HNO}},
  note         = {Machine review of arXiv:1908.08907}
}
read the original abstract

The Riemann surface for polylogarithms of half-integer index, which has the topology of an infinite dimensional hypercube, is studied in relation to one-dimensional KPZ universality in finite volume. Known exact results for fluctuations of the KPZ height with periodic boundaries are expressed in terms of meromorphic functions on this Riemann surface, summed over all the sheets of a covering map to an infinite cylinder. Connections to stationary large deviations, particle-hole excitations and KdV solitons are discussed.

Figures

Figures reproduced from arXiv: 1908.08907 by the authors.

Figure 1
Figure 1. Summary of several useful covering maps between th [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Picture of particle-hole excitations at both edge [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Neighbourhood of a point q = [z∗, C+] = [z∗, C−] in a Riemann surface (right) such that z∗ is a branch point of the function g0 from which the Riemann surface is built. The neighbourhood is formed by gluing together along the branch cut originating from q two half-disks obtained from taking the square root of full disks from the sheets C± (left). The complex numbers z± parametrize half a neighbourhood of q in C±. Th… view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Three different choices of branch cuts (solid lines) [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Two different choices of branch cuts (solid lines) fo [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Neighbourhood of q = [(zj )r , FP ] = [(zj )l , FP ⊖[[1,j]]] = [(zj )r , FP ⊖{j} ] = [(zj )l , FP ⊖[[1,j−1]]], 2 ≤ j ≤ N − 1 in RN , formed by gluing four quarter-disks obtained from taking the square root of half-disks in the sheets FP , FP ⊖[[1,j]], FP ⊖{j} and FP ⊖[…
Figure 7
Figure 7. Figure 7: Compact representation of branch cuts for the shee [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Sphere, torus, and surface with genus g = 5, along with hypercubes of dimensions 1, 2, 3 made of spheres connected with cylinders that can be mapped to them by continuous deformations. isomorphic to the Riemann sphere, while R3 is a torus, see figure 10. More generally…
Figure 9
Figure 9. Figure 9: Representation of the surface corresponding to th [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Representation of the torus homeomorphic to the R [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: Homology class of a loop γ = θjk . . . θj1 ·GP on the Riemann surface RN rewritten as a combination of loops ℓa,b ·GQ. Considering several loops based at a point p of a Riemann surface M, the homotopy class of their product γ may depend on the order of the loops in γ.…
Figure 12
Figure 12. Figure 12: Choice of branch cuts (red, vertical lines) parti [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Two choices for a fundamental domain of R under the action of ˇg, from which Rˇ = R/ˇg is built. The fundamental domain is the hatched portion. How sheets are glued together along branch cuts (red lines, with dots for the branch points) is not represented for clarity.…
Figure 14
Figure 14. Figure 14: Connectivity of the infinite strips S 0 P , P ⊏ Z + 1/2 partitioning the Riemann surface Rˇ. The red dots represent ramification points for the covering map ˇρ sending all the strips to the infinite cylinder C. [v, ∅] [v + 2iπ, ∅] 4iπ 2iπ 2iπ 4iπ 5iπ 3iπ iπ −iπ −3iπ −…
Figure 15
Figure 15. Figure 15: Examples of paths on R which are also closed loops on Rˇ. The solid curves belong to C∅ and the dashed lines to CBm, m = 2, 1, −1, −2 from top to bottom. Alternatively, we consider a partition of R into half-infinite strips S l,m P = {[v, P], Re v ≤ 0, 2π(m − 1/2) < I…
Figure 16
Figure 16. Figure 16: Choice of a fundamental domain of R under the action of g ∆ with ∆ = {a} (left) and ∆ = {a, b} (right). The sheets CP , partitioned along dashed lines into pairs of half-infinite strips S m P = S l,m P ∪ Sr,m P from (39), are grouped together according to the value of…
Figure 17
Figure 17. Figure 17: Two choices for the fundamental domain of the coll [PITH_FULL_IMAGE:figures/full_fig_p034_17.png]
Figure 18
Figure 18. Figure 18: Two possible choices of branch cuts for the functi [PITH_FULL_IMAGE:figures/full_fig_p037_18.png]
Figure 19
Figure 19. Figure 19: Examples of paths of integration in D for I0(ν) in (68) and for JP (ν) in (74), so that the functions are analytic in D. The vertical, red lines represent the branch cuts C \ D. The bigger, red dots are the branch points 2iπa, a ∈ Z + 1/2. From (67), the function JP t…
Figure 20
Figure 20. Figure 20: Paths β2 (solid curve) and β−1 (dashed curve) from −∞ to −∞ in C. Here, |P|+ (respectively |P|−) denotes the number of positive (resp. negative) elements of P. The identity (76) leads to JP (ν−2iπn) = J(P +n)⊖Bn (ν)+W(P +n)⊖Bn −WP when Re ν > 0. In terms of the transl…
Figure 21
Figure 21. Figure 21: Decomposition of a loop ℓa,b ·P, b < 0 < a in terms of paths of the form βn ·Q. Paths on various sheets CP are represented differently. e 2I (p), e2J (p), eI+J (p) in terms of meromorphic differentials integrated on a path of Rˇ between [−∞, ∅] and p. Defining a merom…
Figure 22
Figure 22. Figure 22: Possible choices for the path of integration in th [PITH_FULL_IMAGE:figures/full_fig_p049_22.png]
Figure 23
Figure 23. Figure 23: Path γn,m in (102) plotted for some choices of ν, µ ∈ D, n, m ∈ Z. From left to right, top to bottom, the graphs represent γ2,−1 = β2 −ν for Re µ < 0 < Re ν, γ2,−1 = β−1 −µ for Re ν < 0 < Re µ, γ2,1 for 0 < Re µ < Re ν, and γ2,−1 for 0 < Re µ < Re ν. The smaller, blac…
Figure 20
Figure 20. Figure 20: figure 20. At this point, [PITH_FULL_IMAGE:figures/full_fig_p066_20.png]
Figure 20
Figure 20. Figure 20: figure 20. Additionally, if Re [PITH_FULL_IMAGE:figures/full_fig_p068_20.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

92 extracted references · 42 canonical work pages

  1. [39]

    Prolhac, Finite-time fluctuations for the totally asymmetric exclus ion process, Phys

    S. Prolhac, Finite-time fluctuations for the totally asymmetric exclus ion process, Phys. Rev. Lett. 116, 090601 (2016), doi:10.1103/PhysRevLett.116.090601

    doi:10.1103/physrevlett.116.090601 2016

  2. [40]

    Baik and Z

    J. Baik and Z. Liu, Fluctuations of TASEP on a ring in relaxation time scale , Comm. Pure Appl. Math. 71, 0747 (2018), doi:10.1002/cpa.21702

    doi:10.1002/cpa.21702 2018

  3. [42]

    Baik and Z

    J. Baik and Z. Liu, Multi-point distribution of periodic TASEP , J. Amer. Math. Soc. 32, 609-674 (2019), doi:10.1090/jams/915

    doi:10.1090/jams/915 2019

  4. [1]

    Halpin-Healy and Y.-C

    T. Halpin-Healy and Y.-C. Zhang, Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplina ry statistical mechanics , Phys. Rep. 254, 215 (1995), doi:10.1016/0370-1573(94)00087-J

    doi:10.1016/0370-1573(94)00087-j 1995

  5. [2]

    Kriecherbauer and J

    T. Kriecherbauer and J. Krug, A pedestrian ’s view on interacting particle systems, KPZ universality and random matrices , J. Phys. A: Math. Theor. 43, 403001 (2010), doi:10.1088/1751-8113/43/40/403001

    doi:10.1088/1751-8113/43/40/403001 2010

  6. [3]

    Corwin, The Kardar-Parisi-Zhang equation and universality class , Random Matrices: Theory and Applications 1, 1130001 (2011), doi:10.1142/S2010326311300014

    I. Corwin, The Kardar-Parisi-Zhang equation and universality class , Random Matrices: Theory and Applications 1, 1130001 (2011), doi:10.1142/S2010326311300014

    doi:10.1142/s2010326311300014 2011

  7. [4]

    Borodin and V

    A. Borodin and V. Gorin, Lectures on integrable probability (2012), https://arxiv.org/abs/1212.3351

    arXiv 2012

  8. [5]

    Quastel and H

    J. Quastel and H. Spohn, The one-dimensional KPZ equation and its universality class , J. Stat. Phys. 160, 965 (2015), doi:10.1007/s10955-015-1250-9

    doi:10.1007/s10955-015-1250-9 2015

Show all 92 references
  1. [6]

    Halpin-Healy and K

    T. Halpin-Healy and K. Takeuchi, A KPZ cocktail-shaken, not stirred... , J. Stat. Phys. 160, 794 (2015), doi:10.1007/s10955-015-1282-1

    2015 doi

  2. [7]

    Spohn, The Kardar-Parisi-Zhang equation - a statistical physics pe rspective (2016), https://arxiv.org/abs/1601.00499

    H. Spohn, The Kardar-Parisi-Zhang equation - a statistical physics pe rspective (2016), https://arxiv.org/abs/1601.00499

    2016 arXiv

  3. [8]

    Takeuchi, An appetizer to modern developments on the Kardar-Parisi-Z hang univer- sality class , Physica A 504, 77 (2018), doi:10.1016/j.physa.2018.03.009

    K. Takeuchi, An appetizer to modern developments on the Kardar-Parisi-Z hang univer- sality class , Physica A 504, 77 (2018), doi:10.1016/j.physa.2018.03.009

    2018 doi

  4. [9]

    Saenz, The KPZ universality class and related topics (2019), https://arxiv.org/abs/1904.03319

    A. Saenz, The KPZ universality class and related topics (2019), https://arxiv.org/abs/1904.03319

    2019 arXiv

  5. [10]

    Takeuchi, M

    K. Takeuchi, M. Sano, T. Sasamoto and H. Spohn, Growing interfaces uncover universal fluctuations behind scale invariance , Sci. Rep. 1, 34 (2011), doi:10.1038/srep00034. 75 SciPost Physics Submission

    2011 doi

  6. [11]

    Somoza, M

    A. Somoza, M. Ortu˜ no and J. Prior, Universal distribution functions in two-dimensional localized systems , Phys. Rev. Lett. 99, 116602 (2007), doi:10.1103/PhysRevLett.99.116602

    2007 doi

  7. [12]

    van Beijeren, Exact results for anomalous transport in one-dimensional H amiltonian systems, Phys

    H. van Beijeren, Exact results for anomalous transport in one-dimensional H amiltonian systems, Phys. Rev. Lett. 108, 180601 (2012), doi:10.1103/PhysRevLett.108.180601

    2012 doi

  8. [13]

    Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains , J

    H. Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains , J. Stat. Phys. 154, 1191 (2014), doi:10.1007/s10955-014-0933-y

    2014 doi

  9. [14]

    Popkov, A

    V. Popkov, A. Schadschneider, J. Schmidt and G. Sch¨ utz , Fibonacci family of dynamical universality classes , Proc. Natl. Acad. Sci. USA 112, 12645 (2015), doi:10.1073/pnas.1512261112

    2015 doi

  10. [15]

    Kulkarni, D

    M. Kulkarni, D. Huse and H. Spohn, Fluctuating hydrodynamics for a discrete Gross- Pitaevskii equation: Mapping onto the Kardar-Parisi-Zhan g universality class , Phys. Rev. A 92, 043612 (2015), doi:10.1103/PhysRevA.92.043612

    2015 doi

  11. [16]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay and J. Haah, Quantum entanglement growth under ran- dom unitary dynamics , Phys. Rev. X 7, 031016 (2017), doi:10.1103/PhysRevX.7.031016

    2017 doi

  12. [17]

    Ljubotina, M

    M. Ljubotina, M. ˇZnidariˇ c and T. Prosen, Kardar-Parisi-Zhang physics in the quantum Heisenberg magnet , Phys. Rev. Lett. 122, 210602 (2019), doi:10.1103/PhysRevLett.122.210602

    2019 doi

  13. [18]

    de Gier, A

    J. de Gier, A. Schadschneider, J. Schmidt and G. Sch¨ utz , Kardar-Parisi-Zhang universality of the Nagel-Schreckenberg model , Phys. Rev. E 100, 052111 (2019), doi:10.1103/PhysRevE.100.052111

    2019 doi

  14. [19]

    Kardar, G

    M. Kardar, G. Parisi and Y.-C. Zhang, Dynamic scaling of growing interfaces , Phys. Rev. Lett. 56, 889 (1986), doi:10.1103/PhysRevLett.56.889

    1986 doi

  15. [20]

    Hairer, Solving the KPZ equation , Ann

    M. Hairer, Solving the KPZ equation , Ann. Math. 178, 559 (2013), doi:10.4007/annals.2013.178.2.4

    2013 doi

  16. [21]

    Kupiainen, Renormalization group and stochastic PDEs , Annales Henri Poincar´ e17, 497 (2016), doi:10.1007/s00023-015-0408-y

    A. Kupiainen, Renormalization group and stochastic PDEs , Annales Henri Poincar´ e17, 497 (2016), doi:10.1007/s00023-015-0408-y

    2016 doi

  17. [22]

    Gubinelli and N

    M. Gubinelli and N. Perkowski, KPZ reloaded, Commun. Math. Phys. 349, 165 (2017), doi:10.1007/s00220-016-2788-3

    2017 doi

  18. [23]

    Johansson, Shape fluctuations and random matrices , Commun

    K. Johansson, Shape fluctuations and random matrices , Commun. Math. Phys. 209, 437 (2000), doi:10.1007/s002200050027

    2000 doi

  19. [24]

    Baik and E

    J. Baik and E. Rains, Limiting distributions for a polynuclear growth model with e xternal sources, J. Stat. Phys. 100, 523 (2000), doi:10.1023/A:1018615306992

    2000 doi

  20. [25]

    Pr¨ ahofer and H

    M. Pr¨ ahofer and H. Spohn, Scale invariance of the PNG droplet and the Airy process , J. Stat. Phys. 108, 1071 (2002), doi:10.1023/A:1019791415147

    2002 doi

  21. [26]

    Sasamoto, Spatial correlations of the 1D KPZ surface on a flat substrate , J

    T. Sasamoto, Spatial correlations of the 1D KPZ surface on a flat substrate , J. Phys. A: Math. Gen. 38, L549 (2005), doi:10.1088/0305-4470/38/33/L01. 76 SciPost Physics Submission

    2005 doi

  22. [27]

    Ferrari and H

    P. Ferrari and H. Spohn, Scaling limit for the space-time covariance of the station- ary totally asymmetric simple exclusion process , Commun. Math. Phys. 265, 1 (2006), doi:10.1007/s00220-006-1549-0

    2006 doi

  23. [28]

    Borodin, P

    A. Borodin, P. Ferrari, M. Pr¨ ahofer and T. Sasamoto, Fluctuation properties of the TASEP with periodic initial configuration , J. Stat. Phys. 129, 1055 (2007), doi:10.1007/s10955-007-9383-0

    2007 doi

  24. [29]

    Tracy and H

    C. Tracy and H. Widom, Total current fluctuations in the asymmetric simple exclusi on process, J. Math. Phys. 50, 095204 (2009), doi:10.1063/1.3136630

    2009 doi

  25. [30]

    J. Baik, P. Ferrari and S. P´ ech´ e,Limit process of stationary TASEP near the character- istic line , Comm. Pure Appl. Math. 63, 1017 (2010), doi:10.1002/cpa.20316

    2010 doi

  26. [31]

    Borodin, I

    A. Borodin, I. Corwin and V. Gorin, Stochastic six-vertex model , Duke Math. J. 165, 563 (2016), doi:10.1215/00127094-3166843

    2016 doi

  27. [32]

    Matetski, J

    K. Matetski, J. Quastel and D. Remenik, The KPZ fixed point (2017), https://arxiv.org/abs/1701.00018

    2017 arXiv

  28. [33]

    Johansson and M

    K. Johansson and M. Rahman, Multi-time distribution in discrete polynuclear growth (2019), https://arxiv.org/abs/1906.01053

    2019 arXiv

  29. [34]

    Liu, Multi-time distribution of TASEP (2019), https://arxiv.org/abs/1907.09876

    Z. Liu, Multi-time distribution of TASEP (2019), https://arxiv.org/abs/1907.09876

    2019 arXiv

  30. [35]

    Tracy and H

    C. Tracy and H. Widom, Level-spacing distributions and the Airy kernel , Commun. Math. Phys. 159, 151 (1994), doi:10.1007/BF02100489

    1994 doi

  31. [36]

    Derrida and J

    B. Derrida and J. Lebowitz, Exact large deviation function in the asymmetric exclusion process, Phys. Rev. Lett. 80, 209 (1998), doi:10.1103/PhysRevLett.80.209

    1998 doi

  32. [37]

    Brunet and B

    E. Brunet and B. Derrida, Probability distribution of the free energy of a directed po lymer in a random medium , Phys. Rev. E 61, 6789 (2000), doi:10.1103/PhysRevE.61.6789

    2000 doi

  33. [38]

    Mallick and S

    K. Mallick and S. Prolhac, Brownian bridges for late time asymptotics of KPZ fluctua- tions in finite volume , J. Stat. Phys. 173, 322 (2018), doi:10.1007/s10955-018-2136-4

    2018 doi

  34. [41]

    Liu, Height fluctuations of stationary TASEP on a ring in relaxati on time scale , Ann

    Z. Liu, Height fluctuations of stationary TASEP on a ring in relaxati on time scale , Ann. Inst. H. Poincar´ e Probab. Statist. 54, 1031 (2018), doi:10.1214/17-AIHP831

    2018 doi

  35. [43]

    Gorissen, A

    M. Gorissen, A. Lazarescu, K. Mallick and C. Vanderzand e, Exact current statistics of the asymmetric simple exclusion process with open boundari es, Phys. Rev. Lett. 109, 170601 (2012), doi:10.1103/PhysRevLett.109.170601. 77 SciPost Physics Submission

    2012 doi

  36. [44]

    Gwa and H

    L.-H. Gwa and H. Spohn, Six-vertex model, roughened surfaces, and an asymmetric sp in Hamiltonian, Phys. Rev. Lett. 68, 725 (1992), doi:10.1103/PhysRevLett.68.725

    1992 doi

  37. [45]

    Prolhac, Spectrum of the totally asymmetric simple exclusion proces s on a pe- riodic lattice - first excited states , J

    S. Prolhac, Spectrum of the totally asymmetric simple exclusion proces s on a pe- riodic lattice - first excited states , J. Phys. A: Math. Theor. 47, 375001 (2014). doi:10.1088/1751-8113/47/37/375001

    2014 doi

  38. [46]

    Blythe and M

    R. Blythe and M. Evans, Nonequilibrium steady states of matrix-product form: a sol ver’s guide, J. Phys. A: Math. Theor. 40, R333 (2007), doi:10.1088/1751-8113/40/46/R01

    2007 doi

  39. [47]

    Lazarescu and K

    A. Lazarescu and K. Mallick, An exact formula for the statistics of the current in the TASEP with open boundaries , J. Phys. A: Math. Theor. 44, 315001 (2011), doi:10.1088/1751-8113/44/31/315001

    2011 doi

  40. [48]

    Gauti´ e, P

    T. Gauti´ e, P. Le Doussal, S. Majumdar and G. Schehr, Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary , J. Stat. Phys. (2019), doi:10.1007/s10955-019-02388-z

    2019 doi

  41. [49]

    Bender and T

    C. Bender and T. Wu, Anharmonic oscillator , Phys. Rev. 184, 1231 (1969), doi:10.1103/PhysRev.184.1231

    1969 doi

  42. [50]

    Klassen and E

    T. Klassen and E. Melzer, The thermodynamics of purely elastic scattering theories and conformal perturbation theory , Nucl. Phys. B 350, 635 (1991), doi:10.1016/0550-3213(91)90159-U

    1991 doi

  43. [51]

    Dorey and R

    P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations , Nucl. Phys. B 482, 639 (1996), doi:10.1016/S0550-3213(96)00516-0

    1996 doi

  44. [52]

    Voros, The return of the quartic oscillator

    A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’I.H.P. Physique th´ eorique 39, 211 (1983). http://www.numdam.org/item/AIHPA_1983__39_3_211_0/

    1983

  45. [53]

    Prolhac, Current fluctuations and large deviations for periodic TASEP on the relaxation scale , J

    S. Prolhac, Current fluctuations and large deviations for periodic TASEP on the relaxation scale , J. Stat. Mech. 2015, P11028 (2015), doi:10.1088/1742-5468/2015/11/P11028

    2015 doi

  46. [54]

    Motegi, K

    K. Motegi, K. Sakai and J. Sato, Long time asymptotics of the totally asym- metric simple exclusion process , J. Phys. A: Math. Theor. 45, 465004 (2012), doi:10.1088/1751-8113/45/46/465004

    2012 doi

  47. [55]

    Belokolos, A

    E. Belokolos, A. Bobenko, V. Enolski, A. Its and V. Matve ev, Algebro-geometric Approach in the Theory of Integrable Equations , Springer Series in Nonlinear Dynamics. Springer, Berlin, doi:10.1007/s00220-007-0235-1 (1994)

    1994 doi

  48. [56]

    Babelon, D

    O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems , Cambridge University Press, doi:10.1017/CBO97805115350 24 (2003)

    2003 doi

  49. [57]

    Zabrodin, Lectures on nonlinear integrable equations and their solut ions (2018), https://arxiv.org/abs/1812.11830

    A. Zabrodin, Lectures on nonlinear integrable equations and their solut ions (2018), https://arxiv.org/abs/1812.11830

    2018 arXiv

  50. [58]

    Fogedby, Nonequilibrium dynamics of a growing interface , J

    H. Fogedby, Nonequilibrium dynamics of a growing interface , J. Phys.: Condens. Matter 14, 1557 (2002), doi:10.1088/0953-8984/14/7/313. 78 SciPost Physics Submission

    2002 doi

  51. [59]

    Janas, A

    M. Janas, A. Kamenev and B. Meerson, Dynamical phase transition in large-deviation statistics of the Kardar-Parisi-Zhang equation , Phys. Rev. E 94, 032133 (2016), doi:10.1103/PhysRevE.94.032133

    2016 doi

  52. [60]

    Baik and Z

    J. Baik and Z. Liu, TASEP on a ring in sub-relaxation time scale , J. Stat. Phys. 165, 1051 (2016), doi:10.1007/s10955-016-1665-y

    2016 doi

  53. [61]

    Joshi, The second Painlev´ e hierarchy and the stationary KdV hierar chy, Publ

    N. Joshi, The second Painlev´ e hierarchy and the stationary KdV hierar chy, Publ. Res. Inst. Math. Sci. 40, 1039 (2004), doi:10.2977/prims/1145475502

    2004

  54. [62]

    A. Zabrodin, Canonical and grand canonical partition functions of Dyson g ases as tau- functions of integrable hierarchies and their fermionic re alization, Complex Analysis and Operator Theory 4, 497 (2010), doi:10.1007/s11785-010-0063-8

    2010 doi

  55. [63]

    Quastel and D

    J. Quastel and D. Remenik, KP governs random growth off a one dimensional substrate (2019), https://arxiv.org/abs/1908.10353

    2019 arXiv

  56. [64]

    Feldman, H

    J. Feldman, H. Kn¨ orrer and E. Trubowitz, Riemann Surfaces of Infinite Genus, CRM Monograph Series 20. American Mathematical Society, P rovidence, doi:10.1090/crmm/020 (2003)

    2003 doi

  57. [65]

    Krajenbrink, P

    A. Krajenbrink, P. Le Doussal and S. Prolhac, Systematic time expansion for the Kardar- Parisi-Zhang equation, linear statistics of the GUE at the e dge and trapped fermions , Nucl. Phys. B 936, 239 (2018), doi:10.1016/j.nuclphysb.2018.09.019

    2018 doi

  58. [66]

    Le Doussal, S

    P. Le Doussal, S. Majumdar, A. Rosso and G. Schehr, Exact short-time height distri- bution in the one-dimensional Kardar-Parisi-Zhang equati on and edge fermions at high temperature, Phys. Rev. Lett. 117, 070403 (2016), doi:10.1103/PhysRevLett.117.070403

    2016 doi

  59. [67]

    Prolhac, Tree structures for the current fluctuations in the exclusio n process, J

    S. Prolhac, Tree structures for the current fluctuations in the exclusio n process, J. Phys. A: Math. Theor. 43, 105002 (2010), doi:10.1088/1751-8113/43/10/105002

    2010 doi

  60. [68]

    Prolhac, Perturbative solution for the spectral gap of the weakly asy mmetric exclusion process, J

    S. Prolhac, Perturbative solution for the spectral gap of the weakly asy mmetric exclusion process, J. Phys. A: Math. Theor. 50, 315001 (2017), doi:10.1088/1751-8121/aa77de

    2017 doi

  61. [69]

    Prolhac, Extrapolation methods and Bethe ansatz for the asymmetric e xclusion process, J

    S. Prolhac, Extrapolation methods and Bethe ansatz for the asymmetric e xclusion process, J. Phys. A: Math. Theor. 49, 454002 (2016), doi:10.1088/1751-8113/49/45/454002

    2016 doi

  62. [70]

    Z. Liu, A. Saenz and D. Wang, Integral formulas of ASEP and q-TAZRP on a ring (2019), https://arxiv.org/abs/1905.02987

    2019 arXiv

  63. [71]

    Kenyon and D

    R. Kenyon and D. Wilson, Critical resonance in the non-intersecting lattice path mod el, Probab. Theor. Rel. Fields 130, 289 (2004), doi:10.1007/s00440-003-0293-z

    2004 doi

  64. [72]

    Poplavskyi, R

    M. Poplavskyi, R. Tribe and O. Zaboronski, On the distribution of the largest real eigenvalue for the real Ginibre ensemble , Ann. Appl. Probab. 27, 1395 (2017), doi:10.1214/16-AAP1233

    2017 doi

  65. [73]

    Baik and T

    J. Baik and T. Bothner, The largest real eigenvalue in the real Ginibre ensemble and i ts relation to the Zakharov-Shabat system (2018), https://arxiv.org/abs/1808.02419. 79 SciPost Physics Submission

    2018 arXiv

  66. [74]

    Derrida and A

    B. Derrida and A. Gerschenfeld, Current fluctuations of the one dimensional symmet- ric simple exclusion process with step initial condition , J. Stat. Phys. 136, 1 (2009), doi:10.1007/s10955-009-9772-7

    2009 doi

  67. [75]

    St´ ephan, Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain , J

    J.-M. St´ ephan, Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain , J. Stat. Mech. 2017, 103108 (2017), doi:10.1088/1742-5468/aa8c19

    2017 doi

  68. [76]

    Brattain, N

    E. Brattain, N. Do and A. Saenz, The completeness of the Bethe ansatz for the periodic ASEP (2015), https://arxiv.org/abs/1511.03762

    2015 arXiv

  69. [77]

    Langlands and Y

    R. Langlands and Y. Saint-Aubin, Algebro-geometric aspects of the Bethe equations , In Strings and Symmetries , vol. 447 of Lecture Notes in Physics, pp. 40–53. Berlin: Springer, doi:10.1007/3-540-59163-X 254 (1995)

    1995 doi

  70. [78]

    Langlands and Y

    R. Langlands and Y. Saint-Aubin, Aspects combinatoires des ´ equations de Bethe , In Advances in Mathematical Sciences: CRM’s 25 Years , vol. 11 of CRM Proceedings and Lecture Notes, pp. 231–302. Amer. Math. Soc., doi:10.1090/crmp/011 (199 7)

    doi

  71. [79]

    Motegi and K

    K. Motegi and K. Sakai, Vertex models, TASEP and Grothendieck polynomials , J. Phys. A: Math. Theor. 46, 355201 (2013), doi:10.1088/1751-8113/46/35/355201

    2013 doi

  72. [80]

    Motegi and K

    K. Motegi and K. Sakai, K-theoretic boson-fermion correspondence and melting cry stals, J. Phys. A: Math. Theor. 47, 445202 (2014), doi:10.1088/1751-8113/47/44/445202

    2014 doi

  73. [81]

    Bobenko, Introduction to Compact Riemann Surfaces , vol

    A. Bobenko, Introduction to Compact Riemann Surfaces , vol. 2013 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, doi:10.1007/978-3-642-1 7413-1 1 (2013)

    2013 doi

  74. [82]

    Cavalieri and E

    R. Cavalieri and E. Miles, Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory, Cambridge University Press, doi:10.1017/CBO9781316569 252 (2016)

    2016 doi

  75. [83]

    Eynard, Lectures notes on compact Riemann surfaces (2018), https://arxiv.org/abs/1805.06405

    B. Eynard, Lectures notes on compact Riemann surfaces (2018), https://arxiv.org/abs/1805.06405

    2018 arXiv

  76. [84]

    Beineke and F

    L. Beineke and F. Harary, The genus of the n-cube , Canadian Journal of Mathematics 17, 494 (1965), doi:10.4153/CJM-1965-048-6

    1965 doi

  77. [85]

    Springer, Introduction to Riemann surfaces , Addison-Wesley (1957)

    G. Springer, Introduction to Riemann surfaces , Addison-Wesley (1957)

    1957

  78. [86]

    Costin and S

    O. Costin and S. Garoufalidis, Resurgence of the fractional polylogarithms , Math. Res. Lett. 16, 817 (2009), doi:10.4310/MRL.2009.v16.n5.a5

    2009 doi

  79. [87]

    Golinelli and K

    O. Golinelli and K. Mallick, Spectral gap of the totally asymmetric exclusion process at ar- bitrary filling , J. Phys. A: Math. Gen. 38, 1419 (2005), doi:10.1088/0305-4470/38/7/001

    2005 doi

  80. [88]

    N. Bogoliubov, Determinantal representation of the time-dependent stati onary correla- tion function for the totally asymmetric simple exclusion m odel, SIGMA 5, 052 (2009), doi:10.3842/SIGMA.2009.052

    2009 doi

  81. [89]

    Motegi, K

    K. Motegi, K. Sakai and J. Sato, Exact relaxation dynamics in the totally asymmetric sim- ple exclusion process, Phys. Rev. E 85, 042105 (2012), doi:10.1103/PhysRevE.85.042105

    2012 doi

  82. [90]

    Prolhac, Asymptotics for the norm of Bethe eigenstates in the periodi c totally asym- metric exclusion process, J

    S. Prolhac, Asymptotics for the norm of Bethe eigenstates in the periodi c totally asym- metric exclusion process, J. Stat. Phys. 160, 926 (2015), doi:10.1007/s10955-015-1230-0. 80 SciPost Physics Submission

    2015 doi

  83. [91]

    Priezzhev, Exact nonstationary probabilities in the asymmetric exclu sion process on a ring , Phys

    V. Priezzhev, Exact nonstationary probabilities in the asymmetric exclu sion process on a ring , Phys. Rev. Lett 91, 050601 (2003), doi:10.1103/PhysRevLett.91.050601

    2003 doi

  84. [92]

    Tracy and H

    C. Tracy and H. Widom, Integral formulas for the asymmetric simple exclusion proce ss, Commun. Math. Phys. 279, 815 (2008), doi:10.1007/s00220-008-0443-3. 81

    2008 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.