The multi-species asymmetric simple exclusion process (ASEP) with long-range jumps on Z\mathbb{Z}, introduced by Lee in 2024, combines rightward drop-push moves, in which a particle jumps to the nearest site occupied by a lower-labeled particle and pushes the block ahead, with leftward multi-species TASEP moves. Lee proved it is Bethe-ansatz integrable and expressed its transition probabilities as a sum over permutations σ∈SN\sigma\in S_N of NN-fold contour integrals whose integrands contain matrix-valued coefficients Aσ\mathbf{A}_\sigma. Several questions left open by that work remain unresolved: whether the transition probabilities with matching initial and final species orders (π=ν\pi=\nu) admit a determinant representation, as they do for the nearest-neighbor multi-species ASEP; whether, for special initial permutations such as 12⋯N12\cdots N, 12⋯212\cdots2, and 1⋯121\cdots12, the Aσ\mathbf{A}_\sigma collapse to single products, yielding simplified at-most-N!N!-term formulas; and the construction of an integrable multi-species model making long-range jumps in both directions, a multi-species analogue of the two-parameter family of Alimohammadi, Karimipour, and Khorrami, together with its transition probabilities. Follow-up work by the same author (2025-2026) constructed new integrable multi-species families and a bidirectional long-range swap model, but left the formula-structure questions untouched and did not construct the push-type two-sided analogue.

Background

In one-dimensional stochastic transport, the asymmetric simple exclusion process (ASEP) is the canonical interacting particle model: particles on the integer lattice Z\mathbb{Z} carry exponential clocks and attempt nearest-neighbor jumps, to the right with probability pp and to the left with probability q=1−pq=1-p, subject to the exclusion rule of at most one particle per site. ASEP and its totally asymmetric case (TASEP, the limit p=1p=1) are exactly solvable by the coordinate Bethe ansatz, an ansatz expressing transition probabilities as sums over permutations of free-particle waves glued together by two-particle scattering amplitudes. In the multi-species ASEP (mASEP), particles carry labels 1,…,N1,\dots,N; a higher-labeled particle moving onto a lower-labeled one exchanges their positions, while moves onto equal or higher labels are suppressed.

A different solvable line consists of long-range exclusion dynamics. In the drop-push model, a particle that attempts to jump onto an occupied site pushes the entire block of particles one site forward, the push cascading until it reaches a vacancy. The two-sided single-species model combining rightward drop-push moves with leftward TASEP moves is the PushASEP of Borodin and Ferrari, and the families of Alimohammadi, Karimipour, and Khorrami (1998, 1999) give single-species exclusion processes that make long-range jumps in both directions; for those, a naive two-sided extension of the drop-push rule fails to be Bethe-ansatz solvable, and integrability holds only under constraints on the rates. Multi-species versions of one-sided drop-push dynamics were known, but before 2024 no integrable multi-species model combined long-range jumps with a left-moving species asymmetry.

Lee (2024) filled this gap. In the multi-species ASEP with long-range jumps, each particle of label ll waits an exponential time of rate 11 and then jumps right with probability pp or left with probability q=1−pq=1-p; a rightward move goes to the nearest site to its right occupied by a particle of label l′<ll'<l (a vacancy counts as label 00) and pushes the block ahead, while a leftward move follows the mTASEP exchange rule (multi-species TASEP, the p=1p=1 case of mASEP). Lee proved the model is integrable, in the sense that the dynamics of arbitrarily many particles reduces to two-particle boundary conditions whose scattering matrices satisfy the Yang-Baxter equation (the consistency condition for three-particle scattering), and derived the transition probability for an arbitrary initial state (Y,ν)(Y,\nu) of the NN particles, each carrying a label from {1,…,N}\{1,\dots,N\}, where X=(x1<⋯<xN)X=(x_1<\cdots<x_N) and Y=(y1<⋯<yN)Y=(y_1<\cdots<y_N) list the particle positions from left to right and π,ν\pi,\nu are the corresponding species words, as

P(Y,ν)(X,π;t)=(12πi)N∫CRN⋯∫CR1∑σ∈SN(Aσ)(π,ν)(∏i=1Nξσ(i)xi−yσ(i)−1eε(ξi)t)dξ1⋯dξN,P_{(Y,\nu)}(X,\pi;t)=\Big(\frac{1}{2\pi i}\Big)^{N}\int_{C_{R_N}}\cdots\int_{C_{R_1}}\sum_{\sigma\in S_N}(\mathbf{A}_\sigma)_{(\pi,\nu)}\Big(\prod_{i=1}^{N}\xi_{\sigma(i)}^{x_i-y_{\sigma(i)}-1}e^{\varepsilon(\xi_i)t}\Big)d\xi_1\cdots d\xi_N,

where ε(ξ)=p/ξ+qξ−1\varepsilon(\xi)=p/\xi+q\xi-1, SNS_N is the symmetric group on {1,…,N}\{1,\dots,N\}, each CRiC_{R_i} is the counterclockwise circle of radius RiR_i centered at the origin with 1<R1<⋯<RN1<R_1<\cdots<R_N, and (Aσ)(π,ν)(\mathbf{A}_\sigma)_{(\pi,\nu)} is the (π,ν)(\pi,\nu) entry of a matrix Aσ\mathbf{A}_\sigma given as a product of NN×NNN^N\times N^N matrices built from the two-particle scattering data (equation (78) of the paper). The proof that this formula satisfies the initial condition is not self-contained: it is reduced, by matching matrix denominators, to corollaries established for a multiparticle-hopping model by Lee (2012).

For the nearest-neighbor mASEP the analogous SNS_N-sum collapses dramatically in special cases: Lee (2020) proved that when the final species order equals the initial one (π=ν\pi=\nu) the transition probability is a determinant, and Lee and Raimbekov (2022) proved that for two species with initial orders 2⋯212\cdots21, 12⋯212\cdots2, 1⋯121\cdots12, or 21⋯121\cdots1 it reduces to at most N!N! contour integrals; Kuan (2020) obtained determinantal expressions for multi-species TASEP observables. Whether such simplifications survive for the long-range model is unknown. Lee (2024) anticipated determinant representations for π=ν\pi=\nu and single-product collapses of Aσ\mathbf{A}_\sigma for special initial permutations such as 12⋯N12\cdots N, 12⋯212\cdots2, 1⋯121\cdots12, but did not prove them; the same Discussion leaves open an integrable multi-species model with long-range jumps in both directions, that is, a multi-species analogue of the two-parameter family of Alimohammadi, Karimipour, and Khorrami.

Subsequent work by the same author broadened the integrable multi-species family: Lee and Raimbekov (2025) extended the Alimohammadi-Ahmadi classification of 28 two-species Yang-Baxter-integrable interaction rules to multi-species settings; Lee (2026, SciPost Physics Core) introduced an integrable multi-species TASEP with long-range swap dynamics and derived its contour-integral transition probabilities; and Lee (2026, Journal of Statistical Mechanics) built species-dependent interpolations between TASEP-type and drop-push-type interactions together with a bidirectional long-range swap extension whose Yang-Baxter equation holds for arbitrary species compositions and whose two-particle reducibility is proven in a binary parameter regime. None of these works addresses the determinantal or single-product questions for the original long-range model, and none constructs a push-type two-sided analogue with transition probabilities.

Problem Statement

The problem is to resolve the questions left open by Lee (2024, Symmetry 16:1164) about the exact solution of the multi-species ASEP with long-range jumps on Z\mathbb{Z}:

(1) Determinantal structure. Prove or disprove that the transition probability P(Y,ν)(X,π;t)P_{(Y,\nu)}(X,\pi;t) with π=ν\pi=\nu (final species order equal to initial species order) admits a determinant representation analogous to the one Lee (2020) established for the nearest-neighbor multi-species ASEP; if such a representation exists, state it explicitly.

(2) Single-product collapse. The source anticipates that for certain special initial species words ν\nu (permutations of the multiset of species labels carried by the NN particles), the candidates named being 12⋯N12\cdots N, 12⋯212\cdots2, and 1⋯121\cdots12, each matrix entry (Aσ)(π,ν)(\mathbf{A}_\sigma)_{(\pi,\nu)} in the Bethe-ansatz sum can be expressed as a single product, so that the transition probability becomes a sum of at most N!N! simplified contour integrals. Prove or disprove this single-product collapse for such anticipated special initial species words and derive the corresponding simplified formulas, in analogy with the two-species results of Lee and Raimbekov (2022).

(3) Two-sided long-range model. Construct an integrable multi-species exclusion model whose particles make long-range (drop-push-type) jumps in both directions, a genuine multi-species analogue of the two-parameter family of Alimohammadi, Karimipour, and Khorrami (1999), and derive its transition probabilities.

Items (1)-(2) jointly seek the simplified closed form of the exact solution of the specific model of Lee (2024); item (3) completes the two-sided long-range class that the source's construction only partially realized. The source's remaining suggestion — using the same machinery to construct multi-species versions of other single-species integrable models — is an open-ended research program without a recognizable completion boundary and is not part of this problem. Answers to (1)-(2) must rigorously establish that the claimed formulas give the model's transition probabilities P(Y,ν)(X,π;t)P_{(Y,\nu)}(X,\pi;t); answers to (3) must hold for all species numbers and all initial data, not only for small systems.

References

  1. Eunghyun Lee, Integrability of the Multi-Species Asymmetric Simple Exclusion Processes with Long-Range Jumps on Z, Symmetry 16(9):1164, 2024. DOI: 10.3390/sym16091164 (arXiv:2407.21608). URL: https://doi.org/10.3390/sym16091164
  2. Eunghyun Lee, Exact Formulas of the Transition Probabilities of the Multi-Species Asymmetric Simple Exclusion Process, Symmetry, Integrability and Geometry: Methods and Applications 16:139, 2020. DOI: 10.3842/sigma.2020.139 (arXiv:1809.07362). URL: https://doi.org/10.3842/sigma.2020.139
  3. Eunghyun Lee and Temirlan Raimbekov, Simplified Forms of the Transition Probabilities of the Two-Species ASEP with Some Initial Orders of Particles, Symmetry, Integrability and Geometry: Methods and Applications 18:008, 2022. DOI: 10.3842/sigma.2022.008 (arXiv:2104.06471). URL: https://doi.org/10.3842/sigma.2022.008
  4. Jeffrey Kuan, Determinantal Expressions in Multi-Species TASEP, Symmetry, Integrability and Geometry: Methods and Applications 16:133, 2020. DOI: 10.3842/sigma.2020.133 (arXiv:2007.02913). URL: https://doi.org/10.3842/sigma.2020.133
  5. M. Alimohammadi, V. Karimipour, M. Khorrami, A Two-Parametric Family of Asymmetric Exclusion Processes and Its Exact Solution, Journal of Statistical Physics 97:373-394, 1999. DOI: 10.1023/a:1004631505288. URL: https://doi.org/10.1023/a:1004631505288
  6. M. Alimohammadi, V. Karimipour, M. Khorrami, Exact solution of a one-parameter family of asymmetric exclusion processes, Physical Review E 57:6370-6376, 1998. DOI: 10.1103/physreve.57.6370. URL: https://doi.org/10.1103/physreve.57.6370
  7. M. Alimohammadi, N. Ahmadi, Class of integrable diffusion-reaction processes, Physical Review E 62:1674-1682, 2000. DOI: 10.1103/physreve.62.1674. URL: https://doi.org/10.1103/physreve.62.1674
  8. A. Borodin, P. L. Ferrari, Large time asymptotics of growth models on space-like paths I: PushASEP, Electronic Journal of Probability 13:1380-1418, 2008. DOI: 10.1214/ejp.v13-541. URL: https://doi.org/10.1214/ejp.v13-541
  9. Eunghyun Lee, The Current Distribution of the Multiparticle Hopping Asymmetric Diffusion Model, Journal of Statistical Physics 149:50-72, 2012. DOI: 10.1007/s10955-012-0582-y (arXiv:1203.0501). URL: https://doi.org/10.1007/s10955-012-0582-y
  10. Eunghyun Lee and Temirlan Raimbekov, Integrable Multispecies Totally Asymmetric Stochastic Interacting Particle Systems with Homogeneous Rates, Symmetry 17(9):1510, 2025. DOI: 10.3390/sym17091510 (arXiv:2508.03157). URL: https://doi.org/10.3390/sym17091510
  11. Eunghyun Lee, Multispecies totally asymmetric simple exclusion process with long-range swap, SciPost Physics Core 9(1):008, 2026. DOI: 10.21468/scipostphyscore.9.1.008 (arXiv:2509.00830). URL: https://doi.org/10.21468/scipostphyscore.9.1.008
  12. Eunghyun Lee, Integrability of multispecies long-range swap models with species-dependent interpolation, Journal of Statistical Mechanics: Theory and Experiment 2026(7):073102, 2026. DOI: 10.1088/1742-5468/ae80b5 (arXiv:2604.12136). URL: https://doi.org/10.1088/1742-5468/ae80b5