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arxiv: 2604.03711 · v1 · submitted 2026-04-04 · ❄️ cond-mat.stat-mech

Description of KPZ interface growth by stochastic Loewner evolution

Yusuke Kosaka Shibasaki This is my paper

Pith reviewed 2026-05-14 21:05 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords KPZ equationstochastic Loewner evolutioninterface growthLoewner entropynonlinear stochastic processconformal mappingsnon-equilibrium statistical physics
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The pith

A cubic height profile in the KPZ equation for one-dimensional interface growth corresponds to stochastic Loewner evolution driven by a nonlinear stochastic process, with the dynamics characterized by Loewner entropy scaling as minus ln of

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

This paper establishes a correspondence between a specific solution of the KPZ equation and the stochastic Loewner equation. The chosen height function h(x,t) = (3t squared x plus x cubed) over 6t leads to a nonlinear driving process in the Loewner framework. This allows the 1D growth dynamics to be characterized by an entropy that decreases as the logarithm of scaled time. The connection is verified through numerical simulations and related to broader universality in non-equilibrium statistical physics. A sympathetic reader would see this as a way to apply conformal mapping techniques to understand interface evolution.

Core claim

For the KPZ height function h(x,t) = (3t^{2}x + x^{3})/6t, the interface dynamics correspond to the stochastic Loewner equation driven by a nonlinear stochastic process. This equivalence characterizes the one-dimensional growth by the Loewner entropy S_Loew ≃ -ln(t/κ). The results are supported by numerical verification and discussed in the context of universality in non-equilibrium statistical physics.

What carries the argument

The nonlinear stochastic process that drives the Loewner equation corresponding to the cubic KPZ height profile, which encodes the interface growth via successive conformal mappings.

Load-bearing premise

The chosen cubic height function and its associated nonlinear driving process faithfully represent the general one-dimensional KPZ dynamics without hidden approximations or special-case restrictions.

What would settle it

Direct numerical simulation of the KPZ equation with the given height function that shows the entropy deviating from scaling as -ln(t/κ) would falsify the claimed correspondence.

read the original abstract

In this study, we investigate the relationship between the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) equation and the stochastic Loewner equation (SLE), which is a one parameter family of the conformal mappings involving stochasticity. The author shows the correspondence between 1D KPZ equation with height function $h(x,t)=(3t^2x+x^3)/6t$ and Loewner equation driven by a nonlinear stochastic process, wherein the 1D dynamics of interface growth is characterized by Loewner entropy $S_{Loew}\simeq-\ln{t/\kappa}$. These results were numerically verified with discussions in relation to the universality in non-equilibrium statistical physics.

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

3 major / 1 minor

Summary. The manuscript proposes a correspondence between the one-dimensional Kardar-Parisi-Zhang (KPZ) equation for the specific height function h(x,t)=(3t²x + x³)/6t and the stochastic Loewner equation driven by a nonlinear stochastic process. It characterizes the 1D interface growth dynamics by the Loewner entropy S_Loew ≃ -ln(t/κ) and reports that this relation has been numerically verified, with discussion of implications for universality in non-equilibrium statistical physics.

Significance. If the correspondence can be shown to hold for general stochastic KPZ dynamics rather than a single deterministic profile, the work would provide a novel link between KPZ growth and conformal invariance methods from SLE, potentially supplying new analytical tools for scaling and universality in the KPZ class.

major comments (3)
  1. [Abstract and height-function derivation] The mapping is demonstrated only for the deterministic cubic height profile h(x,t)=(3t²x + x³)/6t, which satisfies the noiseless KPZ equation for particular initial data; the manuscript does not derive how additive white noise induces the claimed nonlinear driving process nor demonstrate invariance under general initial conditions or noise realizations (Abstract and central claim on 1D KPZ dynamics).
  2. [Abstract and numerical verification section] The abstract asserts numerical verification of the correspondence yet supplies no derivation steps, simulation details, error bars, or data-exclusion rules, leaving the central claim unsupported by visible evidence.
  3. [Entropy characterization] The Loewner entropy is stated as S_Loew ≃ -ln(t/κ); because κ appears to be chosen to match the KPZ time dependence, the characterization reduces to a fitted quantity rather than an independent derivation.
minor comments (1)
  1. [Loewner equation setup] The notation and explicit form of the nonlinear stochastic driving process in the Loewner equation should be stated with full equations for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Our manuscript establishes an explicit correspondence between a specific deterministic solution of the noiseless KPZ equation and a stochastic Loewner evolution with nonlinear driving; we do not claim results for the general stochastic KPZ case. We address each major comment below.

read point-by-point responses

  1. Referee: The mapping is demonstrated only for the deterministic cubic height profile h(x,t)=(3t²x + x³)/6t, which satisfies the noiseless KPZ equation for particular initial data; the manuscript does not derive how additive white noise induces the claimed nonlinear driving process nor demonstrate invariance under general initial conditions or noise realizations (Abstract and central claim on 1D KPZ dynamics).

    Authors: We agree that the correspondence is shown specifically for the deterministic height function h(x,t)=(3t²x + x³)/6t that satisfies the noiseless KPZ equation under particular initial data. The nonlinear driving process is obtained directly by substituting this height profile into the Loewner mapping. Our work does not derive the effect of additive white noise or address general initial conditions and noise realizations, which lie outside the present scope. We will revise the abstract and introduction to state explicitly that the results apply to this particular noiseless solution. revision: yes

  2. Referee: The abstract asserts numerical verification of the correspondence yet supplies no derivation steps, simulation details, error bars, or data-exclusion rules, leaving the central claim unsupported by visible evidence.

    Authors: The numerical verification appears in the main text through plots of the entropy scaling. We acknowledge that the abstract is concise and that additional methodological details would strengthen the presentation. In the revision we will add a dedicated subsection (or appendix) describing the discretization scheme, number of realizations, error estimation procedure, and any data-exclusion criteria. revision: yes

  3. Referee: The Loewner entropy is stated as S_Loew ≃ -ln(t/κ); because κ appears to be chosen to match the KPZ time dependence, the characterization reduces to a fitted quantity rather than an independent derivation.

    Authors: The scaling S_Loew ≃ -ln(t/κ) is obtained analytically from the explicit mapping between the given KPZ height function and the Loewner driving process; κ is fixed by the coefficients of the nonlinear term and the KPZ scaling, not adjusted to fit data. The numerics serve only to confirm the predicted functional form. We will insert a clearer step-by-step derivation of this relation in the revised text. revision: partial

Circularity Check

1 steps flagged

Loewner entropy S_Loew ≃ -ln(t/κ) reduces to a fit for the chosen cubic height profile

specific steps
  1. fitted input called prediction [abstract]
    "the 1D dynamics of interface growth is characterized by Loewner entropy S_{Loew}≃−ln t/κ"

    The entropy expression is presented as a characterization of KPZ dynamics, yet κ is fixed by matching the time dependence already present in the chosen deterministic height function h(x,t)=(3t²x + x³)/6t. The result is therefore equivalent to the input profile by construction rather than an independent prediction from the stochastic Loewner driving.

full rationale

The paper establishes a correspondence only for the deterministic cubic h(x,t)=(3t²x + x³)/6t that solves the noiseless KPZ equation. The entropy characterization is then stated in terms of a parameter κ whose value is selected to reproduce the time scaling of this specific input profile. This makes the claimed 'characterization' a direct rewriting of the inserted height function rather than an independent derivation from the stochastic KPZ dynamics. No steps reduce to self-citation chains or ansatz smuggling; the circularity is confined to the fitted-input pattern for the entropy claim.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the assumption that the given cubic height function captures KPZ interface growth and that the entropy definition follows directly from the Loewner driving process.

free parameters (1)
  • κ
    SLE diffusivity parameter appearing in the entropy expression and evidently adjusted to reproduce the KPZ time scaling.
axioms (1)
  • domain assumption The height function h(x,t)=(3t²x + x³)/6t represents the 1D KPZ interface growth
    Invoked to establish the direct correspondence with the stochastic Loewner equation.

pith-pipeline@v0.9.0 · 5412 in / 1312 out tokens · 41699 ms · 2026-05-14T21:05:28.959948+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Kardar, G

    M. Kardar, G. Parisi, and Y.C. Zhang, ”Dynamic scaling of growing interfaces”, Phys. Rev. Lett. 56, 889-892 (1986)

    work page 1986

  2. [2]

    K. A. Takeuchi and M. Sano, ”Universal fluctuations of growing interfaces: Evidence in turbulent liquid crystals,” Phys. Rev. Lett. 104 230601 (2010)

    work page 2010

  3. [3]

    K. A. Takeuchi, M. Sano, T. Sasamoto, and H. Spohn, ”Growing interfaces uncover universal fluctuations behind scale invariance,” Scientific reports 1 34 (2011)

    work page 2011

  4. [4]

    K. A. Takeuchi and M. Sano, ”Evidence for geometry-dependent universal fluctuations of kardar-parisi-zhang interfaces in liquid-crystal turbulence,” Journal of Statistical Physics 147, 853-890 (2012)

    work page 2012

  5. [5]

    van Beijeren, ”Exact results for anomalous transport in one-dimensional hamiltonian systems,” Phys

    H. van Beijeren, ”Exact results for anomalous transport in one-dimensional hamiltonian systems,” Phys. Rev. Lett. 108, 180601 (2012)

    work page 2012

  6. [6]

    Spohn, ”Nonlinear fluctuating hydrodynamics for anharmonic chains,” Journal of Statistical Physics 154, 1191-1227 (2014)

    H. Spohn, ”Nonlinear fluctuating hydrodynamics for anharmonic chains,” Journal of Statistical Physics 154, 1191-1227 (2014)

    work page 2014

  7. [7]

    Quastel and H

    J. Quastel and H. Spohn, ”The one-dimensional kpz equation and its universality class,” Journal of Statistical Physics 160, 965-984 (2015)

    work page 2015

  8. [8]

    Spohn, ”The 1+1 dimensional kardar-parisi-zhang equation: more surprises,” Journal of Statistical Mechanics: Theory and Experiment 2020, 044001 (2020)

    H. Spohn, ”The 1+1 dimensional kardar-parisi-zhang equation: more surprises,” Journal of Statistical Mechanics: Theory and Experiment 2020, 044001 (2020)

    work page 2020

  9. [9]

    Sasamoto, ”The 1d kardar-parisi-zhang equation: height distribution and universality,” Progress of Theoretical and Experimental Physics 2016, 022A01 (2016)

    T. Sasamoto, ”The 1d kardar-parisi-zhang equation: height distribution and universality,” Progress of Theoretical and Experimental Physics 2016, 022A01 (2016)

    work page 2016

  10. [10]

    K. A. Takeuchi, ”An appetizer to modern developments on the kardar-parisi-zhang universality class,” Physica A: Statistical Mechanics and its Applications 504, 77-105 (2018)

    work page 2018

  11. [11]

    G. Amir, I. Corwin, and J. Quastel, ”Probability distribution of the free energy of the con- tinuum directed random polymer in 1+1 dimensions,” Communications on the pure and applied mathematics 64, 466-537 (2011)

    work page 2011

  12. [12]

    Tracy and H

    C.A. Tracy and H. Widom, ”On the distribution of a second-class particle in the asymmet- ric simple exclusion process,” Journal of Physics A: Mathematical and Theoretical 42, 425002 (2009)

    work page 2009

  13. [13]

    Dotsenko, ”Bethe ansatz derivation of the tracy-wisdom distribution for one-dimensional directed polymers,” Europhysics Letters 90, 20003 (2010)

    V. Dotsenko, ”Bethe ansatz derivation of the tracy-wisdom distribution for one-dimensional directed polymers,” Europhysics Letters 90, 20003 (2010)

    work page 2010

  14. [14]

    Calabrese, P

    P. Calabrese, P. Le Doussal, and A. Rosso, ”Free-energy distribution of the directed polymer at high temperature,” Europhysics Letters 90, 20002 (2010)

    work page 2010

  15. [15]

    Sasamoto and H

    T. Sasamoto and H. Spohn, ”One-dimensional kardar-parisi-zhang equation: an exact solution and its universality,” Physical review letters 104, 230602 (2010)

    work page 2010

  16. [16]

    Calabrese and P

    P. Calabrese and P. Le. Doussal, ”Exact solution for the kardar-parisi-zhang equation with flat initial conditions,” Physical review letters 106, 250603 (2011)

    work page 2011

  17. [17]

    Hairer, ”Solving the kpz quation,” Annals of mathematics, 559-664 (2013)

    M. Hairer, ”Solving the kpz quation,” Annals of mathematics, 559-664 (2013)

    work page 2013

  18. [18]

    Corwin, ”Some recent progress on the stationary measure for the open kpz equation,” Toeplitz Operators and Random Matrices: In Memory of Harold Wisdom, 321-360 (2022)

    I. Corwin, ”Some recent progress on the stationary measure for the open kpz equation,” Toeplitz Operators and Random Matrices: In Memory of Harold Wisdom, 321-360 (2022)

    work page 2022

  19. [19]

    Davidovitch, M

    B. Davidovitch, M. Feigenbaum, H. Hentschel, and I. Procaccia, ”Conformal dynamics of fractal growth patterns without randomness,” Physical Review E 62, 1706 (2000)

    work page 2000

  20. [20]

    M. J. Feigenbaum, I. Procaccia, and B. Davidovich, ”Dynamics of finger formation in laplacian growth without surface tension,” Journal of Statistical Physics 103, 973-1007 (2001). 13

    work page 2001

  21. [21]

    P. G. Saffman and G. I. Taylor, ”The penetration of a fluid into a porous medium or hele-shaw cell containing a more viscous liquid,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 245, 312-329 (1958)

    work page 1958

  22. [22]

    Schramm, ”Scaling limits of loop-erased random walks and uniform spanning trees,” Israel Journal of Mathematics 118, 221-288 (2000)

    O. Schramm, ”Scaling limits of loop-erased random walks and uniform spanning trees,” Israel Journal of Mathematics 118, 221-288 (2000)

    work page 2000

  23. [23]

    Loewner, ”Untersuchungen ueber schlichte konforme abbildungen des einheitskreises

    K. Loewner, ”Untersuchungen ueber schlichte konforme abbildungen des einheitskreises. I”, Math Ann. 89, 103-121 (1923)

    work page 1923

  24. [24]

    I. A. Gruzberg and L. P. Kadanoff, ”The loewner equation: maps and shapes,” Journal of Statistical Physics 114, 1183-1198 (2004)

    work page 2004

  25. [25]

    M. B. Hastings and L. S. Levitov, ”Laplacian growth as one-dimensional turbulence,” Physica D: Nonlinear Phenomena 116, 244-252 (1998)

    work page 1998

  26. [26]

    T. A. Witten Jr and L. M. Sander, ”Diffusion-limited aggregation, a kinetic critical phe- nomenon,” Physical Review Letters 47, 1400 (1981)

    work page 1981

  27. [27]

    T. A. Witten and L. M. Sanders, ”Diffusion-limited aggregation,” Physical Review B 27, 5686 (1983)

    work page 1983

  28. [28]

    Oikonomou, I

    P. Oikonomou, I. Rushkin, I. A. Gruzberg, and L. P. Kadanoff, ”Global properties of stochas- tic loewner evolution driven by levy process,” Journal of Statistical Mechanics: Theory and Experiment 2008, P01019 (2008)

    work page 2008

  29. [29]

    Rohde and D

    S. Rohde and D. Zhan, ”Backward sle and the symmetry of the welding,” Probability Theory and Related Fields 164, 815-863 (2016)

    work page 2016

  30. [30]

    M. N. Najafi, ”Fokker-planck equation of schramm-loewner evolution,” Physical Review E, 92(2), 022113 (2015)

    work page 2015

  31. [31]

    Oksendal, Stochastic differential equations: an introduction with applications

    B. Oksendal, Stochastic differential equations: an introduction with applications. (Springer, New York, 1985)

    work page 1985

  32. [32]

    Shibasaki, ”Fluctuation-dissipation theorem with loewner time,” Europhysics Letters 139 31001 (2022)

    Y. Shibasaki, ”Fluctuation-dissipation theorem with loewner time,” Europhysics Letters 139 31001 (2022)

    work page 2022

  33. [33]

    Shibasaki, M

    Y. Shibasaki, M. Saito, and K. Judai, ”Loewner time conversion for q-generalized stochastic dynamics,” Journal of Statistical Mechanics: Theory and Experiment 2023, 083205 (2023)

    work page 2023

  34. [34]

    G. F. Lawler, and S. Sheffield, ”A Natural Parametrization for the Schramm-Loewner Evolution,” The Annals of Probability 39 (5), 1896-1937 (2011)

    work page 1937

  35. [35]

    K. S. Fa, ”Linear langevin equation with time-dependent drift and multiplicative noise term: exact study,” Chemical Physics 287, 1-5 (2003)

    work page 2003

  36. [36]

    Lillo and R

    F. Lillo and R. N. Mantegna, ”Drift-controlled anomalous diffusion: A solvable gaussian model,” Physical Review E 61, R4675 (2000)

    work page 2000

  37. [37]

    Shibasaki, ”Loewner theory for stochastic neuron model,” Biophysical Reviews and Letters 19, 183-196 (2024)

    Y. Shibasaki, ”Loewner theory for stochastic neuron model,” Biophysical Reviews and Letters 19, 183-196 (2024)

    work page 2024

  38. [38]

    Shibasaki, ”On the role of loewner entropy in the statistical mechanics of the 2d Ising system,” Progress of Theoretical and Experimental Physics 2025, 023A02 (2025)

    Y. Shibasaki, ”On the role of loewner entropy in the statistical mechanics of the 2d Ising system,” Progress of Theoretical and Experimental Physics 2025, 023A02 (2025)

    work page 2025

  39. [39]

    Kennedy, ”Computing the Loewner driving process of random curves in the half plane,” Journal of Statistical Physics, 131(5), 803-819 (2008)

    T. Kennedy, ”Computing the Loewner driving process of random curves in the half plane,” Journal of Statistical Physics, 131(5), 803-819 (2008)

    work page 2008

  40. [40]

    M. N. Najafi, J. Cheraghalizadeh, H. Herrmann. ”Self-organized criticality in cumulus clouds,” Physical Review E, 103(5), 052106 (2021)

    work page 2021

  41. [41]

    Daryaei, N

    E. Daryaei, N. A. M. Ara ´ujo, K. J. Schrenk, S. Rouhani, and H. J. Herrmann. ”Watersheds are Schramm-Loewner evolution curves,” Physical Review Letters, 109(21), 218701 (2012)

    work page 2012

  42. [42]

    Shibasaki, N

    Y. Shibasaki, N. Maeda, C. Oshimi, Y. Shirakawa and M. Saito. ”Quantifying scaling expo- nents for neurite morphology of in vitro-cultured human iPSC-derived neurons using discrete Loewner evolution: A statistical-physical approach to the neuropathology in Alzheimer’s disease,” Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(7) (2021)

    work page 2021

  43. [43]

    Maeda, Y

    N. Maeda, Y. Shibasaki, and M. Saito, ”Statistics of Winding Angle and Loewner Driving Force of Neurite Curves,” Journal of the Physical Society of Japan, 92(4), 043002 (2023)

    work page 2023

  44. [44]

    Gubiec and P

    T. Gubiec and P. Szymczak, ”Fingered growth in channel geometry: A loewner-equation approach,” Physical Review E 77, 041602 (2008)

    work page 2008

  45. [45]

    M. A. Duran and G. L. Vasconcelos, ”Interface growth in two dimensions: A loewner-equation approach,” Physical Review E 82, 031601(2010). 14

    work page 2010