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arxiv: 2508.09879 · v2 · pith:Q2ZGHXYNnew · submitted 2025-08-13 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Emergent Hydrodynamics in an Exclusion Process with Long-Range Interactions

Ali Zahra , Jerome Dubail , Gunter M. Sch\"utz This is my paper

Pith reviewed 2026-05-21 23:12 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords symmetric Dyson exclusion processnon-local hydrodynamicsHilbert transformlong-range interactionsexclusion processDoob transformballistic scalingarctic curves
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The pith

A lattice gas with long-range Coulomb interactions shows non-local hydrodynamics where the current is a sine times the hyperbolic sine of the Hilbert transform of the density.

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

The paper examines the symmetric Dyson exclusion process, a lattice gas that combines site exclusion with long-range interactions of Coulomb type. An exact mapping of its stochastic generator to the XX quantum spin chain via the Doob transform allows the authors to conjecture that the large-scale behavior is ballistic and governed by a single hydrodynamic field equation whose current is a genuinely non-local functional of the entire density profile through the Hilbert transform. This description is equivalent to a local two-field complex Hopf system at finite density. The conjecture yields closed-form solutions for the melting of single and double block initial states, producing limit shapes and arctic curves that agree with direct Monte Carlo simulations of the microscopic process.

Core claim

The SDEP displays ballistic scaling and non-local hydrodynamics governed by the continuity equation partial_t rho + partial_x j[rho] = 0, with the current j[rho] = (1/pi) sin(pi rho(x,t)) sinh(pi H rho(x,t)) where H denotes the Hilbert transform. This one-field non-local description is equivalent to a local two-field complex Hopf system for finite particle density.

What carries the argument

The non-local current functional j[rho] that multiplies the sine of pi times the density by the hyperbolic sine of pi times the Hilbert transform of the density, which encodes the long-range interactions at the macroscopic level.

If this is right

  • The melting dynamics of single and double block initial states admit closed evolution formulas that produce explicit limit shapes and arctic curves.
  • The non-local one-field hydrodynamic description is equivalent to a local two-field complex Hopf system at finite density.
  • Large-scale Monte Carlo simulations confirm the predicted shapes arising from the non-local current.
  • The model provides a concrete example of emergent non-local hydrodynamics driven by long-range interactions in an exclusion process.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The appearance of the Hilbert transform suggests that analogous non-local currents could arise in other one-dimensional systems with inverse-square or long-range potentials.
  • A direct hydrodynamic limit derivation from the microscopic rates, independent of the quantum mapping, would provide an independent check of the conjecture.
  • The equivalence to the complex Hopf system opens the possibility of linking the dynamics to integrable hierarchies or complex-fluid models in neighboring contexts.

Load-bearing premise

The exact microscopic mapping to the XX chain via the Doob transform produces the specific non-local macroscopic current equation at large scales.

What would settle it

High-resolution Monte Carlo simulations of the melting of a double-block initial state on large lattices would show whether the observed arctic curve matches the one predicted by the hydrodynamic equation.

Figures

Figures reproduced from arXiv: 2508.09879 by Ali Zahra, Gunter M. Sch\"utz, Jerome Dubail.

Figure 1
Figure 1. Figure 1: Evolution of the density profile from a single block initial condition [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Emergence of a limit-shape phenomenon as the system size grows: the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Arctic curve delineates the frozen regions, where the density is one [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
read the original abstract

We study the symmetric Dyson exclusion process (SDEP) - a lattice gas with exclusion and long-range, Coulomb-type interactions that emerge both as the maximal-activity limit of the symmetric exclusion process and as a discrete version of Dyson's Brownian motion on the unitary group. Exploiting an exact ground-state (Doob) transform, we map the stochastic generator of the SDEP onto the spin-$1/2$ XX quantum chain, which in turn admits a free-fermion representation. At macroscopic scales we conjecture that the SDEP displays ballistic (Eulerian) scaling and non-local hydrodynamics governed by the equation $\partial_t \rho+\partial_x j[\rho]=0$ with $j[\rho]=(1/\pi)\sin(\pi\rho(x,t))\sinh(\pi\mathcal{H}\rho(x,t))$, where $\mathcal{H}$ is the Hilbert transform, making the current a genuinely non-local functional of the density. This non-local one-field description is equivalent to a local two-field "complex Hopf" system for finite particle density. Closed evolution formulas allow us to solve the melting of single and double block initial states, producing limit shapes and arctic curves that agree with large-scale Monte Carlo simulations. The model thus offers a tractable example of emergent non-local hydrodynamics driven by long-range interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces the symmetric Dyson exclusion process (SDEP), a lattice gas with exclusion and long-range Coulomb-type interactions. It establishes an exact mapping of the stochastic generator to the spin-1/2 XX quantum chain via the Doob transform, which admits a free-fermion representation. At macroscopic scales the authors conjecture ballistic Eulerian scaling with non-local hydrodynamics governed by the continuity equation ∂_t ρ + ∂_x j[ρ] = 0, where the current is the non-local functional j[ρ] = (1/π) sin(πρ) sinh(π ℋ ρ) and ℋ denotes the Hilbert transform. Closed-form solutions are derived for the melting of single- and double-block initial states, producing limit shapes and arctic curves that are shown to agree with large-scale Monte Carlo simulations.

Significance. If the conjectured hydrodynamic equation is valid, the work supplies a tractable, exactly mappable example of emergent non-local hydrodynamics driven by long-range interactions. The exact Doob transform to the XX chain and the free-fermion representation constitute clear technical strengths, as do the closed evolution formulas that permit explicit limit-shape calculations and direct comparison with simulations.

major comments (1)
  1. [hydrodynamic conjecture paragraph] The specific non-local current j[ρ] = (1/π) sin(πρ) sinh(π ℋ ρ) is introduced only as a conjecture (abstract and the paragraph immediately following the XX-chain mapping). No intermediate scaling analysis, mode expansion, or explicit limit from the microscopic generator (or from the fermionic dispersion of the XX chain) is supplied to derive the Hilbert-transform non-locality in the one-field current. This step is load-bearing for the central claim of emergent non-local hydrodynamics.
minor comments (1)
  1. The equivalence between the non-local one-field description and the local two-field complex Hopf system is stated but not derived in detail; a short appendix sketching the transformation would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for recognizing the technical strengths of the exact Doob transform to the XX chain and the free-fermion representation. We address the single major comment below.

read point-by-point responses

  1. Referee: The specific non-local current j[ρ] = (1/π) sin(πρ) sinh(π ℋ ρ) is introduced only as a conjecture (abstract and the paragraph immediately following the XX-chain mapping). No intermediate scaling analysis, mode expansion, or explicit limit from the microscopic generator (or from the fermionic dispersion of the XX chain) is supplied to derive the Hilbert-transform non-locality in the one-field current. This step is load-bearing for the central claim of emergent non-local hydrodynamics.

    Authors: We agree that the hydrodynamic equation is introduced as a conjecture without a complete scaling-limit derivation or mode expansion from the microscopic generator. The form of the current is motivated by the long-range Coulomb interactions of the original SDEP and by the structure that emerges from the exact mapping to the XX chain: the free-fermion representation of the XX Hamiltonian naturally produces non-local functionals involving the Hilbert transform when the continuum limit is taken for the conserved density. While we do not supply the full intermediate analysis in the present manuscript, the conjecture is strongly supported by the closed-form solutions for single- and double-block initial states, which generate explicit limit shapes and arctic curves that agree quantitatively with large-scale Monte Carlo simulations. In the revised version we will expand the discussion immediately after the XX-chain mapping to include a heuristic argument based on the fermionic dispersion and the expected non-local current arising from the long-range interactions, while making the conjectural status more explicit. revision: partial

Circularity Check

0 steps flagged

Exact Doob mapping to XX chain supports independent conjecture for non-local hydrodynamics

full rationale

The paper establishes an exact microscopic mapping of the SDEP generator to the XX quantum chain via Doob transform, which is independent of the macroscopic description. The specific non-local current j[ρ] = (1/π) sin(πρ) sinh(π ℋ ρ) is introduced explicitly as a conjecture for the Eulerian scaling limit rather than derived, fitted, or defined in terms of itself. No self-citations, ansatzes smuggled via prior work, or reductions of the central PDE to input data by construction are present in the provided derivation chain. The subsequent closed-form solutions for block initial conditions and their agreement with Monte Carlo simulations provide external checks, rendering the overall chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the exact Doob transform mapping to the XX chain (standard in the field) and the conjecture for the hydrodynamic limit; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The symmetric Dyson exclusion process admits an exact ground-state (Doob) transform mapping its stochastic generator onto the spin-1/2 XX quantum chain.
    Stated as exploited in the abstract to enable the free-fermion representation.
  • ad hoc to paper At macroscopic scales the SDEP displays ballistic scaling and is governed by the non-local hydrodynamic equation with current involving the Hilbert transform.
    This is the central conjecture of the paper.

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Forward citations

Cited by 1 Pith paper

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

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Works this paper leans on

74 extracted references · 74 canonical work pages · cited by 1 Pith paper · 5 internal anchors

  1. [1]

    Large Scale Dynamics of Interacting Particles

    H. Spohn, Large Scale Dynamics of Interacting Particles , Theoretical and Math- ematical Physics. Springer, Berlin \slash Heidelberg, doi:10.1007/978-3-642-84371-6 (1991)

    work page doi:10.1007/978-3-642-84371-6 1991

  2. [2]

    Kipnis and C

    C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems , vol. 320 of Grundlehren der mathematischen Wissenschaften , Springer, Berlin \slash Heidelberg, doi:10.1007/978-3-662-03752-2 (1999)

    work page doi:10.1007/978-3-662-03752-2 1999

  3. [3]

    T. M. Liggett, Continuous Time Markov Processes: An Introduction , vol. 113 of Graduate Studies in Mathematics , American Mathematical Society, Providence, RI, doi:10.1090/gsm/113 (2010)

    work page doi:10.1090/gsm/113 2010

  4. [4]

    P. L. Garrido, J. L. Lebowitz, C. Maes and H. Spohn, Long-range correlations for con- servative dynamics, Physical Review A 42(4) (1990), doi:10.1103/PhysRevA.42.1954

    work page doi:10.1103/physreva.42.1954 1990

  5. [5]

    G. L. Eyink, J. L. Lebowitz and H. Spohn, Lattice gas models in contact with stochas- tic reservoirs: Local equilibrium and relaxation to the steady state , Communications in Mathematical Physics 140, 119 (1991), doi:10.1007/BF02099293

    work page doi:10.1007/bf02099293 1991

  6. [6]

    C. Bahadoran, Hydrodynamics and Hydrostatics for a Class of Asymmetric Particle Systems with Open Boundaries , Communications in Mathematical Physics 310(1) (2012), doi:10.1007/s00220-011-1395-6

    work page doi:10.1007/s00220-011-1395-6 2012

  7. [7]

    Gon¸ calves,Hydrodynamics for Symmetric Exclusion in Contact with Reservoirs , In Stochastic Dynamics Out of Equilibrium , vol

    P. Gon¸ calves,Hydrodynamics for Symmetric Exclusion in Contact with Reservoirs , In Stochastic Dynamics Out of Equilibrium , vol. 282 of Springer Proceedings in Math- ematics & Statistics . Springer, Cham (2019)

    work page 2019

  8. [8]

    T. M. Liggett, Long-range exclusion processes, Annals of Probability 8, 861 (1980), doi:10.1214/aop/1176994618

    work page doi:10.1214/aop/1176994618 1980

  9. [9]

    E. D. Andjel and H. Guiol, Long-range exclusion processes, genera- tor and invariant measures , Annals of Probability 33(6), 2314 (2005), doi:10.1214/009117905000000486

    work page doi:10.1214/009117905000000486 2005

  10. [10]

    Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization

    P. Gon¸ calves and M. Jara, Density fluctuations for exclusion processes with long jumps, Probability Theory and Related Fields 170, 311 (2018), doi:10.1007/s00440- 017-0758-0

    work page doi:10.1007/s00440- 2018

  11. [11]

    Belitsky, N

    V. Belitsky, N. P. N. Ngoc and G. M. Sch¨ utz, Asymmetric exclusion process with long-range interactions, doi:10.48550/arXiv.2409.05017 (2024), 2409.05017. 20 SciPost Physics Submission

    work page doi:10.48550/arxiv.2409.05017 2024

  12. [12]

    G. M. Sch¨ utz,Exactly Solvable Models for Many-Body Systems Far from Equilibrium, In Phase Transitions and Critical Phenomena , vol. 19. Academic Press, London, doi:10.1016/S1062-7901(01)80015-X (2001)

    work page doi:10.1016/s1062-7901(01)80015-x 2001

  13. [13]

    F. C. Alcaraz, M. Droz, M. Henkel and V. Rittenberg, Reaction–Diffusion Pro- cesses, Critical Dynamics and Quantum Chains , Annals of Physics 230 (1994), doi:10.1006/aphy.1994.1026

    work page doi:10.1006/aphy.1994.1026 1994

  14. [14]

    Spohn, Bosonization, vicinal surfaces, and hydrodynamic fluctuation theory, Phys- ical Review E 60(5) (1999), doi:10.1103/PhysRevE.60.6411

    H. Spohn, Bosonization, vicinal surfaces, and hydrodynamic fluctuation theory, Phys- ical Review E 60(5) (1999), doi:10.1103/PhysRevE.60.6411

    work page doi:10.1103/physreve.60.6411 1999

  15. [16]

    G. M. Sch¨ utz, The Space–Time Structure of Extreme Current and Activity Events in the ASEP , In Nonlinear Mathematical Physics and Natural Hazards , vol. 163 of Springer Proceedings in Physics, pp. 13–28. Springer, Cham, doi:10.1007/978-3-319- 14328-6 2 (2015)

    work page doi:10.1007/978-3-319- 2015

  16. [17]

    R. L. Jack and P. Sollich, Large deviations and ensembles of trajectories in stochastic models , Progress of Theoretical Physics Supplement 184 (2010), doi:10.1143/PTPS.184.304

    work page doi:10.1143/ptps.184.304 2010

  17. [18]

    Chetrite and H

    R. Chetrite and H. Touchette, Nonequilibrium Markov processes conditioned on large deviations, Annales Henri Poincar´ e16(9) (2015), doi:10.1007/s00023-014-0375-8

    work page doi:10.1007/s00023-014-0375-8 2015

  18. [19]

    F. J. Dyson, A Brownian-Motion Model for the Eigenvalues of a Random Matrix , Journal of Mathematical Physics 3 (1962), doi:10.1063/1.1703862

    work page doi:10.1063/1.1703862 1962

  19. [20]

    Calogero, Ground State of a One-Dimensional N-Body System , Journal of Math- ematical Physics 10 (1969), doi:10.1063/1.1664821

    F. Calogero, Ground State of a One-Dimensional N-Body System , Journal of Math- ematical Physics 10 (1969), doi:10.1063/1.1664821

    work page doi:10.1063/1.1664821 1969

  20. [21]

    Sutherland, Exact Results for a Quantum Many-Body Problem in One Dimension

    B. Sutherland, Exact Results for a Quantum Many-Body Problem in One Dimension. II, Physical Review A 5 (1972), doi:10.1103/PhysRevA.5.1372

    work page doi:10.1103/physreva.5.1372 1972

  21. [22]

    A. G. Abanov, E. Bettelheim and P. Wiegmann, Integrable hydrodynamics of Calogero–Sutherland model: bidirectional Benjamin–Ono equation, Journal of Physics A: Mathematical and Theoretical 42 (2009), doi:10.1088/1751-8113/42/13/135201

    work page doi:10.1088/1751-8113/42/13/135201 2009

  22. [23]

    I. M. Krichever, Elliptic solutions of nonlinear integrable equations and related topics, Acta Applicandae Mathematicae 36 (1994), doi:10.1007/BF01001540

    work page doi:10.1007/bf01001540 1994

  23. [24]

    Lecomte, J

    V. Lecomte, J. P. Garrahan and F. v. Wijland, Inactive dynamical phase of a symmet- ric exclusion process on a ring , Journal of Physics A: Mathematical and Theoretical 45 (2012), doi:10.1088/1751-8113/45/17/175001

    work page doi:10.1088/1751-8113/45/17/175001 2012

  24. [25]

    R. L. Jack, I. R. Thompson and P. Sollich, Hyperuniformity and Phase Separation in Biased Ensembles of Trajectories for Diffusive Systems , Physical Review Letters 114 (2015), doi:10.1103/PhysRevLett.114.060601

    work page doi:10.1103/physrevlett.114.060601 2015

  25. [26]

    Conformal invariance in driven diffusive systems at high currents

    D. Karevski and G. M. Sch¨ utz, Conformal invariance in driven dif- fusive systems at high currents , Physical Review Letters 118 (2017), doi:10.1103/PhysRevLett.118.030601, eprint: arXiv:1606.04248. 21 SciPost Physics Submission

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.118.030601 2017

  26. [27]

    Interaction of Markov processes

    F. Spitzer, Interaction of Markov processes , Advances in Mathematics 5(2) (1970), doi:10.1016/0001-8708(70)90034-4

    work page doi:10.1016/0001-8708(70)90034-4 1970

  27. [28]

    T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Pro- cesses, vol. 324 of Grundlehren der mathematischen Wissenschaften , Springer, Berlin/Heidelberg, doi:10.1007/978-3-662-03990-8 (1999)

    work page doi:10.1007/978-3-662-03990-8 1999

  28. [29]

    E. Lieb, T. Schultz and D. Mattis, Two Soluble Models of an Antiferromagnetic Chain, Annals of Physics 16, 407 (1961), doi:10.1016/0003-4916(61)90115-4

    work page doi:10.1016/0003-4916(61)90115-4 1961

  29. [30]

    Niemeijer, Some exact calculations on a chain of spins 1/2 , Physica 36 (1967), doi:10.1016/0031-8914(67)90235-2

    T. Niemeijer, Some exact calculations on a chain of spins 1/2 , Physica 36 (1967), doi:10.1016/0031-8914(67)90235-2

    work page doi:10.1016/0031-8914(67)90235-2 1967

  30. [31]

    Doyon, and J

    I. Bouchoule, B. Doyon and J. Dubail, The effect of atom losses on the distribu- tion of rapidities in the one-dimensional Bose gas , SciPost Physics 9(4) (2020), doi:10.21468/SciPostPhys.9.4.044

    work page doi:10.21468/scipostphys.9.4.044 2020

  31. [32]

    A. G. Abanov, Hydrodynamics of correlated systems , In ´E. Br´ ezin, V. Kazakov, D. Serban, P. Wiegmann and A. Zabrodin, eds., Applications of Random Matrices in Physics, vol. 221 of NATO Science Series II: Mathematics, Physics and Chemistry , pp. 139–161. Springer, Dordrecht, doi:10.1007/1-4020-4531-X 5 (2006)

    work page doi:10.1007/1-4020-4531-x 2006

  32. [33]

    Antal, Z

    T. Antal, Z. R´ acz, A. R´ akos and G. M. Sch¨ utz,Transport in the XX chain at zero temperature: Emergence of flat magnetization profiles , Physical Review E 59 (1999), doi:10.1103/PhysRevE.59.4912

    work page doi:10.1103/physreve.59.4912 1999

  33. [34]

    Bettelheim, A

    E. Bettelheim, A. G. Abanov and P. Wiegmann, Orthogonality catastrophe and shock waves in a nonequilibrium Fermi gas , Physical Review Letters 97, 246402 (2006), doi:10.1103/PhysRevLett.97.246402

    work page doi:10.1103/physrevlett.97.246402 2006

  34. [35]

    Bettelheim and L

    E. Bettelheim and L. Glazman, Quantum ripples over a semiclassical shock , Physical Review Letters 109 (2012), doi:10.1103/PhysRevLett.109.260602

    work page doi:10.1103/physrevlett.109.260602 2012

  35. [36]

    Ruggiero, Y

    P. Ruggiero, Y. Brun and J. Dubail, Conformal field theory on top of a breathing one-dimensional gas of hard core bosons , SciPost Physics 6(4) (2019), doi:10.21468/SciPostPhys.6.4.051

    work page doi:10.21468/scipostphys.6.4.051 2019

  36. [37]

    S. Scopa et al., Exact entanglement growth of a one-dimensional hard-core quantum gas during a free expansion , Journal of Physics A: Mathematical and Theoretical 54(40) (2021), doi:10.1088/1751-8121/ac20ee

    work page doi:10.1088/1751-8121/ac20ee 2021

  37. [38]

    P.-G. d. Gennes, Soluble model for fibrous structures with steric constraints , Journal of Chemical Physics 48(5) (1968), doi:10.1063/1.1669420

    work page doi:10.1063/1.1669420 1968

  38. [39]

    Kenyon and A

    R. Kenyon and A. Okounkov, Limit shapes and the complex burgers equation , Acta Mathematica 199(2), 263 (2007), doi:10.1007/s11511-007-0021-0

    work page doi:10.1007/s11511-007-0021-0 2007

  39. [40]

    Allegra, J

    N. Allegra, J. Dubail, J.-M. St´ ephan and J. Viti, Inhomogeneous field theory inside the arctic circle, Journal of Statistical Mechanics: Theory and Experiment (5) (2016), doi:10.1088/1742-5468/2016/05/053108

    work page doi:10.1088/1742-5468/2016/05/053108 2016

  40. [41]

    Gorin, Lectures on Random Lozenge Tilings , vol

    V. Gorin, Lectures on Random Lozenge Tilings , vol. 193 of Cambridge Studies in Advanced Mathematics , Cambridge University Press, Cambridge, doi:10.1017/9781108921183 (2021). 22 SciPost Physics Submission

    work page doi:10.1017/9781108921183 2021

  41. [42]

    A. M. Matytsin, Large-\emphN limit of the Itzykson–Zuber integral , Nuclear Physics B 411 (1994), doi:10.1016/0550-3213(94)90471-5

    work page doi:10.1016/0550-3213(94)90471-5 1994

  42. [43]

    V. E. Korepin, A. G. Izergin and N. M. Bogoliubov, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, doi:10.1017/CBO9780511628832 (1993)

    work page doi:10.1017/cbo9780511628832 1993

  43. [44]

    A. G. Abanov and F. Franchini, Emptiness formation probability for the anisotropic XY spin chain in a magnetic field , Physics Letters A 316(5), 342 (2003), doi:10.1016/j.physleta.2003.07.009

    work page doi:10.1016/j.physleta.2003.07.009 2003

  44. [45]

    Franchini and A

    F. Franchini and A. G. Abanov, Asymptotics of Toeplitz determinants and the empti- ness formation probability for the XY spin chain , Journal of Physics A: Mathematical and General 38(23), 5069 (2005), doi:10.1088/0305-4470/38/23/002

    work page doi:10.1088/0305-4470/38/23/002 2005

  45. [46]

    St´ ephan,Emptiness formation probability, toeplitz determinants, and conformal field theory, Journal of Statistical Mechanics: Theory and Experiment p

    J.-M. St´ ephan,Emptiness formation probability, toeplitz determinants, and conformal field theory, Journal of Statistical Mechanics: Theory and Experiment p. P05010 (2014), doi:10.1088/1742-5468/2014/05/P05010

    work page doi:10.1088/1742-5468/2014/05/p05010 2014

  46. [47]

    J. S. Pallister, S. H. Pickering, D. M. Gangardt and A. G. Abanov, Phase transitions in full counting statistics of free fermions and directed polymers , Physical Review Research 7(2) (2025), doi:10.1103/PhysRevResearch.7.L022008

    work page doi:10.1103/physrevresearch.7.l022008 2025

  47. [48]

    D. G. Crowdy, Viscous Marangoni flow driven by insoluble surfactant and the complex Burgers equation , SIAM Journal on Applied Mathematics 81(6) (2021), doi:10.1137/21M1400316

    work page doi:10.1137/21m1400316 2021

  48. [49]

    Andraus and M

    S. Andraus and M. Katori, Characterizations of the hydrodynamic limit of the Dyson model, Published: arXiv:1602.00449 (2016)

    work page arXiv 2016

  49. [50]

    Dandekar, P

    R. Dandekar, P. L. Krapivsky and K. Mallick, Dynamical fluctuations in the Riesz gas, Physical Review E 107(4) (2023), doi:10.1103/PhysRevE.107.044129

    work page doi:10.1103/physreve.107.044129 2023

  50. [51]

    Dandekar, P

    R. Dandekar, P. L. Krapivsky and K. Mallick, Current fluctuations in the Dyson gas , Physical Review E 110(6) (2024), doi:10.1103/PhysRevE.110.064153

    work page doi:10.1103/physreve.110.064153 2024

  51. [52]

    P. L. Krapivsky and K. Mallick, Expansion into the vacuum of stochas- tic gases with long-range interactions , Physical Review E 111(6) (2025), doi:10.1103/PhysRevE.111.064109

    work page doi:10.1103/physreve.111.064109 2025

  52. [53]

    Rottman and M

    C. Rottman and M. Wortis, Statistical mechanics of equilibrium crystal shapes: interfacial phase diagrams and phase transitions , Physics Reports 103 (1984), doi:10.1016/0370-1573(84)90066-8

    work page doi:10.1016/0370-1573(84)90066-8 1984

  53. [54]

    Nienhuis, H

    B. Nienhuis, H. J. Hilhorst and H. W. J. Bl¨ ote, Triangular SOS models and cubic-crystal shapes , Journal of Physics A: Mathematical and General 17 (1984), doi:10.1088/0305-4470/17/18/025

    work page doi:10.1088/0305-4470/17/18/025 1984

  54. [55]

    Random Domino Tilings and the Arctic Circle Theorem

    W. Jockusch, J. Propp and P. Shor, Random domino tilings and the arctic circle theorem, doi:10.48550/arXiv.math/9801068 (1998)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/9801068 1998

  55. [56]

    Elkies, G

    N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating-sign matrices and domino tilings. Part I , Journal of Algebraic Combinatorics 1, 111 (1992), doi:10.1023/A:1022420103267. 23 SciPost Physics Submission

    work page doi:10.1023/a:1022420103267 1992

  56. [57]

    Elkies, G

    N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating-sign matrices and domino tilings. Part II , Journal of Algebraic Combinatorics 1, 219 (1992), doi:10.1023/A:1022483817303

    work page doi:10.1023/a:1022483817303 1992

  57. [58]

    B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux , Advances in Mathematics 26 (1977), doi:10.1016/0001-8708(77)90030-5

    work page doi:10.1016/0001-8708(77)90030-5 1977

  58. [59]

    Korepin and P

    V. Korepin and P. Zinn-Justin, Thermodynamic limit of the six-vertex model with domain wall boundary conditions , Journal of Physics A: Mathematical and General 33 (2000), doi:10.1088/0305-4470/33/40/304

    work page doi:10.1088/0305-4470/33/40/304 2000

  59. [60]

    Colomo and A

    F. Colomo and A. G. Pronko, The arctic circle revisited , In Integrable Sys- tems and Random Matrices: In Honor of Percy Deift , vol. 458 of Contempo- rary Mathematics , pp. 361–376. American Mathematical Society, Providence, RI, doi:10.1090/conm/458/08947 (2008)

    work page doi:10.1090/conm/458/08947 2008

  60. [61]

    The 6-vertex model with fixed boundary conditions

    K. Palamarchuk and N. Reshetikhin, The 6-vertex model with fixed boundary condi- tions, doi:10.48550/arXiv.1010.5011 (2010), 1010.5011

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1010.5011 2010

  61. [62]

    St´ ephan, Extreme boundary conditions and random tilings , SciPost Physics Lecture Notes 26 (2021), doi:10.21468/SciPostPhysLectNotes.26

    J.-M. St´ ephan, Extreme boundary conditions and random tilings , SciPost Physics Lecture Notes 26 (2021), doi:10.21468/SciPostPhysLectNotes.26

    work page doi:10.21468/scipostphyslectnotes.26 2021

  62. [63]

    Lectures on Dimers

    R. Kenyon, Lectures on dimers, doi:10.48550/arXiv.0910.3129 (2009)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.0910.3129 2009

  63. [64]

    Colomo and A

    F. Colomo and A. G. Pronko, The arctic curve of the domain-wall six-vertex model , Journal of Statistical Physics 138(4), 662 (2010), doi:10.1007/s10955-009-9902-2

    work page doi:10.1007/s10955-009-9902-2 2010

  64. [65]

    Colomo, A

    F. Colomo, A. G. Pronko and P. Zinn-Justin, The arctic curve of the domain-wall six- vertex model in its antiferroelectric regime , Journal of Statistical Mechanics: Theory and Experiment (3) (2010), doi:10.1088/1742-5468/2010/03/L03002

    work page doi:10.1088/1742-5468/2010/03/l03002 2010

  65. [66]

    Granet, L

    E. Granet, L. Budzynski, J. Dubail and J. L. Jacobsen, Inhomogeneous Gaussian free field inside the interacting arctic curve , Journal of Statistical Mechanics: Theory and Experiment (1) (2019), doi:10.1088/1742-5468/aaf71b

    work page doi:10.1088/1742-5468/aaf71b 2019

  66. [67]

    P. D. Francesco and E. Guitter, The arctic curve for Aztec rectangles with defects via the tangent method, Journal of Statistical Physics 176(3) (2019), doi:10.1007/s10955- 019-02315-2

    work page doi:10.1007/s10955- 2019

  67. [68]

    Imamura, M

    T. Imamura, M. Mucciconi and T. Sasamoto, New approach to KPZ models through free fermions at positive temperature , Journal of Mathematical Physics 64, 083301 (2023), doi:10.1063/5.0089778

    work page doi:10.1063/5.0089778 2023

  68. [69]

    Kechagia, Y

    P. Kechagia, Y. C. Yortsos and P. Lichtner, Nonlocal Kardar-Parisi-Zhang equation to model interface growth , Physical Review E 64(1), 016315 (2001), doi:10.1103/PhysRevE.64.016315, Publisher: APS

    work page doi:10.1103/physreve.64.016315 2001

  69. [70]

    Bernardin and R

    C. Bernardin and R. Chetrite, Macroscopic Fluctuation Theory for Ginzburg–Landau Dynamics with Long-Range Interactions , Journal of Statistical Physics 192(1), 7 (2025), doi:10.1007/s10955-024-03384-8, Publisher: Springer

    work page doi:10.1007/s10955-024-03384-8 2025

  70. [71]

    Hager, J

    J. Hager, J. Krug, V. Popkov and G. Sch¨ utz, Minimal current phase and universal boundary layers in driven diffusive systems , Physical Review E 63(5), 056110 (2001), doi:10.1103/PhysRevE.63.056110, Publisher: APS. 24 SciPost Physics Submission

    work page doi:10.1103/physreve.63.056110 2001

  71. [72]

    R. Boccagna, Stationary currents in long-range interacting magnetic systems , Math- ematical Physics, Analysis and Geometry 23(3), 30 (2020), doi:10.1007/s11040-020- 09354-2, Publisher: Springer

    work page doi:10.1007/s11040-020- 2020

  72. [73]

    M. Mourragui, Large deviations of the empirical current for the boundary driven Kawasaki process with long range interaction , ALEA, Latin Amer- ican Journal of Probability and Mathematical Statistics 11(2), 643 (2014), doi:10.48550/arXiv.1406.1463, 1406.1463

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1406.1463 2014

  73. [74]

    R. J. Harris and G. M. Sch¨ utz,Fluctuation theorems for stochastic dynamics, Journal of Statistical Mechanics: Theory and Experiment (07) (2007), doi:10.1088/1742- 5468/2007/07/P07020

    work page doi:10.1088/1742- 2007

  74. [75]

    Monthus, Microcanonical conditioning of Markov processes on time-additive ob- servables, Journal of Statistical Mechanics: Theory and Experiment p

    C. Monthus, Microcanonical conditioning of Markov processes on time-additive ob- servables, Journal of Statistical Mechanics: Theory and Experiment p. 023207 (2022), doi:10.1088/1742-5468/ac4e81. 25

    work page doi:10.1088/1742-5468/ac4e81 2022