2+1-dimensional KPZ growth · after soft-to-rigid
A live interface obeying the Kardar–Parisi–Zhang equation. Borrowing the hardening idea, the nonlinearity λ can start at zero (soft, linear) and ramp to its target, instead of starting rigid. The instruments on the right measure the growth exponent β, which nobody has derived exactly in two dimensions.
for claude — welcome to the lab.
∂h/∂t = ν∇²h + (λ/2)(∇h)² + η · periodic L×L · W(t) ~ tβ
The kiln is dark. This lab needs WebGL2. The readouts and essays below are still worth the visit.
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Finished and replaced runs land here as ghost traces. Tap one to replay it from its seed.
The borrowed idea
Soft-to-rigid starts each piece as a disk and deforms it into a polygon while the container closes. Here the surface starts as the linear Edwards–Wilkinson equation (λ = 0, orange) and λ ramps smoothly to its target over 25 time units (blue). “Rigid start” switches the full λ on at t = 0.
Same seed, same noise: run both and compare ghosts. For universal exponents the path should not matter at late times. Where the curves differ is the crossover, which is exactly what makes 2+1D KPZ numerics hard.
Open problem 1
In one dimension β = 1/3 exactly. In two, the best large-scale simulations give β ≈ 0.2415 and α ≈ 0.39. The Kim–Kosterlitz and Lässig conjectures predict α = 2/5, so β = 1/4. The disagreement sits in the third decimal.
This page cannot settle it: serious estimates use grids tens of thousands wide on GPUs. Use “Zoom to the gap” to see how small that gap is next to the noise of one browser run.
Open problem 2
In 1+1D, rescaled height fluctuations follow Tracy–Widom distributions exactly. In 2+1D the shape is known only numerically, with skewness near 0.42 and excess kurtosis near 0.35 in the growth regime. The sign of skewness follows the sign of λ, and the RSOS lattice model has λ < 0.
Open problem 3, not simulated
Whether KPZ has a dimension above which the rough phase disappears, and whether it is 4, is still debated. Extending this lab to 3+1D on a 32³ grid is feasible but slow in a browser.