Pith. sign in

REVIEW 2 major objections 8 minor 1 cited by

Bethe roots for periodic TASEP and algebraic curve

T0 review · 2 major / 8 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Counting poles on a Riemann surface proves that the periodic TASEP Bethe equations have exactly the expected number of physical roots, with multiplicities governed by the curve's connected components.

desk verdict A solid new algebraic proof of Bethe-root counting for periodic TASEP, with a fixable monodromy gap that should be settled before the component/genus formulas are taken as established. read the letter →

arxiv 2504.19690 v3 pith:NMAIBWTR submitted 2025-04-28 math-ph math.MP

Shinsuke Iwao , Kohei Motegi This is my paper · ORCID
classification math-phmath.MP MSC 82B2360J2782C22
keywords BetheansatzperiodicTASEPRiemannsurfacealgebraiccurvecompletenessmonodromyfive-vertexmodelfreeenergy
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 sets out to prove that the Bethe ansatz for the periodic totally asymmetric exclusion process (TASEP) is complete: for $L$ sites and $N$ particles, the Bethe equations should possess exactly $\binom{L}{N}$ physical solutions, counted with multiplicity, where 'physical' means excluding the singular all-zeros root that does not correspond to an eigenstate. The proof translates the equations into one algebraic equation on a Riemann surface: the inverse branches of the rational map $\varphi(t)=t^N(1-t)^L$ are glued into sheets labelled by $N$-element subsets, and the product $v$ of the selected branches becomes a single-valued function. On this surface the Bethe conditions reduce to intersecting the surface with a line $w=(-1)^{N+1}e^{L\gamma}v$, and the number of intersections is obtained by counting poles of the meromorphic function $h=v/w$, all located over $w=0$. The pole count is exactly $\binom{L}{N}$, giving completeness for generic fugacity. When $\gcd(L,N)>1$, the surface splits into connected components whose orbit structure under a monodromy group explains how often a given product value occurs, reproducing spectral degeneracies of the Markov matrix, and the same geometric input yields explicit formulas for the number of components, ramification points, and total genus, as well as partition functions of the five-vertex model whose thermodynamic free energy is $7\zeta(3)/(16\pi^2)$.

What carries the argument

The central object is the Riemann surface $X$, obtained by analytically continuing the $L$ inverse branches $t_1(w),\dots,t_L(w)$ of $\varphi(t)=t^N(1-t)^L$ and realized algebraically as the (possibly singular, non-reduced) plane curve $C_0\subset\mathbb{C}^2$ with coordinates $v=\prod_{i\in I}t_i(w)$ and $w$, defined by the equation $\prod_{I\in\Omega}(v-\prod_{i\in I}t_i(w))=0$ whose coefficients are elementary symmetric polynomials in the branches. The counting argument is carried by the meromorphic function $h=v/w$: its zero set on $X$ outside the origin matches the Bethe line, and its poles are concentrated on the fiber $w=0$, where the local expansion of $v$ is governed by the cyclic monodromy $\hat\sigma$ of order $\mathrm{lcm}(N,L-N)$. The component classification is carried by the larger monodromy group $G_2=\langle\hat\sigma,\hat\tau\rangle$, where $\hat\tau$ shifts all $L$ indices cyclically; orbits of $G_2$ on $N$-subsets are in bijection with cyclic orbits of the package-counting tuples $(A_1,\dots,A_e)$, and these orbits determine the connected components, their ramification data, and, through the Riemann-Hurwitz formula, the genus.

What would settle it

A direct way to settle the claim is to compute the actual permutation induced on the sheets of $w:X\to\mathbb{P}^1$ by a small loop around $w=c^*$ for a case like $(L,N)=(6,3)$ or $(8,4)$ and compare it with $\hat\sigma^{-1}\circ\hat\tau$; a mismatch would falsify the component/genus classification, whereas the pole-counting proof of completeness can be checked independently by computing the total pole order of $h=v/w$ on the fiber $w=0$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.7: the Bethe equations $z_i^N(1-z_i)^L = (-1)^{N+1}e^{L\gamma}\prod_j z_j$ for $i=1,\dots,N$ have $\binom{L}{N}$ physical roots counted with multiplicity. The argument realizes the equations on a compact Riemann surface $X$ formed from the $L$ branches of the equation $w=t^N(1-t)^L$. Each sheet is labelled by an $N$-subset $I\subset\{1,\dots,L\}$, and the function $v=\prod_{i\in I}t_i(w)$ is single-valued on $X$; the Bethe line $w=(-1)^{N+1}e^{L\gamma}v$ cuts the corresponding plane curve $C_0$ exactly at the physical roots. The proof counts poles of $h=v/w$ at the fiber $w=0$, using the cyclic monodromy $\hat\sigma$ that permutes the first $N$ branches among themselves and the last $L-N$ branches among themselves; summing the pole orders over all sheets gives $\binom{L}{N}$, so for generic $\gamma$ every root is simple and the Bethe ansatz is complete (Corollary 2.8). For $\gcd(L,N)=e>1$, the paper classifies the connected components of $X$ by orbits of the full monodromy group $G_2=\langle\hat\sigma,\hat\tau\rangle$ on $N$-subsets, equivalently by cyclic orbits of the $e$-tuple $(A_1,\dots,A_e)$ counting how many selected indices lie in each residue class modulo $e$. Components that differ by replacing one full residue class with an empty one map to the same irreducible component of $C_0$, which is the algebro-geometric mechanism behind Golinelli-Mallick-type eigenvalue degeneracies. Explicit formulas for the number of components, ramification indices, and the total genus follow from Riemann-Hurwitz plus character computations, and for half-filling the special roots $t_k+t_{N+k}=2+\omega_N^{1/2-k}$, $t_kt_{N+k}=1$ evaluate five-vertex partition functions in terms of roots of unity.

Load-bearing premise

The load-bearing assumption is that the monodromy of the covering $w:X\to\mathbb{P}^1$ around the critical value $c^*$ is generated by the permutation $\hat\sigma^{-1}\circ\hat\tau$ (stated in Section 3.1 without proof); if that monodromy were different, the classification of connected components, the degeneracy multiplicities, and the genus formulas would change, while the pole count of Theorem 2.7 would remain valid.

Editorial extensions

If this is right

  • For generic fugacity $\gamma$, the Bethe ansatz is complete at the level of eigenvectors: the Bethe vectors built from the roots span the full $\binom{L}{N}$-dimensional space, with the zero-root steady state handled separately at $\gamma=0$.
  • When $L$ and $N$ share a common factor, the Markov matrix exhibits spectral degeneracies exactly when two connected components of the curve are related by replacing a full package (residue class) by an empty one; in the most degenerate case the component is defined by a power of $v-1$ and the common eigenvalue is explicitly $h(-L'+N')$.
  • The closed formulas for the number of connected components, the ramification structure, and the total genus give complete algebro-geometric data for every $(L,N)$, reproducing all previously computed examples.
  • At half-filling $L=2N$, the special Golinelli-Mallick-type roots $t_k+t_{N+k}=2+\omega_N^{1/2-k}$, $t_kt_{N+k}=1$ make the on-shell Bethe-vector norm an explicit product over roots of unity; its thermodynamic-limit free energy per site is $7\zeta(3)/(16\pi^2)$.
  • Particle-hole duality holds at the level of the curve: the Bethe equations for $N$ and $L-N$ particles give the same values of $v$ and the same eigenvalues $E=\sum_i z_i/(1-z_i)$, and overlaps with alternating initial states vanish for Golinelli-Mallick type tuples at half- and one-third filling.

Reading between the lines

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

  • The completeness theorem depends only on monodromy around $w=0$, so the unproved statement about monodromy around $c^*$ has no bearing on Theorem 2.7; the classification of components, degeneracies, and genus formulas are the parts that would need revisiting if that monodromy statement failed.
  • The same pole-counting scheme should transfer to the ASEP with $q\neq 1$, where the rational map $w=t^N/(1-t)^L$ is replaced by its two-parameter ASEP analogue; a natural test is whether the component classification reproduces the known degeneracies of the ASEP Markov matrix.
  • The appearance of $\zeta(3)$ in the half-filling free energy suggests a possible numerical probe: computing the same free energy through independent transfer-matrix methods for larger $N$ should approach $0.05328\ldots$, a check not performed in the paper.
  • The explicit factorizations in Appendix B identify the roots missed by the earlier Cassini-oval ansatz at $(L,N)=(10,5)$; one could test the completeness machinery by verifying numerically that those roots (coming from the degree-15 factor in $f_1$) satisfy the Bethe equations and give independent eigenvectors.
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 / 8 minor

Summary. The paper develops an algebro-geometric method for the Bethe ansatz equations of the periodic totally asymmetric simple exclusion process (TASEP) with L sites and N particles. The authors embed the Bethe equations into a plane curve C0 whose coordinates are the product v of Bethe roots and the auxiliary variable w, via the rational map t^N/(1-t)^L. They prove (Theorem 2.7) that the Bethe equations have binom(L,N) physical roots counted with multiplicity, by counting poles of the meromorphic function h=v/w on the associated Riemann surface X. They then study the connected-component decomposition of X in terms of the monodromy group G2 = <σ,τ> acting on the set of N-element subsets of [L], and derive explicit formulas for the number of connected components, the genus of each component, and the total genus (Lemmas 3.12, Eq. (33), Eq. (39)), recovering Prolhac's table. They also give a Golinelli-Mallick-type degeneracy analysis (Propositions 5.1, 5.2, 5.5) and apply special Bethe roots with v=1 to evaluate the norm (partition function) of the five-vertex model, obtaining a free energy F = 7ζ(3)/(16π²) in the thermodynamic limit.

Significance. The pole-counting proof of completeness is a clear advance: it is rigorous, self-contained in the algebraic formulation, and does not rely on the numerical-ansatz approach of earlier works. The connected-component and genus formulas, together with the degeneracy classification, provide a systematic framework that explains and extends the examples of Prolhac and Golinelli-Mallick. The free-energy evaluation is a new exact result. I also note the paper's positive features: the algebraic curve realization is explicitly computable for small cases (Appendix B), and the counting argument in Theorem 2.7 is machine-checkable. However, the component/genus/degeneracy results depend on an unproved assertion about the monodromy around the third critical value c*, which is the main obstacle to accepting the paper in its current form.

major comments (2)
  1. [Section 3.1] The statement "We also show that the monodromy action around c∗ is generated by ˆσ−1 ◦ˆτ" is not followed by a proof. This is a load-bearing assertion: it defines the group G2, whose orbit decomposition gives the connected components (Prop. 3.4) and which enters the Riemann-Hurwitz computation in Eq. (33) and the factorization in Prop. 5.8. Without a proof, the explicit component/genus formulas are not rigorously established. The claim is plausible and can be verified by a short local analysis at the double critical point t = -N/(L-N) (where the map has ramification index 2, forcing the monodromy to be a transposition, and the product with the known monodromies around 0 and ∞ determines the transposition to be (N L) = σ^{-1}τ). The authors should supply this derivation in the manuscript.
  2. [Section 5.1, Proposition 5.2] The proof contains the step "By the identity theorem in complex analysis, we can replace I′,J′ with g(I′),g(J′), where g is an arbitrary element of the monodromy group G2." As written, this step is not justified because the identity theorem applies to single-valued functions on a connected Riemann surface, whereas the branches t_i(w) are multivalued functions on the base; the proposed replacement changes the branch and may move the equality to a different sheet. The intended argument likely uses the Galois action on the field of algebraic functions generated by the t_i, and it should be written out carefully. This is load-bearing because Proposition 5.2 is the converse half of the degeneracy criterion (eq. (41) and the classification in Section 5.1).
minor comments (8)
  1. [Eqs. (5), (6), (31)] These equations are missing the division sign: the correct map is ϕ(t) = t^N/(1-t)^L, and similarly w = t_i(w)^N/(1-t_i(w))^L and ω_e^{k-1} w^{1/e} = t^{N'}/(1-t)^{L'}.
  2. [Lemma 3.1] The condition for injectivity of the permutation action on the N-subsets is misprinted: "if 0<L<N" should be "if 0<N<L".
  3. [Section 3.4, Lemma 3.12] The symbol N is used for both the number of particles and the number of G2-orbits; please use a different notation (e.g., 𝒩) for the latter.
  4. [Example 4.2] The characters are written with subscript O11, but the orbit being discussed is O20; the notation is inconsistent.
  5. [Section 6.2] The statement that the norm is "real positive when N = 4 (mod 2) and real negative when N = 2 (mod 2)" should be phrased as N ≡ 0 (mod 4) and N ≡ 2 (mod 4), respectively.
  6. [General] There are several typographical errors, e.g., "Bethe anzats" in Section 1, "irredicuble" in Lemma 6.1, and "T ASEP" in the abstract.
  7. [Section 6.2, Eqs. (90)-(93)] The evaluation of the free energy is quite terse; a brief justification for exchanging sums and integrals (e.g., by absolute convergence of the series for Li_2) would improve readability.
  8. [Section 2.1] The definition of D for odd N is written as "U \ {w ; w ∈ R≤c∗} ∪ {w ; w ∈ R≥0}"; it should be clarified that the union of the two removed rays is subtracted from U.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the completeness proof counts poles from local expansions, and the component/genus formulas are derived from monodromy and character computations rather than from fitted inputs.

full rationale

The central completeness theorem (Theorem 2.7) is established by counting poles of the meromorphic function h = v/w on the Riemann surface X. The proof uses the local expansions (7) and (8) to compute the pole order at each G1-orbit in the fiber over w = 0, sums the resulting orders combinatorially, and obtains exactly (L choose N) without ever assuming that this is the number of solutions. Thus the target count is not used as an input. The subsequent corollaries and the eigenvector-level completeness statements (Propositions 6.2 and 6.3) are consequences of this count, not assumptions. The irreducible-decomposition and genus formulas in Sections 3 and 4 are derived from the monodromy group G2 = <sigma, tau> and explicit character computations; the numerical checks against Prolhac's table and the explicit factorizations in Appendix B are verifications, not fitted parameters renamed as predictions. The norm and overlap formulas used in Section 6 are quoted from published results, including work coauthored by one of the present authors, but they are applied as lemmas in an application and are not load-bearing for the completeness or genus claims. The only notable gap is the unproved assertion in Section 3.1 that the monodromy around the critical value c* is generated by sigma^{-1} circ tau; this gap could affect the component and genus classification, but it is a missing proof rather than a circular dependency, and it does not enter the pole-counting proof of Theorem 2.7. Overall, no step in the paper reduces by construction to its own inputs.

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

The paper's proofs rest on the standard Bethe-ansatz-to-spectrum correspondence and on Prolhac's Riemann surface construction, plus one unproved monodromy statement at c*. No numerical constants are fitted; L,N,gamma are the model parameters.

assumptions (4)
  • domain assumption Solutions of the Bethe ansatz equations (2) correspond to eigenvectors of the (gamma-deformed) Markov matrix, excluding the singular solution z_i = 0.
    Used throughout as the physical-to-mathematical dictionary; cited to Gwa-Spohn and Golinelli-Mallick, not re-derived in this paper.
  • domain assumption The Riemann surface X obtained by analytic continuation of the branches t_i is compact, has C(L,N) sheets, and has branch locus exactly {0, c*, infinity}.
    Property (2-a),(2-b) in Section 2.1; justified by standard complex analysis and Prolhac's construction [38,39], but the full construction is not reproduced.
  • ad hoc to paper The monodromy action around c* is generated by \hat{\sigma}^{-1} \circ \hat{\tau}.
    Stated in Section 3.1 without proof; used to define G2, compute Y2 characters, and derive genus and degeneracy formulas. This is the weakest unproved input.
  • domain assumption The norm formula (73) and overlap determinant formulas (102),(106) hold after the variable changes described in Section 6.
    Taken from prior work [49,50,51]; these supply the partition function factorizations used in the free energy and overlap results.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bethe roots for periodic TASEP and algebraic curve." pith.science (2026). https://pith.science/paper/NMAIBWTR

@misc{pith2026250419690,
  author       = {Pith},
  title        = {Pith review of: Bethe roots for periodic TASEP and algebraic curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMAIBWTR}},
  note         = {Machine review of arXiv:2504.19690}
}
read the original abstract

We present an algebraic method for solving the Bethe ansatz equations for the periodic totally asymmetric exclusion process (TASEP) with an arbitrary number of sites and particles. The Bethe ansatz equations are realized as an algebraic equation on a certain Riemann surface. While our Riemann surface is essentially the same as the one introduced by Prolhac, we focus on its algebraic realization as a (singular) plane curve. Through a counting argument on the Riemann surface, we establish a rigorous proof that the Bethe ansatz equation has the expected number of solutions when counted with multiplicity. Consequently, under appropriate generic conditions, the completeness of the Bethe ansatz follows. The decomposition of the Riemann surface into connected components determines how often each value of the product of Bethe roots appears. We classify the connected components and their multiplicities using a similar argument to the spectral degeneracy of the Markov matrix discussed by Golinelli-Mallick. As a result, we give an algebro-geometric characterization of Golinelli-Mallick-type spectral degeneracy of the Markov matrix. We also give explicit formulas for the number of connected components, the number of ramification points, and the total genus of the Riemann surface. These formulas recover the table of examples presented by Prolhac. Moreover, we explore applications of the special type of Bethe roots that appear in this case to partition functions of the five-vertex model. We introduce a version of the free energy and evaluate the thermodynamic limit to find the explicit form in terms of the Riemann zeta function.

Figures

Figures reproduced from arXiv: 2504.19690 by the authors.

Figure 1
Figure 1. The L-operator of the five vertex model (49). For each of the five non￾zero local configurations (there are sixteen local configurations in total, and eleven of them not displayed are all assigned weight zero), the corresponding weight is denoted below. Note that the commutativity for the B- and C-operators [B(t), B(r)] = [C(t), C(r)] = 0 follow from the intertwining relation (50), and the ordering of the spectral p… view at source ↗
Figure 2
Figure 2. The on-shell Bethe vector norm (73), which can be regarded as the par￾tition function of the five vertex model under a scalar product boundary condition. For each row, we insert into the spectral variable of the L-operators the same value of t denoted on the left. For the rest of this section, we consider the case with no fugacity γ = 0. We evaluate the norm of the on-shell Bethe vector for the Golinelli-Mallick typ… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Scaling Properties of Current Fluctuations in Periodic TASEP

    cond-mat.stat-mech 2025-07 conditional novelty 5.0 of 10

    For periodic TASEP, the current fluctuation function grows linearly with system size for positive tilt and saturates at -1 for negative tilt, while the relaxation gap decays polynomially or exponentially, respectively.

Reference graph

Works this paper leans on

63 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bethe, Zur Theorie der Metalle, Z

    H. Bethe, Zur Theorie der Metalle, Z. Phys. 71, 205 (1931)

    1931

  2. [2]

    Baxter, Exactly solved models in statistical mecha nics, Academic Press, Cambridge (1982)

    R.J. Baxter, Exactly solved models in statistical mecha nics, Academic Press, Cambridge (1982)

    work page 1982

  3. [3]

    Derrida, An exactly soluble non-equilibrium system: The asymmetric simple exclusion process, Phys

    B. Derrida, An exactly soluble non-equilibrium system: The asymmetric simple exclusion process, Phys. Rep. 301, 65 (1998)

    1998

  4. [4]

    Liggett, Stochastic interacting systems: Contact , voter and exclusion pro cesses, Springer, Berlin (1999)

    T.M. Liggett, Stochastic interacting systems: Contact , voter and exclusion pro cesses, Springer, Berlin (1999)

    work page 1999

  5. [5]

    Kirillov, Combinatorial identities and the comple teness of states for Heisenberg magnet, J

    A.N. Kirillov, Combinatorial identities and the comple teness of states for Heisenberg magnet, J. Soviet Math. 30, 2298 (1985)

    work page 1985

  6. [6]

    Kirillov and N.Yu

    A.N. Kirillov and N.Yu. Reshetikhin, The Bethe ansatz an d the combinatorics of Young tableaux, J. Soviet Math. 41, 925 (1988), 925

    work page 1988

  7. [7]

    Langlands and Y

    R.P. Langlands and Y. Saint-Aubin, Combinatorial aspec ts of the Bethe equations, CRM Proc. Lect. Notes 11, 231-301 (1997)

    work page 1997

  8. [8]

    Tarasov and A

    V. Tarasov and A. Varchenko, Completeness of Bethe vecto rs and difference equations with regular singular points, Int. Math. Res. Not. 13, 637 (1995)

    work page 1995

Show all 63 references
  1. [9]

    Baxter, Completeness of the Bethe Ansatz for the six and eight-vertex models, J

    R.J. Baxter, Completeness of the Bethe Ansatz for the six and eight-vertex models, J. Stat. Phys. 108, 1 (2002)

    2002

  2. [10]

    Mukhin, V

    E. Mukhin, V. Tarasov, and A. Varchenko, Bethe algebra o f homogeneous XXX Heisenberg model has simple spectrum, Commun. Math. Phys. 288, 1 (2009)

    2009

  3. [11]

    Tarasov, Completeness of the Bethe Ansatz for the per iodic isotropic Heisenberg model, Rev

    V. Tarasov, Completeness of the Bethe Ansatz for the per iodic isotropic Heisenberg model, Rev. Math. Phys. 30, 1840018 (2018) BETHE ROOTS FOR PERIODIC TASEP AND ALGEBRAIC CUR VE 37

    2018

  4. [12]

    Borodin, I

    A. Borodin, I. Corwin, L. Petrov, and T. Sasamoto, Spect ral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz, Commun. Math. Phys. 339, 1167 (2 015)

  5. [13]

    Gwa and H

    L.-H. Gwa and H. Spohn, Bethe solution for the dynamical -scaling exponent of the noisy Burgers equation, Phys. Rev. A 46, 844 (1992)

    1992

  6. [14]

    Gwa and H

    L.-H. Gwa and H. Spohn, Six-vertex model, roughened sur faces, and an asymmetric spin Hamiltonian, Phys. Rev. Lett. 68, 725 (1992)

    1992

  7. [15]

    Kim, Bethe Ansatz solution for crossover scaling fun ctions of the asymmetric XXZ chain and the Kardar- Parisi-Zhang-type growth model, Phys

    D. Kim, Bethe Ansatz solution for crossover scaling fun ctions of the asymmetric XXZ chain and the Kardar- Parisi-Zhang-type growth model, Phys. Rev. E 52, 3512 (1995 )

    1995

  8. [16]

    Kim, Asymmetric XXZ chain at the antiferromagnetic t ransition: Spectra and partition functions, J

    D. Kim, Asymmetric XXZ chain at the antiferromagnetic t ransition: Spectra and partition functions, J. Phys. A: Math. Gen. 30, 3817 (1997)

    1997

  9. [17]

    de Gier and F

    J. de Gier and F. H. L. Essler, Bethe Ansatz solution of th e asymmetric exclusion process with open boundaries, Phys. Rev. Lett. 95, 240601 (2005)

    2005

  10. [18]

    Golinelli and K

    O. Golinelli and K. Mallick, Bethe Ansatz calculation o f the spectral gap of the asymmetric exclusion process, J. Phys. A: Math. Gen. 37, 3321 (2004)

    2004

  11. [19]

    Golinelli and K

    O. Golinelli and K. Mallick, Spectral gap of the totally asymmetric exclusion process at arbitrary filling, J. Phys. A: Math. Gen. 38, 1419 (2005)

    2005

  12. [20]

    Golinelli and K

    O. Golinelli and K. Mallick, Hidden symmetries in the as ymmetric exclusion process, J. Stat. Mech.: Theor. Exp. P12001 (2004)

    2004

  13. [21]

    Golinelli and K

    O. Golinelli and K. Mallick, Spectral Degeneracies in t he Totally Asymmetric Exclusion Process, J. Stat. Phys. 120, 779 (2005)

    2005

  14. [22]

    Golinelli and K

    O. Golinelli and K. Mallick, Family of commuting operat ors for the totally asymmetric exclusion process, J. Phys. A: Math. Theor. 40, 5795 (2007)

    2007

  15. [23]

    Prolhac, Spectrum of the totally asymmetric simple e xclusion process on a periodic lattice-first excited states , J

    S. Prolhac, Spectrum of the totally asymmetric simple e xclusion process on a periodic lattice-first excited states , J. Phys. A: Math. Theor. 47, 375001 (2014)

    2014

  16. [24]

    Prolhac, Asymptotics for the norm of Bethe eigenstat es in the periodic totally asymmetric exclusion process, J

    S. Prolhac, Asymptotics for the norm of Bethe eigenstat es in the periodic totally asymmetric exclusion process, J. Stat. Phys. 160, 926 (2015)

    2015

  17. [25]

    Prolhac, Current fluctuations for totally asymmetri c exclusion on the relaxation scale, J

    S. Prolhac, Current fluctuations for totally asymmetri c exclusion on the relaxation scale, J. Phys. A: Math. Theor. 48, 06FT02 (2015)

    2015

  18. [26]

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

    S. Prolhac, Current fluctuations and large deviations f or periodic TASEP on the relaxation scale, J. Stat. Mech.: Theory Exp. P11028 (2015)

    2015

  19. [27]

    Prolhac, Finite-time fluctuations for the totally as ymmetric exclusion process, Phys

    S. Prolhac, Finite-time fluctuations for the totally as ymmetric exclusion process, Phys. Rev. Lett. 116, 090601 (2016)

    2016

  20. [28]

    Kardar, G

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

    1986

  21. [29]

    Corwin, The Kardar-Parisi-Zhang equation and unive rsality class, Random Matrices: Theory Appl

    I. Corwin, The Kardar-Parisi-Zhang equation and unive rsality class, Random Matrices: Theory Appl. 01, 1130001 (2012)

    2012

  22. [30]

    H. Spohn, The Kardar-Parisi-Zhang equation: A statist ical physics perspective, in Stochastic processes and random matrices, Oxford University Press, Oxford, UK, ISBN 0198797311 (2017)

    2017

  23. [31]

    Takeuchi, An appetizer to modern developments on t he Kardar-Parisi Zhang universality class, Phys

    K.A. Takeuchi, An appetizer to modern developments on t he Kardar-Parisi Zhang universality class, Phys. A: Stat. Mech. Appl. 504, 77 (2018)

    2018

  24. [32]

    Johansson, Shape fluctuations and random matrices, C omm

    K. Johansson, Shape fluctuations and random matrices, C omm. Math. Phys. 209, 437 (2000)

    2000

  25. [33]

    Johansson, Discrete polynuclear growth and determi nantal processes, Comm

    K. Johansson, Discrete polynuclear growth and determi nantal processes, Comm. Math. Phys. 242, 277 (2003)

    2003

  26. [34]

    Borodin, P.L

    A. Borodin, P.L. Ferrari, M. Pr¨ ahofer, and T. Sasamoto , Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys. 129, 1055 (2007)

    2007

  27. [35]

    Tracy and H

    C.A. Tracy and H. Widom, Asymptotics in ASEP with step in itial condition, Comm. Math. Phys. 290, 129 (2009)

    2009

  28. [36]

    Sasamoto and H

    T. Sasamoto and H. Spohn, One-dimensional Kardar-Pari si-Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104, 230602 (2010)

    2010

  29. [37]

    G. Amir, I. Corwin, and J. Quastel, Probability distrib ution of the free energy of the continuum directed random polymer in 1+ 1 dimensions, Communications on Pure and Appli ed Mathematics 64, 466 (2011)

    2011

  30. [38]

    Prolhac, Riemann surfaces for KPZ with periodic boun daries, SciPost Phys

    S. Prolhac, Riemann surfaces for KPZ with periodic boun daries, SciPost Phys. 8, 008 (2020)

    2020

  31. [39]

    Prolhac, Riemann surface for TASEP with periodic bou ndaries, J

    S. Prolhac, Riemann surface for TASEP with periodic bou ndaries, J. Phys. A: Math. Theor. 53, 445003 (2020)

    2020

  32. [40]

    Prolhac, From the Riemann surface of TASEP to ASEP J

    S. Prolhac, From the Riemann surface of TASEP to ASEP J. P hys. A: Math. Theor. 54, 395002 (2021)

    2021

  33. [41]

    Prolhac, KPZ fluctuations in finite volume, SciPost Ph ys

    S. Prolhac, KPZ fluctuations in finite volume, SciPost Ph ys. Lect. Notes 81 (2024)

    2024

  34. [42]

    de Gier, R

    J. de Gier, R. Kenyon, and S.S. W atson, Limit shapes for t he asymmetric five vertex model, Commun. Math. Phys. 385, 793 (2021). 38 SHINSUKE IW AO AND KOHEI MOTEGI

    2021

  35. [43]

    Nakerst, T

    G. Nakerst, T. Prosen, and M. Haque, Spectral boundary o f the asymmetric simple exclusion process: Free fermions, Bethe ansatz, and random matrix theory Phys. Rev. E 110, 014110 (2024)

    2024

  36. [44]

    Brattain, N

    E. Brattain, N. Do, and A. Saenz, The completeness of the Bethe Ansatz for the periodic ASEP, arXiv:1511.03762

    arXiv

  37. [45]

    Ishiguro, J

    Y. Ishiguro, J. Sato, and K. Nishinari, Asymmetry-indu ced delocalization transition in the integrable non- Hermitian spin chain, Physical Review Research 5, 033102 (2 023)

  38. [46]

    Baik and Z

    J. Baik and Z. Liu, Fluctuations of TASEP on a ring in rela xation time scale. Comm. Pure Appl. Math., 71, 0747 (2018)

    2018

  39. [47]

    Baik and Z

    J. Baik and Z. Liu, Multipoint distribution of periodic TASEP, J. Amer. Math. Soc. 32, 609 (2019)

    2019

  40. [48]

    Arita, A

    C. Arita, A. Kuniba, K. Sakai, and T. Sawabe, Spectrum of a multi-species asymmetric simple exclusion process on a ring, J. Phys. A: Math. Theor. 42, 345002 (2009)

    2009

  41. [49]

    Bogoliubov, Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally Asymmetric Simple Exclusion Model, SIGMA 5 052 (200 9)

    N.M. Bogoliubov, Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally Asymmetric Simple Exclusion Model, SIGMA 5 052 (200 9)

  42. [50]

    Motegi, K

    K. Motegi, K. Sakai, and J. Sato, Long time asymptotics o f the totally asymmetric simple exclusion process, J. Phys. A: Math. Theor. 45, 465004 (2012)

    2012

  43. [51]

    Motegi and K

    K. Motegi and K. Sakai, Vertex models, TASEP and Grothen dieck polynomials, J. Phys. A. 46, 355201 (2013)

    2013

  44. [52]

    Korepin, Calculation of norms of Bethe wave functi ons, Commun

    V.E. Korepin, Calculation of norms of Bethe wave functi ons, Commun. Math. Phys. 86, 391 (1982)

    1982

  45. [53]

    Slavnov, Algebraic Bethe Ansatz and Correlation F unctions, An Advanced Course, W orld Scientific (2022)

    N.A. Slavnov, Algebraic Bethe Ansatz and Correlation F unctions, An Advanced Course, W orld Scientific (2022)

    2022

  46. [54]

    Korepin, N.M

    V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin, Quant um Inverse Scattering Method and Correlation Func- tions, Cambridge University Press (1993)

    1993

  47. [55]

    Slavnov, Calculation of scalar products of wave fu nctions and form factors in the framework of the algebraic Bethe Ansatz, Theor

    N.A. Slavnov, Calculation of scalar products of wave fu nctions and form factors in the framework of the algebraic Bethe Ansatz, Theor. Math. Phys. 79, 502 (1989)

    1989

  48. [56]

    Burenev and A.G

    I.N. Burenev and A.G. Pronko, Thermodynamics of the five -vertex model with scalar-product boundary condi- tions, Commun. Math. Phys. 405, 148 (2024)

    2024

  49. [57]

    de Gier and V.E

    J. de Gier and V.E. Korepin, Six-vertex model with domai n wall boundary conditions: variable inhomogeneities, J. Phys. A: Math. Gen. 34, 8135 (2001)

    2001

  50. [58]

    Korepin and P

    V.E. Korepin and P. Zinn-Justin, Thermodynamic limit o f the Six-Vertex Model with Domain W all Boundary Conditions, J. Phys. A 33, 7053 (2000)

    2000

  51. [59]

    Bleher and V.V

    P.M. Bleher and V.V. Fokin, Exact Solution of the Six-Ve rtex Model with Domain W all Boundary Conditions. Disordered Phase, Commun. Math. Phys. 268, 223 (2006)

    2006

  52. [60]

    Tavares, G.A.P

    T.S. Tavares, G.A.P. Ribeiro, and V.E. Korepin, The ent ropy of the six-vertex model with variety of different boundary conditions, J. Stat. Mech. P06016 (2015)

    2015

  53. [61]

    A. Lazarescu, Exact Large Deviations of the Current in t he Asymmetric Simple Exclusion Process with Open Boundaries, Doctor thesis (Universit´ e Pierre et Marie Cur ie), arXiv:1311.7370

    arXiv

  54. [62]

    Pronko and Yu.G

    G.P. Pronko and Yu.G. Stroganov, Bethe equations ‘on th e wrong side of the equator’, J. Phys. A: Math. Gen. 32, 2333 (1999)

    1999

  55. [63]

    W eyl, and G

    H. W eyl, and G. R. MacLane. The concept of a Riemann surfa ce. Courier Corporation, (2009). Appendix A. Computation of symmetric polynomials A.1. Ω -symmetric functions. Given L indeterminates t1,...,t L, the Ω-elementary symmetric polynomials E1,E 2,...,E ( L N) and the Ω-po...

    2009

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.