Extremely large scale simulation of a Kardar-Parisi-Zhang model using graphics cards

The octahedron model introduced recently has been implemented onto graphics cards, which permits extremely large scale simulations via binary lattice gases and bit coded algorithms. We confirm scaling behaviour belonging to the 2d Kardar-Parisi-Zhang universality class and find a surface growth exponent: beta=0.2415(15) on 2^17 x 2^17 systems, ruling out beta=1/4 suggested by field theory. The maximum speed-up with respect to a single CPU is 240. The steady state has been analysed by finite size scaling and a growth exponent alpha=0.393(4) is found. Correction to scaling exponents are computed and the power-spectrum density of the steady state is determined. We calculate the universal scaling functions, cumulants and show that the limit distribution can be obtained by the sizes considered. We provide numerical fitting for the small and large tail behaviour of the steady state scaling function of the interface width.

💡 Research Summary

The authors present a large‑scale GPU implementation of the octahedron model, a discrete representation of the two‑dimensional Kardar‑Parisi‑Zhang (KPZ) equation. By encoding the two local slopes at each lattice site into two bits, a 4 × 4 block fits into a single 32‑bit word, allowing highly efficient memory usage and massive parallelism. Two levels of domain decomposition are employed: a “dead‑border” scheme at the device level to avoid inter‑block conflicts, and a block‑local 8 × 8 thread cell that maximizes the number of active threads per CUDA block. Random numbers are generated per thread with a 32‑bit linear congruential generator, periodically reseeded to prevent correlations. The code runs on an NVIDIA C2070 GPU with 6 GB of memory, enabling simulations up to L = 2¹⁷ (≈1.3 × 10⁵) lattice sites per dimension, i.e., a 2⁴⁰‑site system. Benchmarks show a speed‑up of up to 240× compared with a single‑core CPU implementation, while the simulation time per Monte‑Carlo step remains in the millisecond range even for the largest systems.

The physical observables are the interface width W(L, t) and its scaling properties. In the growth regime the effective exponent β_eff(t) is obtained from the logarithmic derivative of W with respect to time. For the largest system (L = 2¹⁷) the authors find β = 0.2415 ± 0.0015, clearly different from the one‑loop renormalisation‑group prediction β = ¼. Corrections to scaling are quantified by fitting β_eff(t) = β + b₁ t^{φ₁}, yielding φ₁ ≈ 1.05 and b₁ ≈ −0.125. In the steady state the width saturates as W(∞, L) ∝ L^{α} with α = 0.393 ± 0.004. Finite‑size corrections are described by α_eff(L) = α + a₁ L^{−ω₁}, giving ω₁ ≈ 1.16 and a₁ ≈ −1.24. The dynamic exponent follows as z = α/β ≈ 1.627 ± 0.026, satisfying the Galilean invariance relation α + z ≈ 2.02.

The power‑spectral density S(k) of the interface in the steady state follows the expected KPZ scaling S(k) ∝ k^{−2−2α} over a decade of wave numbers (0.002 < k < 0.1). At higher k the lattice regularisation produces a deviation, as anticipated. The authors also analyse the full probability distribution of the squared width, P_L(W²), and its cumulants κ_n (n = 1…4). The cumulants scale as κ_n ∝ L^{2nα} with negligible subleading corrections, indicating that the universal limit distribution Ψ(x) can be approximated accurately even for the finite sizes studied. The small‑x and large‑x tails of Ψ(x) show deviations due to limited statistics, but the central part collapses well for system sizes from L = 2⁸ up to L = 2¹³.

Overall, the paper demonstrates that bit‑coded GPU simulations enable unprecedented system sizes for KPZ‑type growth models, providing high‑precision estimates of the critical exponents and confirming the universality class in 2 + 1 dimensions. The work rules out the β = ¼ conjecture, quantifies correction‑to‑scaling exponents, and supplies detailed information on the steady‑state spectrum and width distribution. The methodology offers a powerful tool for future investigations of non‑equilibrium surface growth, driven lattice gases, and related stochastic partial differential equations in higher dimensions.