Exact Results for the Symmetric Dyson Exclusion Process

Abstract

The symmetric Dyson exclusion process (SDEP) is an exclusion process on the lattice with a long-range logarithmic Coulomb-type interaction. It appears in several equivalent forms: as symmetric random walkers conditioned, in the Doob-transform sense, not to collide; as the maximal-activity limit of a conditioned SSEP; and as a ground-state transform of the spin-1/2 XX chain. In this work, we exploit this latter representation to obtain exact evolution formulas from deterministic initial configurations. The resulting determinantal kernel is expressed through a finite interpolation expression in terms of Lagrange polynomials, leading to explicit density evolution in finite and infinite lattices. For the melting of a densely packed block, we show that the full hierarchy of density moments is governed by a finite-dimensional polynomial algebra, and we identify Catalan numbers in the leading time coefficients. We also derive the Euler-scale density profile and the arctic curve separating frozen and liquid regions, and show that they coincide with conjectured hydrodynamic results obtained in a previous work.

Interacting Quantum Symmetric Exclusion Process
Denis Bernard, Friedrich Hübner, Stefano Scopa
HAL30 Jul 2026PreprintLicence not statedFull textMathematical physics Condensed matterHAL
Canonical Local Equilibrium and Cutoff Profiles for the Symmetric Exclusion Process on Discrete Tori
Joe P. Chen
arXiv9 Sep 2026PreprintCC BY-SA 4.0Full textMathematical physics ProbabilityarXiv
Scaling Limits of Density Fluctuation Fields of the Symmetric Partial Exclusion Processes Approaching the Zero Range Process
Kanta Nozawa
arXiv21 Oct 2026PreprintCC BY 4.0Full textProbabilityarXiv
Higher-Order Gaussian Approximations on Path Space for the Simple Symmetric Exclusion Process
Benjamin Gess, Vitalii Konarovskyi
arXiv9 Oct 2026PreprintarXiv licenceFull textProbabilityarXiv
Non-symmetric vector dyson equations
Jiaoyang Huang, Zhonggen Su, Ruizhe Xu
arXiv21 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEs Mathematical physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{zahra2026exact,
  title = {{Exact Results for the Symmetric Dyson Exclusion Process}},
  author = {Ali Zahra and Jerome Dubail and G M Schütz},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05768996v1},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Exact Results for the Symmetric Dyson Exclusion Process
AU  - Ali Zahra
AU  - Jerome Dubail
AU  - G M Schütz
PY  - 2026
DA  - 2026/09/29
PB  - HAL
UR  - https://hal.science/hal-05768996v1
AB  - The symmetric Dyson exclusion process (SDEP) is an exclusion process on the lattice with a long-range logarithmic Coulomb-type interaction. It appears in several equivalent forms: as symmetric random walkers conditioned, in the Doob-transform sense, not to collide; as the maximal-activity limit of a conditioned SSEP; and as a ground-state transform of the spin-1/2 XX chain. In this work, we exploit this latter representation to obtain exact evolution formulas from deterministic initial configurations. The resulting determinantal kernel is expressed through a finite interpolation expression in terms of Lagrange polynomials, leading to explicit density evolution in finite and infinite lattices. For the melting of a densely packed block, we show that the full hierarchy of density moments is governed by a finite-dimensional polynomial algebra, and we identify Catalan numbers in the leading time coefficients. We also derive the Euler-scale density profile and the arctic curve separating frozen and liquid regions, and show that they coincide with conjectured hydrodynamic results obtained in a previous work.
ER  -

Abstract

Related

Cite