Fig. 5: Mode decoupling along Cole–Hopf line.

a For \({{\mathbb{Z}}}_{2}\) coupled KPZ with only cross-coupling in noise, the fixed point in the (W, Z)-plane (initial condition T = 1, Y = W) is decoupled KPZ in the original \({{\mathbb{Z}}}_{2}\) coordinates. Here \(W={\tilde{\Delta }}_{22}/{\tilde{\Delta }}_{11}\) is the noise-strength ratio, and the fixed point W^* = 1 shows that noise cross-coupling is irrelevant. b The flow in the \((T,{\tilde{\Gamma }}_{22}^{1}{\tilde{\Delta }}_{22}/{\tilde{\Gamma }}_{11}^{1}{\tilde{\Delta }}_{11})\)-plane approaches (1, 1), allowing for a decoupling transformation. c, d Fluctuation distributions χ for initial condition (X, Y) = (1, 2). c The field \({\tilde{\theta }}_{1}\) converges to the sum of rescaled TW-GOE distributions (yellow). For comparison, we plot the rescaled TW-GOE distribution \({2}^{2/3}{F}^{{\prime} }\left.(-{2}^{-2/3}{\chi }_{1})\right)\) (black dashed), and its translation (black solid) so that the means coincide. d The field \({\tilde{\theta }}_{2}\) converges to the difference of TW-GOE distributions (yellow), contrasted with Gaussian (red). Insets (c, d) show time convergence of higher order moments κ_n to their asymptotic values as predicted by Eq. (10) (horizontal lines) with skewness κ₃ = −0.207, and kurtosis κ₄ = 0.0829 and κ₃ = −0.00,  κ₄ = 0.0829 respectively for \({\tilde{\theta }}_{1}\) and \({\tilde{\theta }}_{2}\).