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arxiv: 2606.11254 · v1 · pith:5PFA57LKnew · submitted 2026-06-08 · ❄️ cond-mat.stat-mech · cs.NA· math.NA· math.PR

Numerical simulations of the spread from the mean of the SLE and Multiple SLE dynamics

Phillip Kim , Vlad Margarint This is my paper

Pith reviewed 2026-06-27 14:35 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.NAmath.NAmath.PR
keywords SLEmultiple SLELoewner differential equationnumerical simulationdistribution of spreadBrownian motion driverconformal mappingDyson Brownian motion
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The pith

Numerical experiments predict bimodal distributions for the spread from the mean in SLE when started near the origin, shifting to bell-shaped farther out, with multiple SLE always bell-shaped.

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

The paper uses Euler's method to numerically simulate the Loewner equation for single and multiple SLE at fixed time. It tracks two measures of how much individual paths deviate from their sample average: the modulus difference and the real-part difference. Experiments indicate that these spread distributions are bimodal close to the origin in the single SLE case but can become bell-shaped when started farther away. In contrast, the multiple SLE case with Dyson Brownian motion drivers yields bell-shaped distributions in all tested scenarios. The work also examines how changing the parameters kappa and beta alters these distributions, offering predictions for theoretical analysis.

Core claim

Numerical simulations of the SLE driven by Brownian motion show that the distributions of |g_t(z) - mean| and Re(g_t(z)) - Re(mean) are bimodal when the starting point is close to the origin and can transition to bell-shaped when started further away. For multiple SLE with Dyson Brownian motion drivers, the distributions are bell-shaped in all cases examined. These findings are obtained by applying Euler's method and provide numerical predictions for future theoretical studies on the spread from the mean behavior.

What carries the argument

The quantities |g_t(z) - \g_t(z)\| and Re(g_t(z)) - Re(\g_t(z)\), which measure the deviation of the conformal map from its sample average at fixed time in the Loewner dynamics.

Load-bearing premise

The Euler method discretization with the chosen time step faithfully approximates the underlying continuous Loewner dynamics without introducing artifacts that alter the distribution shapes, particularly near singularities.

What would settle it

Performing simulations with a significantly smaller time step or an alternative integration method such as Runge-Kutta and checking if the bimodal or bell-shaped nature of the distributions remains unchanged.

Figures

Figures reproduced from arXiv: 2606.11254 by Phillip Kim, Vlad Margarint.

Figure 1
Figure 1. Figure 1: An example of Dyson Brownian Motion for β = 2 for N = 10 with all points starting roughly at 0 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The plot of gt(z) starting from 1 + 3i. In this figure, ginfin which is the theoretical map as N → ∞ and g0 which is the map for when κ = 0 are plotted for reference. We observe that as N increases, our simulations indicate the dynamics is approaching the theoretical prediction. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Histogram of Re(g0.25(1.02i)) − Re(g0.25(1.02i)) for κ = 8 3 . 3.2 The SLE maps dynamics started from z0 = 3i When starting our maps further up, we found that the values were much more closely distributed around the sample mean, which suggests a tighter clustering. This is represented by a ”bell shape” in Re(gt(z))−Re(gt(z). This matches the 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Histogram of Re(g0.25(3i)) − Re(g0.25(1.02i)) for κ = 8 3 . 4 Numerical simulations for the Multiple SLE dynamics In the case of the Multiple SLE dynamics the computations cost is much higher given the complexity of both the DBM drivers dynamics and their impact on the multiple SLE. The first thing to note was that for N = 60, that there was a dependence on β for how spread the values were, as seen in 5. T… view at source ↗
Figure 5
Figure 5. Figure 5: Histogram of |gt(z) − gt(z)| for β = 2, 3, 4. These experiments show a good prediction with the theory as when β de￾creases the parameter κ increases (as β = 8/κ) and we see this reflected in the simulations as well according to the image. There is more spread around the sample mean as κ is larger. We ran further experiments with β = 1, 2 for N = 10, 50, 100. To select the initial X0 for our Dyson Brownian… view at source ↗
Figure 6
Figure 6. Figure 6: Histograms for β = 1 and N = 50. 5 Future Directions We hope that these explorations will stimulate theoretical work in the future to check these numerical predictions. In addition, we encourage the study of the 9 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example plot within the Complex Plane with starting point with [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Histograms for Re(g0.25(1.02i)) − Re(g0.25(1.02i)). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Histograms for Re(g0.25(1.02i)) − Re(g0.25(1.02i)). (a) κ = 4 (b) κ = 5 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Histograms for Re(g0.25(1.02i)) − Re(g0.25(1.02i)). B Numerical experiments studying the spread of the dynamics for the SLE maps dynamics started from z0 = 3i (a) κ = 1 (b) κ = 2 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Histogram of Re(g0.25(3i)) − Re(g0.25(3i)). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Histogram of Re(g0.25(3i)) − Re(g0.25(3i)). (a) κ = 4 (b) κ = 5 [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Histogram of Re(g0.25(3i)) − Re(g0.25(3i)). 15 [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Histograms for β = 1 and N = 10. (a) N ·(Re(g0.25(1.02i))−Re(g0.25(1.02i))) (b) N · |g0.25(1.02i) − g0.25(1.02i)| [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Histograms for β = 1 and N = 50. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Histograms for β = 1 and N = 100. (a) N ·(Re(g0.25(1.02i))−Re(g0.25(1.02i))) (b) N · |g0.25(1.02i) − g0.25(1.02i)| [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Histograms for β = 2 and N = 10. (a) N ·(Re(g0.25(1.02i))−Re(g0.25(1.02i))) (b) N · |g0.25(1.02i) − g0.25(1.02i)| [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Histograms for β = 2 and N = 50. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Histograms for β = 2 and N = 100. D Numerical experiments studying the spread for the mulitple SLE maps dynamics started from z0 = 3i (a) N · (Re(g0.25(3i)) − Re(g0.25(3i))) (b) N · |g0.25(3i) − g0.25(3i)| [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Histograms for β = 1 and N = 10. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Histograms for β = 1 and N = 50. (a) N · (Re(g0.25(3i)) − Re(g0.25(3i))) (b) N · |g0.25(3i) − g0.25(3i)| [PITH_FULL_IMAGE:figures/full_fig_p019_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Histograms for β = 1 and N = 100. (a) N · (Re(g0.25(3i)) − Re(g0.25(3i))) (b) N · |g0.25(3i) − g0.25(3i)| [PITH_FULL_IMAGE:figures/full_fig_p019_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Histograms for β = 2 and N = 10. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Histograms for β = 2 and N = 50. (a) N · (Re(g0.25(3i)) − Re(g0.25(3i))) (b) N · |g0.25(3i) − g0.25(3i)| [PITH_FULL_IMAGE:figures/full_fig_p020_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Histograms for β = 2 and N = 100. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_25.png] view at source ↗
read the original abstract

The Schramm-Loewner Evolution (SLE) describes a family of fractal curves that arise in the study of the scaling limits of many planar Statistical Physics models. These curves are modeled using the Loewner Differential Equation for the conformal maps $g_t(z)$ with a Brownian motion driver. Using Euler's Method, in the current work we performed numerical experiments to study at a fixed time the quantities $|g_t(z) - \overline{g_t(z)}|$ and $Re(g_t(z)) - Re(\overline{g_t(z)})$, where $Re$ denotes the real part and $\overline{g_t(z)}$ refers to the sample average. These random variables measure the 'spread' of the dynamics from the average behavior at fixed time. One of the scopes of this work is to give numerical predictions for future theoretical investigations on these quantities. When investigating these quantities in the SLE case our experiments predict that the distribution is bimodal when the dynamics started close to the origin, and it can become bell-shaped if the dynamics is started further from the origin. In the second part, we performed experiments for a Multiple SLE model whose driver is Dyson Brownian Motion. Due to singularity in the dynamics of the drivers and the many data points needed, this part is challenging from a computational perspective. In the multiple SLE case, our experiments predict that the distribution is bell-shaped in all cases. In addition, we check the changes in the distributions as we vary the parameter $\kappa$ in the SLE case and $\beta$ in the Multiple SLE case.

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

2 major / 2 minor

Summary. The paper performs forward Euler numerical integration of the Loewner ODE for standard SLE (Brownian driver) and multiple SLE (Dyson Brownian driver) at fixed time t. It reports the empirical distributions of |g_t(z) − ar g_t(z)| and Re(g_t(z)) − Re(ar g_t(z)) for varying starting points z, claiming these are bimodal for SLE started near the origin and bell-shaped for SLE started farther away or for all multiple-SLE cases; it also examines dependence on κ and β and presents the shapes as predictions for future analytic work.

Significance. If the reported distribution shapes survive proper numerical validation, the work supplies concrete empirical targets that could guide analytic study of deviation statistics in Loewner evolutions. The computational challenges near singularities are acknowledged, but the lack of documented discretization control and statistical error assessment limits the strength of the predictions.

major comments (2)
  1. [Numerical experiments / Methods] Numerical experiments (Euler integration of the Loewner ODE): the time step Δt is never stated and no mesh-refinement study, local truncation-error bound, or comparison with a higher-order integrator is supplied. Because the modality claims rest entirely on the observed histograms, this omission is load-bearing; the shapes could be discretization artifacts, especially when drivers approach the real axis or multiple poles interact.
  2. [Results and figures] Results on distribution shapes: no sample sizes, error bars, or convergence diagnostics with respect to number of Monte Carlo realizations are reported, nor is any benchmark against known SLE properties (e.g., exact moments for small κ or chordal vs. radial cases) provided. Without these, the bimodal-versus-bell-shaped distinction cannot be assessed for statistical reliability.
minor comments (2)
  1. [Abstract] The abstract states that the multiple-SLE part is “challenging from a computational perspective” due to singularities and data volume, yet no quantitative information on realized sample counts, run times, or singularity-handling strategy is given.
  2. [Introduction / Notation] Notation: ar g_t(z) is introduced as the sample average; it should be stated explicitly whether this is an ensemble average over independent realizations or a time average, and how many realizations enter the average.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our numerical study of spread distributions in SLE and multiple SLE. We address each major comment below and will revise the manuscript accordingly to improve documentation and statistical validation.

read point-by-point responses

  1. Referee: [Numerical experiments / Methods] Numerical experiments (Euler integration of the Loewner ODE): the time step Δt is never stated and no mesh-refinement study, local truncation-error bound, or comparison with a higher-order integrator is supplied. Because the modality claims rest entirely on the observed histograms, this omission is load-bearing; the shapes could be discretization artifacts, especially when drivers approach the real axis or multiple poles interact.

    Authors: We agree that the discretization parameters must be explicitly documented. In the revised manuscript we will state the fixed time step Δt used for the forward Euler integration, include a mesh-refinement study for representative starting points and κ values showing that the reported bimodal and bell-shaped features persist under refinement, and briefly justify the choice of Euler scheme given the computational constraints near singularities already noted in the text. revision: yes

  2. Referee: [Results and figures] Results on distribution shapes: no sample sizes, error bars, or convergence diagnostics with respect to number of Monte Carlo realizations are reported, nor is any benchmark against known SLE properties (e.g., exact moments for small κ or chordal vs. radial cases) provided. Without these, the bimodal-versus-bell-shaped distinction cannot be assessed for statistical reliability.

    Authors: We accept that sample sizes, error estimates, and convergence information are required for assessing reliability. The revised version will report the number of Monte Carlo realizations for each histogram, add error bars, and include diagnostics showing stability with increasing sample count. While the spread quantities |g_t(z)−ḡ_t(z)| and Re(g_t(z))−Re(ḡ_t(z)) lack known closed-form distributions, we will add supporting benchmarks such as the deterministic κ=0 limit and small-κ moment comparisons where analytic results are available. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct outputs of numerical integration with no fitted parameters or self-referential derivations

full rationale

The manuscript contains no analytic derivation chain, no parameter fitting, and no self-citations. All reported distribution shapes (bimodal or bell-shaped) are obtained by forward Euler integration of the Loewner ODE driven by Brownian motion or Dyson Brownian motion, followed by direct histogram construction on the simulated trajectories of |g_t(z) - ḡ_t(z)| and Re(g_t(z)) - Re(ḡ_t(z)). These quantities are defined independently of the simulation outputs; the paper simply records their empirical statistics at fixed t. Because no quantity is defined in terms of another or obtained by fitting a subset of the same data, none of the enumerated circularity patterns apply. The work is therefore self-contained against external benchmarks (the continuous Loewner dynamics) and receives the default non-finding.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The claims rest on the accuracy of the chosen numerical scheme and the assumption that finite-sample histograms faithfully represent the underlying continuous distributions; no new entities are postulated.

free parameters (4)
  • Euler time step
    Discretization parameter required to integrate the Loewner equation; its value is not reported and directly affects accuracy near singularities.
  • Number of Monte Carlo samples
    Determines the reliability of the empirical distribution; unspecified in the abstract.
  • Fixed time t and starting point z
    Parameters at which spread is measured; their specific values control the reported bimodality transition.
  • kappa and beta
    Model parameters varied to observe distribution changes.
axioms (2)
  • domain assumption The Loewner differential equation driven by Brownian motion (or Dyson Brownian motion) generates the SLE (multiple SLE) curves.
    This is the modeling assumption that justifies running the numerical experiments at all.
  • standard math Standard properties of Brownian motion and stochastic differential equations hold.
    Invoked implicitly to justify the existence and simulation of the drivers.

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