We report on the first exact solution of the KPZ equation in one dimension, with an initial condition which physically corresponds to the motion of a macroscopically curved height profile. The solution provides a determinantal formula for the probability distribution function of the height $h(x,t)$ for all $t>0$. In particular, we show that for large $t$, on the scale $t^{1/3}$, the statistics is given by the Tracy-Widom distribution, known already from the theory of GUE random matrices. Our solution confirms that the KPZ equation describes the interface motion in the regime of weak driving force. Within this regime the KPZ equation details how the long time asymptotics is approached.
We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate $γ$ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a non-trivial effect on the probability distribution of the random walker. At small $γ$, we show analytically that the time-evolution of the latter can be mapped to the Stochastic Heat Equation (SHE). In this limit, the width of the log probability thus follows a Family-Vicsek scaling law, $N^αf(t/N^{α/β})$, with roughness and growth exponents corresponding to the K…
We study the kinetic roughening of the single-step (SS) growth model with a tunable parameter $p$ in $1+1$ and $2+1$ dimensions by performing extensive numerical simulations. We show that there exists a very slow crossover from an intermediate regime dominated by the Edwards-Wilkinson class to an asymptotic regime dominated by the Kardar-Parisi-Zhang (KPZ) class for any $p <\frac{1}{2}$. We also identify the crossover time, the nonlinear coupling constant, and some nonuniversal parameters in the KPZ equation as a function $p$. The effective nonuniversal parameters are continuously decreasing w…
Consider a deterministically growing surface of any dimension, where the growth at a point is an arbitrary nonlinear function of the heights at that point and its neighboring points. Assuming that this nonlinear function is monotone, invariant under the symmetries of the lattice, equivariant under constant shifts, and twice continuously differentiable, it is shown that any such growing surface approaches a solution of the deterministic KPZ equation in a suitable space-time scaling limit.
The Kardar-Parisi-Zhang (KPZ) equation is a paradigmatic model of nonequilibrium low-dimensional systems with spatiotemporal scale invariance, recently highlighting universal behavior in fluctuation statistics. Its space derivative, namely the noisy Burgers equation, has played a very important role in its study, predating the formulation of the KPZ equation proper, and being frequently held as an equivalent system. We show that, while differences in the scaling exponents for the two equations are indeed due to a mere space derivative, the fluctuations behave in a remarkably different way: whi…
We report on the universality of height fluctuations at the crossing point of two interacting (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) interfaces with curved and flat initial conditions. We introduce a control parameter p as the probability for the initially flat geometry to be chosen and compute the phase diagram as a function of p. We find that the distribution of the fluctuations converges to the Gaussian orthogonal ensemble Tracy-Widom (TW) distribution for p<0.5, and to the Gaussian unitary ensemble TW distribution for p>0.5. For p=0.5 where the two geometries are equally weighted, the…
The Kardar-Parisi-Zhang (KPZ) equation defines the main universality class for nonlinear growth and roughening of surfaces. But under certain conditions, a conserved KPZ equation (cKPZ) is thought to set the universality class instead. This has non-mean-field behavior only in spatial dimension $d<2$. We point out here that cKPZ is incomplete: it omits a symmetry-allowed nonlinear gradient term of the same order as the one retained. Adding this term, we find a parameter regime where the $1$-loop renormalization group flow diverges. This suggests a phase transition to a new growth phase, possibl…
On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph
Universidad de Extremadura（埃斯特雷马杜拉大学）；Instituto de Computación Científica Avanzada de Extremadura (ICCAEx), Universidad de Extremadura（埃斯特雷马杜拉先进计算研究所 (ICCAEx)）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
The front of a compact bacterial colony growing on a Petri dish is a paradigmatic instance of non-equilibrium fluctuations in the celebrated Eden, or Kardar-Parisi-Zhang (KPZ), universality class. While in many experiments the scaling exponents crucially differ from the expected KPZ values, the source of this disagreement has remained poorly understood. We have performed growth experiments with B. subtilis 168 and E. coli ATCC 25922 under conditions leading to compact colonies in the classically-alleged Eden regime, where individual motility is suppressed. Non-KPZ scaling is indeed observed fo…
The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed for over three decades. Some experiments indicated the power law for the interface width, $w(t)\sim t^β$, remains the same as in growth on planar surfaces. Escudero (Phys. Rev. Lett. {\bf 100}, 116101, 2008) argued, however, that for the radial KPZ equations in (1+1)-dimension $w(t)$ should increase as $w(t)\sim [\ln(t)]^{1/2}$ in the long-time limit. Krug (Phys. Rev. Lett. {\bf 102}, 139601, 2009) argued, however, that the dynamics of the interface must remain unchanged with a change in the geometry. Other studies ind…
The statistics of the fluctuations of quantum many-body systems are highly revealing of their nature. In driven-dissipative systems displaying macroscopic quantum coherence, as exciton polariton condensates under incoherent pumping, the phase dynamics can be mapped to the stochastic Kardar-Parisi-Zhang (KPZ) equation. However, in two dimensions (2D), it was theoretically argued that the KPZ regime may be hindered by the presence of vortices, and a non-equilibrium BKT behavior was reported close to condensation threshold. We demonstrate here that, when a discretized 2D polariton system is consi…
Three-Dimensional Kardar--Parisi--Zhang Scaling in Polariton Condensates
Moscow Center for Advanced Studies（莫斯科高级研究中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；Barnard College, Columbia University（哥伦比亚大学巴纳德学院）；Westlake University（西湖大学）；St. Petersburg State University（圣彼得堡国立大学）；Russian Quantum Center（俄罗斯量子中心）
Revealing universal behaviors is a hallmark of statistical physics. Phenomena such as the stochastic growth of crystalline surfaces, of interfaces in bacterial colonies, and spin transport in quantum magnets all belong to the same universality class, despite the great plurality of physical mechanisms they involve at the microscopic level. This universality stems from a common underlying effective dynamics governed by the non-linear stochastic Kardar-Parisi-Zhang (KPZ) equation. Recent theoretical works suggest that this dynamics also emerges in the phase of out-of-equilibrium systems displayin…
The determination of the exact exponents of the KPZ class in any substrate dimension $d$ is one of the most important open issues in Statistical Physics. Based on the behavior of the dimensional variation of some exact exponent differences for other growth equations, I find here that the KPZ growth exponents (related to the temporal scaling of the fluctuations) are given by $β_d = \frac{7}{8d+13}$. These exponents present an excellent agreement with the most accurate estimates for them in the literature. Moreover, they are confirmed here through extensive Monte Carlo simulations of discrete gr…
Dynamical universality classes are distinguished by their dynamical exponent $z$ and unique scaling functions encoding space-time asymmetry for, e.g. slow-relaxation modes or the distribution of time-integrated currents. So far the universality class of the Nagel-Schreckenberg (NaSch) model, which is a paradigmatic model for traffic flow on highways, was not known except for the special case $v_{\text{max}}=1$. Here the model corresponds to the TASEP (totally asymmetric simple exclusion process) that is known to belong to the superdiffusive Kardar-Parisi-Zhang (KPZ) class with $z=3/2$. In this…
The recently developed formalism of nonlinear fluctuating hydrodynamics (NLFH) has been instrumental in unraveling many new dynamical universality classes in coupled driven systems with multiple conserved quantities. In principle, this formalism requires knowledge of the exact expression of locally conserved current in terms of local density of the conserved components. However, for most nonequilibrium systems an exact expression is not available and it is important to know what happens to the predictions of NLFH in these cases. We address this question for the first time here in a system with…
We consider the evolution of interfaces with a diffusive term and a generalized Kardar-Parisi-Zhang (KPZ) non-linearity, which results in a propagation velocity that depends periodically on the tilt of the interface. Using large scale simulations of a model class with these properties in 1+1 dimensions, we show that the fluctuations are in general still in the KPZ universality class, but a new universality class seems to appear in the limit of weak non-linearity. We argue that this is the typical behavior of any interface model with periodic tilt dependence.
We prove that under a particular weak scaling, the 4-parameter interacting particle system introduced by Corwin and Petrov converges to the KPZ equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated.
Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) \textit{stochastic} partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state (NESS) of the one-dimensional Kuramoto-Sivashinsky (KS) \textit{deterministic} PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile $h(x,t)$ for different initial conditions. We es…
We consider the statistical properties of arrival times and balls on first-passage percolation (FPP) $2D$ square lattices with strong disorder in the link-times. A previous work showed a crossover in the weak disorder regime, between Gaussian and Kardar-Parisi-Zhang (KPZ) universality, with the crossover length decreasing as the noise amplitude grows. On the other hand, this work presents a very different behavior in the strong-disorder regime. A new crossover length appears below which the model is described by bond-percolation universality class. This characteristic length scale grows with t…
Exciton-polaritons under driven-dissipative conditions exhibit a condensation transition which belongs to a different universality class than equilibrium Bose-Einstein condensates. By numerically solving the generalized Gross-Pitaevskii equation with realistic experimental parameters, we show that one-dimensional exciton-polaritons display fine features of Kardar-Parisi-Zhang (KPZ) dynamics. Beyond the scaling exponents, we show that their phase distribution follows the Tracy-Widom form predicted for KPZ growing interfaces. We moreover evidence a crossover to the stationary Baik-Rains statisti…
When a growing interface belonging to the KPZ universality class is tilted with average slope $m$, its average velocity increases in $\fracΛ{2}\,m^2$, where $Λ$ is related to the nonlinear coefficient $λ$ of the KPZ equation. Nevertheless, a necessary condition for this association to hold true is that the mean square height-gradient increases in $b\, m^2$ when the interface is tilted. For the continuous KPZ equation $b = 1$ and the relation $Λ=λ$ is achieved. In this work, we study the local fluctuations of the height gradient through an analysis of the values of $b$. We show that, for 1-dime…
