Dynamical phase transition in large-deviation statistics of the Kardar-Parisi-Zhang equation

We study the short-time behavior of the probability distribution $\mathcal{P}(H,t)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension. The process starts from a stationary interface: $h(x,t=0)$ is given by a realization of two-sided Brownian motion constrained by $h(0,0)=0$. We find a singularity of the large deviation function of $H$ at a critical value $H=H_c$. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry $x\leftrightarrow -x$ of optimal paths $h(x,t)$ predicted by the weak-noise theory of the KPZ equation. At $\left|H\right|\gg |H_c|$ the corresponding tail of $\mathcal{P}\left(H\right)$ scales as $-ln\mathcal{P}\sim {\left|H\right|}^{3/2}/t^{1/2}$ and agrees, at any $t>0$, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of $\mathcal{P}$ scales as $-ln\mathcal{P}\sim {\left|H\right|}^{5/2}/t^{1/2}$ and coincides with the corresponding tail for the sharp-wedge initial condition.

Figure 1

The optimal path in the linear approximation, as described by Eqs. (14) and (15) for $H<0$, at $t=0,0.25,0.5,0.75$ and 1 (top panel, from top to bottom) and $t=0,0.5$ and $0.95$ (bottom panel).

Figure 2

The optimal path for $\mathrm{\Lambda }=-6.3$ computed numerically. Shown are $h$ (top) and $\rho$ (bottom) vs $x$ at $t=0$ (solid line), $0.5$ (short dash), and 1 (long dash).

Figure 3

Same as in Fig. 2 but for $\mathrm{\Lambda }=-10$.

Figure 4

Top: $\mathrm{\Delta }$ vs $\left|H\right|$ at $H<0$. Symbols: numerical results, solid line: Eq. (17), dashed line: the $\left|H\right|\gg 1$ asymptotic $\left|\mathrm{\Delta }\right|=\left|H\right|$. Bottom: $s$ vs $\left|H\right|$ at $H<0$: The asymmetric and symmetric branches are shown by the solid and empty symbols, respectively. Also shown are the small- and large-$\left|H\right|$ asymptotics of $s$.

Figure 5

The optimal path, $h(x,t)$ and $\rho (x,t)$, for $H<0$ and $\left|H\right|\gg 1$; see Eqs. (18, 19, 20), for $t=0,1/2$, and 1.

Figure 6

Example of exact two-ramp and two-soliton solutions (B7) and (B8) for $h(x,t)$ and $\rho (x,t)$, respectively. Shown (for $c=8$) are $h$ and $\rho$ vs $x$ for $N=3,c_1=X_1=0,c_3=-c_2=c$, and $X_3=-X_2=-(3/8)c$. Top panel: $t=0$ (solid), $1/4$ (dashed), $3/4$ (dash-dotted), and 1 (solid). Bottom panel: $t=0$ (solid), $1/4$ (dashed), and 1 (dash-dotted). Inset: $\rho$ vs $x$ at $t=1/2,4/5$ and 1. At $c\gg 1$ and $t>\tau ,\rho$ approaches the exact stationary one-soliton solution [28, 31, 34].

Figure 7

Same as in Fig. 6 but for $X_3=-X_2=-c$ (that is, $\tau =1$) and $t=0$ (solid), $1/2$ (dashed), and 1 (dash-dotted). The inset shows a blowup of the collision and merger of the two solitons at $t=0.95,0.965,$ and 1.

Figure 8

Top: The shock position $x_s$ vs time, alongside with its asymptotics (C7) and (C8) for $x>0$. Bottom: $V={\partial }_{x}h$ vs $x$ at $H\gg 1$ at times (from top to bottom) $0.1,0.3,0.5$ (when $V=0$), $0.7$, and $0.9$. Both the uniform-strain solution (C3) and the Hopf solution with the shock are shown for $x>0$. In this limit $H={\left(3\pi \mathrm{\Lambda }\right)}^{2/3}/2$ [35].