Extremal paths, the stochastic heat equation, and the three-dimensional Kardar-Parisi-Zhang universality class

Following our numerical work [Phys. Rev. Lett. 109, 170602 (2012)] focused upon the $2+1$ Kardar-Parisi-Zhang (KPZ) equation with flat initial condition, we return here to study, in depth, the three-dimensional (3D) radial KPZ problem, comparing common scaling phenomena exhibited by the pt-pt directed polymer in a random medium (DPRM), the stochastic heat equation (SHE) with multiplicative noise in three dimensions, and kinetic roughening phenomena associated with 3D Eden clusters. Examining variants of the 3D DPRM, as well as numerically integrating, via the Itô prescription, the constrained SHE for different values of the KPZ coupling, we provide strong evidence for universality within this 3D KPZ class, revealing shared values for the limit distribution skewness and kurtosis, along with universal first and second moments. Our numerical analysis of the 3D SHE, well flanked by the DPRM results, appears without precedent in the literature. We consider, too, the $2+1$ KPZ equation in the deeply evolved kinetically roughened stationary state, extracting the essential limit distribution characterizing fluctuations therein, revealing a higher-dimensional relative of the $1+1$ KPZ Baik-Rains distribution. Complementary, corroborative findings are provided via the Gaussian DPRM, as well as the restricted-solid-on-solid model of stochastic growth, stalwart members of the $2+1$ KPZ class. Next, contact is made with a recent nonperturbative, field-theoretic renormalization group calculation for the key universal amplitude ratio in this context. Finally, in the crossover from transient to stationary-state statistics, we observe a higher dimensional manifestation of the skewness minimum discovered by Takeuchi [Phys. Rev. Lett. 110, 210604 (2013)] in $1+1$ KPZ class liquid-crystal experiments.

Figure 1

Fluctuation PDFs: 3D radial KPZ class.

Figure 2

3D pt-pt KPZ class: skewness $s$ and kurtosis $k$.

Figure 3

3D pt-pt KPZ class: universal second moment. Inset: KM analysis: $g{5}_{1/2}$ DPRM, SHE Itô-6, and $2+1$ Eden A model.

Figure 4

3D pt-pt KPZ class: universal first moment.

Figure 5

Universal limit distribution: 3D pt-pt KPZ class.

Figure 6

Universal first moment, $2+1$ KPZ class: DPRM pt-plane geometry; KPZ stochastic growth with flat interface IC.

Figure 7

Limit distribution: $2+1$ pt-plane KPZ class.

Figure 8

$2+1$ KPZ stationary state: universal second moment. Inset: determination of the KPZ exponent ${\beta }_{2+1}=0.241\left(1\right)$, via the variance of the stationary-state fluctuations.

Figure 9

Universal limit distribution: $2+1$ KPZ stationary-state. Inset: Takeuchi's minimum [42], manifest within the skewness of the $2+1$ $g5$ DPRM.

Figure 10

Limit distribution: 3D pt-line KPZ class.