When fast and slow interfaces grow together: Connection to the half-space problem of the Kardar-Parisi-Zhang class

We study height fluctuations of interfaces in the $(1+1)$-dimensional Kardar-Parisi-Zhang (KPZ) class, growing at different speeds in the left half and the right half of space. Carrying out simulations of the discrete polynuclear growth model with two different growth rates, combined with the standard setting for the droplet, flat, and stationary geometries, we find that the fluctuation properties at and near the boundary are described by the KPZ half-space problem developed in the theoretical literature. In particular, in the droplet case, the distribution at the boundary is given by the largest-eigenvalue distribution of random matrices in the Gaussian symplectic ensemble, often called the GSE Tracy-Widom distribution. We also characterize crossover from the full-space statistics to the half-space one, which arises when the difference between the two growth speeds is small.

Figure 1

Sketch of the time evolution rules of the discrete PNG model. (a) An example of the interface evolution from time $t$ (dashed line) to $t+1$ (solid line). The vertical arrows indicate elevation by random nucleations and the horizontal arrows show plateau expansion. The growth parameter is $q=q_{\mathrm{L}}$ for $x<0$ and $q=q_{\mathrm{R}}$ for $x\ge 0$. (b) When two plateaus encounter, the higher one overrides.

Figure 2

An example of interfaces in the flat (a) and circular (b) geometries, with $q_{\mathrm{L}}=0.2$ and $q_{\mathrm{R}}=0.25$. The height profiles recorded every 20 time steps are shown.

Figure 3

The mean and the variance of the rescaled height $H(0,t)$ at the boundary for the droplet case. Different colors and symbols correspond to different values of $q_{\mathrm{L}}$, as shown in the legend of panel (c). The horizontal lines indicate the mean and the variance of ${\chi }_{\mathrm{GUE}}$ (dashed) and ${\chi }_{\mathrm{GSE}}^{\prime }$ (dash-dotted). The raw data in (a) and (b) are plotted against ${\left(\mathrm{\Delta }q\right)}^{3/2}t$ in (c) and (d). The insets show the approach to the GSE-TW values. To improve statistical accuracy, the data for $q_{\mathrm{L}}=0.240$ were obtained from 500 000 realizations. In view of the alternating character of the updating, only data at even times are shown.

Figure 4

The mean and the variance of the rescaled height $H(0,t)$ at the boundary for the flat case. The horizontal lines indicate the mean and the variance of ${\chi }_{\mathrm{GOE}}^{\prime }$ (dashed) and ${\chi }_{\mathrm{GUE}}$ (dash-dotted).

Figure 5

The mean and the variance of the rescaled height $H(0,t)$ at the boundary for the stationary case. The horizontal lines indicate the mean and the variance of ${\chi }_{\mathrm{BR}}$ (dashed) and ${\chi }_{\mathrm{GOE}}$ (dash-dotted).

Figure 6

The rescaled spatial correlation function $C_{\mathrm{s}}^{\prime }(X,t)$ for the droplet (a)–(c) and the flat (d)–(f) cases. In (a) and (d), $t$ is fixed ($t=10000$) and $q_{\mathrm{L}}$ is varied. In (b) and (e), $q_{\mathrm{L}}$ is fixed ($q_{\mathrm{L}}=0.246$) and $t$ is varied. In (c) and (f), pairs of $q_{\mathrm{L}}$ and $t$ that give the same value of ${\left(\mathrm{\Delta }q\right)}^{3/2}t$ are used. The panel (g) shows the asymptotic forms of the correlation function for the half-space problem, compared with the ${\mathrm{Airy}}_1$ and ${\mathrm{Airy}}_2$ correlation for the full-space problem. The curves for the ${\mathrm{Airy}}_2$ and ${\mathrm{Airy}}_1$ processes were numerically evaluated by Bornemann [26].