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

Abstract

We study the short-time behavior of the probability distribution 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 = Hc. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry x ↔ −x of optimal paths h(x, t) predicted by the weak-noise theory of the KPZ equation. At |H| ≫ |Hc| the corresponding tail of P(H) scales as − ln P ∼ |H| 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 P scales as − ln P ∼ |H| 5/2 /t 1/2 and coincides with the corresponding tail for the sharp-wedge initial condition.

Exact short-time height distribution in the one-dimensional Kardar-Parisi-Zhang equation with Brownian initial condition

“…A new feature arises at the value H = H c2 (w) where the validity of the first analytic continuations ends. We obtain H c2 (0) ≈ 1.85316 consistent with the numerical estimate of [24], which suggests that this is the same critical point. We propose two continuations for Φ(H) for H > H c2 (w), given in (30), an analytic one which leads to c + = 4/3, apparently corresponding to the symmetric WNT solution and a non-analytic one which leads to c + = 2/3, corresponding to the asymmetric WNT solution.…”

“…A surprising feature arises on the positive H side, where for H > H c2 a spontaneous symmetry breaking of reflection invariance occurs, leading to the coexistence of symmetric and asymmetric solutions. The value H c2 ≈ 1.85 was obtained numerically in [24]. While the symmetric solution gives c + = 4/3 the asymmetric ones gives c + = 2/3, i.e.…”

“…Recently, the short time behaviour of the KPZ equation has been investigated [15,[20][21][22][23][24]. It was found that the probability distribution function (PDF) of the height H at a given point, see below, takes the following large deviation form P (H, t) ∼ exp − Φ(H) √ t , H fixed and t 1 (2) where the rate function Φ(H) depends on the initial condition [25].…”

“…The weak noise theory (WNT) uses a saddle point evaluation of the dynamical action associated to the KPZ equation (1), using 1/ √ t as a large parameter [21,23,24,27]. Until now these saddle point equations have been solved analytically only (i) near the center of the distribution (ii) in the two tails.…”

“…On the positive H side, the form P (H, t) ∼ exp(− Recently, Janas, Kamenev and Meerson [24] studied stationary initial conditions using the WNT. On the negative H side they found c − = 4 15π .…”