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

Kardar-Parisi-Zhang growth in ɛ dimensions and beyond

We examine anew the relationship of directed polymers in random media on traditional hypercubic versus hierarchical lattices, with the goal of understanding the dimensionality dependence of the essential scaling index β at the heart of the Kardar-Parisi-Zhang universality class. A seemingly accurate, but entirely empirical, ansatz due to Perlsman and Schwartz, proposed many years ago, can be put in proper context by anchoring the connection between these distinct lattice types at vanishing dimensionality. We graft together complementary perturbative field-theoretic and nonperturbative real-space renormalization group tools to establish the necessary connection, thereby elucidating the central mystery underlying the ansatz's uncanny apparent success, but also revealing its intrinsic limitations. Furthermore, we perform an extensive Euler integration of the KPZ equation in 3+1 dimensions which, bolstered by a separate directed polymer simulation, allows us an estimate for the critical exponent β_{3+1}^{KPZ}=0.1845(4) that greatly improves upon all previous Monte Carlo calculations in this regard and rules out the Perlsman-Schwartz value, 0.1882^{+}, in that dimension. Finally, leveraging this hybrid RG partnership permits us a versatile, more potent, tool to explore the general KPZ problem across dimensions, as well as a conjecture for its key critical exponent, β=1/2-0.22967ɛ, as ɛ→0, testable in a three-loop calculation.

Dimensional crossover in Kardar-Parisi-Zhang growth

Two-dimensional (2D) Kardar-Parisi-Zhang (KPZ) growth is usually investigated on substrates of lateral sizes L_{x}=L_{y}, so that L_{x} and the correlation length (ξ) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates L_{x}≠L_{y} and, thus, the surfaces can become correlated in a single direction, when ξ∼L_{x}≪L_{y}. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling as W∼t^{β_{2D}} for t≪t_{c} and W∼t^{β_{1D}} for t≫t_{c}, where t_{c}∼L_{x}^{1/z_{2D}}. The height distributions (HDs) also cross over from the 2D flat (cylindrical) HD to the asymptotic Tracy-Widom Gaussian orthogonal ensemble (Gaussian unitary ensemble) distribution. Moreover, 2D to one-dimensional (1D) crossovers are found also in the asymptotic growth velocity and in the steady-state regime of flat systems, where a family of universal HDs exists, interpolating between the 2D and 1D ones as L_{y}/L_{x} increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.

Unpredicted Scaling of the One-Dimensional Kardar-Parisi-Zhang Equation

The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree. Yet recent numerical simulations in the tensionless (or inviscid) limit of the KPZ equation [C. Cartes et al., The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalized to Kardar-Parisi-Zhang scaling, Phil. Trans. R. Soc. A 380, 20210090 (2022).PTRMAD1364-503X10.1098/rsta.2021.0090; E. Rodríguez-Fernández et al., Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation, Phys. Rev. E 106, 024802 (2022).PRESCM2470-004510.1103/PhysRevE.106.024802] unveiled a new scaling, with a critical dynamical exponent z=1 different from the KPZ one z=3/2. In this Letter, we show that this scaling is controlled by a fixed point which had been missed so far and which corresponds to an infinite nonlinear coupling. Using the functional renormalization group (FRG), we demonstrate the existence of this fixed point and show that it yields z=1. We calculate the correlation function and associated scaling function at this fixed point, providing both a numerical solution of the FRG equations within a reliable approximation, and an exact asymptotic form obtained in the limit of large wave numbers. Both scaling functions accurately match the one from the numerical simulations.

Universal interface fluctuations in the contact process

We study the interface representation of the contact process at its directed-percolation critical point, where the scaling properties of the interface can be related to those of the original particle model. Interestingly, such a behavior happens to be intrinsically anomalous and more complex than that described by the standard Family-Vicsek dynamic scaling Ansatz of surface kinetic roughening. We expand on a previous numerical study by Dickman and Muñoz [Phys. Rev. E 62, 7632 (2000)10.1103/PhysRevE.62.7632] to fully characterize the kinetic roughening universality class for interface dimensions d=1,2, and 3. Beyond obtaining scaling exponent values, we characterize the interface fluctuations via their probability density function (PDF) and covariance, seen to display universal properties which are qualitatively similar to those recently assessed for the Kardar-Parisi-Zhang (KPZ) and other important universality classes of kinetic roughening. Quantitatively, while for d=1 the interface covariance seems to be well described by the KPZ, Airy_{1} covariance, no such agreement occurs in terms of the fluctuation PDF or the scaling exponents.

One-point height fluctuations and two-point correlators of (2+1) cylindrical KPZ systems

While the one-point height distributions (HDs) and two-point covariances of (2+1) Kardar-Parisi-Zhang (KPZ) systems have been investigated in several recent works for flat and spherical geometries, for the cylindrical one the HD was analyzed for few models and nothing is known about the spatial and temporal covariances. Here, we report results for these quantities, obtained from extensive numerical simulations of discrete KPZ models, for three different setups yielding cylindrical growth. Beyond demonstrating the universality of the HD and covariances, our results reveal other interesting features of this geometry. For example, the spatial covariances measured along the longitudinal and azimuthal directions are different, with the former being quite similar to the curve for flat (2+1) KPZ systems, while the latter resembles the Airy_{2} covariance of circular (1+1) KPZ interfaces. We also argue (and present numerical evidence) that, in general, the rescaled temporal covariance A(t/t_{0}) decays asymptotically as A(x)∼x^{-λ[over ¯]} with an exponent λ[over ¯]=β+d^{*}/z, where d^{*} is the number of interface sides kept fixed during the growth (being d^{*}=1 for the systems analyzed here). Overall, these results complete the picture of the main statistics for the (2+1) KPZ class.

Kardar-Parisi-Zhang universality class in (d+1)-dimensions

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}=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 growth models and real-space renormalization group (RG) calculations for directed polymers in random media (DPRM), up to d=15. The left-tail exponents of the probability density functions for the DPRM energy provide another striking verification of the analytical result above.

Height distributions in interface growth: The role of the averaging process

Height distributions (HDs) are key quantities to uncover universality and geometry-dependence in evolving interfaces. To quantitatively characterize HDs, one uses adimensional ratios of their first central moments (m_{n}) or cumulants (κ_{n}), especially the skewness S and kurtosis K, whose accurate estimate demands an averaging over all L^{d} points of the height profile at a given time, in translation-invariant interfaces, and over N independent samples. One way of doing this is by calculating m_{n}(t) [or κ_{n}(t)] for each sample and then carrying out an average of them for the N interfaces, with S and K being calculated only at the end. Another approach consists in directly calculating the ratios for each interface and, then, averaging the N values. It turns out, however, that S and K for the growth regime HDs display strong finite-size and -time effects when estimated from these "interface statistics," as already observed in some previous works and clearly shown here, through extensive simulations of several discrete growth models belonging to the EW and KPZ classes on one- and two-dimensional substrates of sizes L=const. and L∼t. Importantly, I demonstrate that with "1-point statistics," i.e., by calculating m_{n}(t) [or κ_{n}(t)] once for all NL^{d} heights together, these corrections become very weak, so that S and K attain values very close to the asymptotic ones already at short times and for small L's. However, I find that this "1-point" (1-pt) approach fails in uncovering the universality of the HDs in the steady-state regime (SSR) of systems whose average height, h[over ¯], is a fluctuating variable. In fact, as demonstrated here, in this regime the 1-pt height evolves as h(t)=h[over ¯](t)+s_{λ}A^{1/2}L^{α}ζ+⋯-where P(ζ) is the underlying SSR HD-and the fluctuations in h[over ¯] yield S_{1-pt}∼t^{-1/2} and K_{1-pt}∼t^{-1}. Nonetheless, by analyzing P(h-h[over ¯]), the cumulants of P(ζ) can be accurately determined. I also show that different, but universal, asymptotic values for S and K (related, so, to different HDs) can be found from the "interface statistics" in the SSR. This reveals the importance of employing the various complementary approaches to reliably determine the universality class of a given system through its different HDs.

Emergent Kardar-Parisi-Zhang Phase in Quadratically Driven Condensates

In bosonic gases at thermal equilibrium, an external quadratic drive can induce a Bose-Einstein condensation described by the Ising transition, as a consequence of the explicitly broken U(1) phase rotation symmetry down to Z_{2}. However, in physical realizations such as exciton polaritons and nonlinear photonic lattices, thermal equilibrium is lost and the state is rather determined by a balance between losses and external drive. A fundamental question is then how nonequilibrium fluctuations affect this transition. Here, we show that in a two-dimensional driven-dissipative Bose system the Ising phase is suppressed and replaced by a nonequilibrium phase featuring Kardar-Parisi-Zhang (KPZ) physics. Its emergence is rooted in a U(1)-symmetry restoration mechanism enabled by the strong fluctuations in reduced dimensionality. Moreover, we show that the presence of the quadratic drive term enhances the visibility of the KPZ scaling, compared to two-dimensional U(1)-symmetric gases, where it has remained so far elusive.

Seascape origin of Richards growth

First proposed as an empirical rule over half a century ago, the Richards growth equation has been frequently invoked in population modeling and pandemic forecasting. Central to this model is the advent of a fractional exponent γ, typically fitted to the data. While various motivations for this nonanalytical form have been proposed, it is still considered foremost an empirical fitting procedure. Here, we find that Richards-like growth laws emerge naturally from generic analytical growth rules in a distributed population, upon inclusion of (i) migration (spatial diffusion) among different locales, and (ii) stochasticity in the growth rate, also known as "seascape noise." The latter leads to a wide (power law) distribution in local population number that, while smoothened through the former, can still result in a fractional growth law for the overall population. This justification of the Richards growth law thus provides a testable connection to the distribution of constituents of the population.

Population extinction on a random fitness seascape

We explore the role of stochasticity and noise in the statistical outcomes of commonly studied population dynamics models within a space-independent (mean-field) perspective. Specifically, we consider a distributed population with logistic growth at each location, subject to "seascape" noise, wherein the population's fitness randomly varies with location and time. Despite its simplicity, the model actually incorporates variants of directed percolation, and directed polymers in random media, within a mean-field perspective. Probability distributions of the population can be computed self-consistently, and the extinction transition is shown to exhibit novel critical behavior with exponents dependent on the ratio of the strengths of migration and noise amplitudes. The results are compared and contrasted with the more conventional choice of demographic noise due to stochastic changes in reproduction.

Direct Evidence for Universal Statistics of Stationary Kardar-Parisi-Zhang Interfaces

The nonequilibrium steady state of the one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) universality class has been studied in-depth by exact solutions, yet no direct experimental evidence of its characteristic statistical properties has been reported so far. This is arguably because, for an infinitely large system, infinitely long time is needed to reach such a stationary state and also to converge to the predicted universal behavior. Here we circumvent this problem in the experimental system of growing liquid-crystal turbulence, by generating an initial condition that possesses a long-range property expected for the KPZ stationary state. The resulting interface fluctuations clearly show characteristic properties of the 1D stationary KPZ interfaces, including the convergence to the Baik-Rains distribution. We also identify finite-time corrections to the KPZ scaling laws, which turn out to play a major role in the direct test of the stationary KPZ interfaces. This paves the way to explore unsolved properties of the stationary KPZ interfaces experimentally, making possible connections to nonlinear fluctuating hydrodynamics and quantum spin chains as recent studies unveiled relation to the stationary KPZ.

Universality and crossover behavior of single-step growth models in 1+1 and 2+1 dimensions

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<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 with p but not in a linear fashion. Our results provide complete and conclusive evidence that the SS model for p≠1/2 belongs to the KPZ universality class in 2+1 dimensions.

Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach

We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with δ-correlated noise (denoted SR-KPZ). Thus, it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using nonperturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short-range one with a typical timescale τ, and a power-law one with a varying exponent θ. We show that for the short-range noise with any finite τ, the symmetries (the Galilean symmetry, and the time-reversal one in 1+1 dimension) are dynamically restored at large scales, such that the long-distance and long-time properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for θ below a critical value θ_{th}, in accordance with previous renormalization group results, while a long-range fixed point controls the critical scaling for θ>θ_{th}, and we evaluate the θ-dependent critical exponents at this long-range fixed point, in both 1+1 and 2+1 dimensions. While the results in 1+1 dimension can be compared with previous studies, no other prediction was available in 2+1 dimension. We finally report in 1+1 dimension the emergence of anomalous scaling in the long-range phase.

Geometry dependence in linear interface growth

The effect of geometry in the statistics of nonlinear universality classes for interface growth has been widely investigated in recent years, and it is well known to yield a split of them into subclasses. In this work, we investigate this for the linear classes of Edwards-Wilkinson and of Mullins-Herring in one and two dimensions. From comparison of analytical results with extensive numerical simulations of several discrete models belonging to these classes, as well as numerical integrations of the growth equations on substrates of fixed size (flat geometry) or expanding linearly in time (radial geometry), we verify that the height distributions (HDs) and the spatial and the temporal covariances are universal but geometry-dependent. In fact, the HDs are always Gaussian, and, when defined in terms of the so-called "KPZ ansatz" [h≃v_{∞}t+(Γt)^{β}χ], their probability density functions P(χ) have mean null, so that all their cumulants are null, except by their variances, which assume different values in the flat and radial cases. The shape of the (rescaled) covariance curves is analyzed in detail and compared with some existing analytical results for them. Overall, these results demonstrate that the splitting of such university classes is quite general, being not restricted to the nonlinear ones.

Stochastic entropies and fluctuation theorems for a discrete one-dimensional Kardar-Parisi-Zhang system

The statistical properties of stochastic entropies in a discrete stationary one-dimensional Kardar-Parisi-Zhang system are numerically studied. As the usual time-independent solution of the associated Fokker-Planck equation is not strictly stationary, it is necessary to transform the current variables to other variables with zero spatial mean. We resorted to discrete representations in order to prove the statistical properties of entropies, and we performed a direct test of the fluctuation theorem. We discuss the connection between fluctuation theorems and the deviation from Gaussian distribution for the entropy.

Circular Kardar-Parisi-Zhang interfaces evolving out of the plane

Circular KPZ interfaces spreading radially in the plane have Gaussian unitary ensemble (GUE) Tracy-Widom (TW) height distribution (HD) and Airy_{2} spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a mountain, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as 〈L(t)〉=L_{0}+ωt^{γ}, while their mean height 〈h〉 increases as usual [〈h〉∼t]. We show that the competition between the L enlargement and the correlation length (ξ≃ct^{1/z}) plays a key role in the asymptotic statistics of the interfaces. While systems with γ>1/z have HDs given by GUE and the interface width increasing as w∼t^{β}, for γ<1/z the HDs are Gaussian, in a correlated regime where w∼t^{αγ}. For the special case γ=1/z, a continuous class of distributions exists, which interpolate between Gaussian (for small ω/c) and GUE (for ω/c≫1). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for ω/c≈10. Despite the GUE HDs for γ>1/z, the spatial covariances present a strong dependence on the parameters ω and γ, agreeing with Airy_{2} only for ω≫1, for a given γ, or when γ=1, for a fixed ω. These results considerably generalize our knowledge on 1D KPZ systems, unveiling the importance of the background space on their statistics.

Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates

Recent experimental works on one-dimensional (1D) circular Kardar-Parisi-Zhang (KPZ) systems whose radii decrease in time have reported controversial conclusions about the statistics of their interfaces. Motivated by this, here we investigate several one-dimensional KPZ models on substrates whose size changes in time as L(t)=L_{0}+ωt, focusing on the case ω<0. From extensive numerical simulations, we show that for L_{0}≫1 there exists a transient regime in which the statistics is consistent with that of flat KPZ systems (the ω=0 case), for both ω<0 and ω>0. Actually, for a given model, L_{0} and |ω|, we observe that a difference between ingrowing (ω<0) and outgrowing (ω>0) systems arises only at long times (t∼t_{c}=L_{0}/|ω|), when the expanding surfaces cross over to the statistics of curved KPZ systems, whereas the shrinking ones become completely correlated. A generalization of the Family-Vicsek scaling for the roughness of ingrowing interfaces is presented. Our results demonstrate that a transient flat statistics is a general feature of systems starting with large initial sizes, regardless of their curvature. This is consistent with their recent observation in ingrowing turbulent liquid crystal interfaces, but it is in contrast with the apparent observation of curved statistics in colloidal deposition at the edge of evaporating drops. A possible explanation for this last result, as a consequence of the very small number of monolayers analyzed in this experiment, is given. This is illustrated in a competitive growth model presenting a few-monolayer transient and an asymptotic behavior consistent, respectively, with the curved and flat statistics.

Kardar-Parisi-Zhang modes in d-dimensional directed polymers

We define a stochastic lattice model for a fluctuating directed polymer in d≥2 dimensions. This model can be alternatively interpreted as a fluctuating random path in two dimensions, or a one-dimensional asymmetric simple exclusion process with d-1 conserved species of particles. The deterministic large dynamics of the directed polymer are shown to be given by a system of coupled Kardar-Parisi-Zhang (KPZ) equations and diffusion equations. Using nonlinear fluctuating hydrodynamics and mode coupling theory we argue that stationary fluctuations in any dimension d can only be of KPZ type or diffusive. The modes are pure in the sense that there are only subleading couplings to other modes, thus excluding the occurrence of modified KPZ-fluctuations or Lévy-type fluctuations, which are common for more than one conservation law. The mode-coupling matrices are shown to satisfy the so-called trilinear condition.

Front roughening of flames in discrete media

The morphology of flame fronts propagating in reactive systems composed of randomly positioned, pointlike sources is studied. The solution of the temperature field and the initiation of new sources is implemented using the superposition of the Green's function for the diffusion equation, eliminating the need to use finite-difference approximations. The heat released from triggered sources diffuses outward from each source, activating new sources and enabling a mechanism of flame propagation. Systems of 40000 sources in a 200×200 two-dimensional domain were tracked using computer simulations, and statistical ensembles of 120 realizations of each system were averaged to determine the statistical properties of the flame fronts. The reactive system of sources is parameterized by two nondimensional values: the heat release time (normalized by interparticle diffusion time) and the ignition temperature (normalized by adiabatic flame temperature). These two parameters were systematically varied for different simulations to investigate their influence on front propagation. For sufficiently fast heat release and low ignition temperature, the front roughness [defined as the root mean square deviation of the ignition temperature contour from the average flame position] grew following a power-law dependence that was in excellent agreement with the Kardar-Parisi-Zhang (KPZ) universality class (β=1/3). As the reaction time was increased, lower values of the roughening exponent were observed, and at a sufficiently great value of reaction time, reversion to a steady, constant-width thermal flame was observed that matched the solution from classical combustion theory. Deviation away from KPZ scaling was also observed as the ignition temperature was increased. The features of this system that permit it to exhibit both KPZ and non-KPZ scaling are discussed.

Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables

We investigate the stationary-state fluctuations of a growing one-dimensional interface described by the Kardar-Parisi-Zhang (KPZ) dynamics with a noise featuring smooth spatial correlations of characteristic range ξ. We employ nonperturbative functional renormalization group methods to resolve the properties of the system at all scales. We show that the physics of the standard (uncorrelated) KPZ equation emerges on large scales independently of ξ. Moreover, the renormalization group flow is followed from the initial condition to the fixed point, that is, from the microscopic dynamics to the large-distance properties. This provides access to the small-scale features (and their dependence on the details of the noise correlations) as well as to the universal large-scale physics. In particular, we compute the kinetic energy spectrum of the stationary state as well as its nonuniversal amplitude. The latter is experimentally accessible by measurements at large scales and retains a signature of the microscopic noise correlations. Our results are compared to previous analytical and numerical results from independent approaches. They are in agreement with direct numerical simulations for the kinetic energy spectrum as well as with the prediction, obtained with the replica trick by Gaussian variational method, of a crossover in ξ of the nonuniversal amplitude of this spectrum.

Local average height distribution of fluctuating interfaces

Height fluctuations of growing surfaces can be characterized by the probability distribution of height in a spatial point at a finite time. Recently there has been spectacular progress in the studies of this quantity for the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions. Here we notice that, at or above a critical dimension, the finite-time one-point height distribution is ill defined in a broad class of linear surface growth models unless the model is regularized at small scales. The regularization via a system-dependent small-scale cutoff leads to a partial loss of universality. As a possible alternative, we introduce a local average height. For the linear models, the probability density of this quantity is well defined in any dimension. The weak-noise theory for these models yields the "optimal path" of the interface conditioned on a nonequilibrium fluctuation of the local average height. As an illustration, we consider the conserved Edwards-Wilkinson (EW) equation, where, without regularization, the finite-time one-point height distribution is ill defined in all physical dimensions. We also determine the optimal path of the interface in a closely related problem of the finite-time height-difference distribution for the nonconserved EW equation in 1+1 dimension. Finally, we discuss a UV catastrophe in the finite-time one-point distribution of height in the (nonregularized) KPZ equation in 2+1 dimensions.

Zero-temperature directed polymer in random potential in 4+1 dimensions

Zero-temperature directed polymer in random potential in 4+1 dimensions is described. The fluctuation ΔE(t) of the lowest energy of the polymer varies as t^{β} with β=0.159±0.007 for polymer length t and ΔE follows ΔE(L)∼L^{α} at saturation with α=0.275±0.009, where L is the system size. The dynamic exponent z≈1.73 is obtained from z=α/β. The estimated values of the exponents satisfy the scaling relation α+z=2 very well. We also monitor the end to end distance of the polymer and obtain z independently. Our results show that the upper critical dimension of the Kardar-Parisi-Zhang equation is higher than d=4+1 dimensions.

Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation

We report extensive numerical simulations of growth models belonging to the nonlinear molecular beam epitaxy (nMBE) class, on flat (fixed-size) and expanding substrates (ES). In both d=1+1 and 2+1, we find that growth regime height distributions (HDs), and spatial and temporal covariances are universal, but are dependent on the initial conditions, while the critical exponents are the same for flat and ES systems. Thus, the nMBE class does split into subclasses, as does the Kardar-Parisi-Zhang (KPZ) class. Applying the "KPZ ansatz" to nMBE models, we estimate the cumulants of the 1+1 HDs. Spatial covariance for the flat subclass is hallmarked by a minimum, which is not present in the ES one. Temporal correlations are shown to decay following well-known conjectures.

Universality of (2+1)-dimensional restricted solid-on-solid models

Extensive dynamical simulations of restricted solid-on-solid models in D=2+1 dimensions have been done using parallel multisurface algorithms implemented on graphics cards. Numerical evidence is presented that these models exhibit Kardar-Parisi-Zhang surface growth scaling, irrespective of the step heights N. We show that by increasing N the corrections to scaling increase, thus smaller step-sized models describe better the asymptotic, long-wave-scaling behavior.

Keldysh field theory for driven open quantum systems

Recent experimental developments in diverse areas-ranging from cold atomic gases to light-driven semiconductors to microcavity arrays-move systems into the focus which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in equilibrium condensed matter physics. This concerns both their non-thermal stationary states and their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.

Probability distributions for directed polymers in random media with correlated noise

The probability distribution for the free energy of directed polymers in random media (DPRM) with uncorrelated noise in d=1+1 dimensions satisfies the Tracy-Widom distribution. We inquire if and how this universal distribution is modified in the presence of spatially correlated noise. The width of the distribution scales as the DPRM length to an exponent β, in good (but not full) agreement with previous renormalization group and numerical results. The scaled probability is well described by the Tracy-Widom form for uncorrelated noise, but becomes symmetric with increasing correlation exponent. We thus find a class of distributions that continuously interpolates between Tracy-Widom and Gaussian forms.

Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixed-point solution

We investigate the regime of fully developed homogeneous and isotropic turbulence of the Navier-Stokes (NS) equation in the presence of a stochastic forcing, using the nonperturbative (functional) renormalization group (NPRG). Within a simple approximation based on symmetries, we obtain the fixed-point solution of the NPRG flow equations that corresponds to fully developed turbulence both in d=2 and 3 dimensions. Deviations to the dimensional scalings (Kolmogorov in d=3 or Kraichnan-Batchelor in d=2) are found for the two-point functions. To further analyze these deviations, we derive exact flow equations in the large wave-number limit, and show that the fixed point does not entail the usual scale invariance, thereby identifying the mechanism for the emergence of intermittency within the NPRG framework. The purpose of this work is to provide a detailed basis for NPRG studies of NS turbulence; the determination of the ensuing intermittency exponents is left for future work.

Scaling, cumulant ratios, and height distribution of ballistic deposition in 3+1 and 4+1 dimensions

We investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by Alves et al. [Phys. Rev. E 90, 052405 (2014)PLEEE81539-375510.1103/PhysRevE.90.052405] in d=2+1 dimensions, where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account. In the present work, we show that this concept holds for d=3+1 and 4+1 dimensions. We have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models, presenting small finite-time corrections to the scaling, that in principle belong to the Kardar-Parisi-Zhang (KPZ) universality class in both d=3+1 and 4+1. Our results constitute further evidence that the upper critical dimension of the KPZ class, if it exists, is larger than 4.

Width and extremal height distributions of fluctuating interfaces with window boundary conditions

We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size l, for interfaces in several universality classes, in substrate dimensions d_{s}=1 and 2. We show that their cumulants follow a Family-Vicsek-type scaling, and, at early times, when ξ≪l (ξ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their nth cumulant scaling as (ξ/l)^{(n-1)d_{s}}. This gives rise to an interesting temporal scaling for such cumulants as 〈w_{n}〉_{c}∼t^{γ_{n}}, with γ_{n}=2nβ+(n-1)d_{s}/z=[2n+(n-1)d_{s}/α]β. This scaling is analytically proved for the Edwards-Wilkinson (EW) and random deposition interfaces and numerically confirmed for other classes. In general, it is featured by small corrections, and, thus, it yields exponents γ_{n} (and, consequently, α,β and z) in good agreement with their respective universality class. Thus, it is a useful framework for numerical and experimental investigations, where it is usually hard to estimate the dynamic z and mainly the (global) roughness α exponents. The stationary (for ξ≫l) SLRDs and LEHDs of the Kardar-Parisi-Zhang (KPZ) class are also investigated, and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidence of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large l. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.

Ground-state statistics of directed polymers with heavy-tailed disorder

In this mostly numerical study, we reconsider the statistical properties of the ground state of a directed polymer in a d=1+1 "hilly" disorder landscape, i.e., when the quenched disorder has power-law tails. When disorder is Gaussian, the polymer minimizes its total energy through a collective optimization, where the energy of each visited site only weakly contributes to the total. Conversely, a hilly landscape forces the polymer to distort and explore a larger portion of space to reach some particularly deep energy sites. As soon as the fifth moment of the disorder diverges, this mechanism radically changes the standard Kardar-Parisi-Zhang scaling behavior of the directed polymer, and new exponents prevail. After confirming again that the Flory argument accurately predicts these exponents in the tail-dominated phase, we investigate several other statistical features of the ground state that shed light on this unusual transition and on the accuracy of the Flory argument. We underline the theoretical challenge posed by this situation, which paradoxically becomes even more acute above the upper critical dimension.

Strong-coupling phases of the anisotropic Kardar-Parisi-Zhang equation

We study the anisotropic Kardar-Parisi-Zhang equation using nonperturbative renormalization group methods. In contrast to a previous analysis in the weak-coupling regime, we find the strong-coupling fixed point corresponding to the isotropic rough phase to be always locally stable and unaffected by the anisotropy even at noninteger dimensions. Apart from the well-known weak-coupling and the now well-established isotropic strong-coupling behavior, we find an anisotropic strong-coupling fixed point for nonlinear couplings of opposite signs at noninteger dimensions.

Origins of scaling corrections in ballistic growth models

We study the ballistic deposition and the grain deposition models on two-dimensional substrates. Using the Kardar-Parisi-Zhang (KPZ) ansatz for height fluctuations, we show that the main contribution to the intrinsic width, which causes strong corrections to the scaling, comes from the fluctuations in the height increments along deposition events. Accounting for this correction in the scaling analysis, we obtain scaling exponents in excellent agreement with the KPZ class. We also propose a method to suppress these corrections, which consists in dividing the surface in bins of size ɛ and using only the maximal height inside each bin to do the statistics. Again, scaling exponents in remarkable agreement with the KPZ class are found. The binning method allows the accurate determination of the height distributions of the ballistic models in both growth and steady-state regimes, providing the universal underlying fluctuations foreseen for KPZ class in 2 + 1 dimensions. Our results provide complete and conclusive evidences that the ballistic model belongs to the KPZ universality class in 2+1 dimensions. Potential applications of the methods developed here, in both numerics and experiments, are discussed.

Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension

We show that the theoretical machinery developed for the Kardar-Parisi-Zhang (KPZ) class in low dimensions is obeyed by the restricted solid-on-solid model for substrates with dimensions up to d=6. Analyzing different restriction conditions, we show that the height distributions of the interface are universal for all investigated dimensions. It means that fluctuations are not negligible and, consequently, the system is still below the upper critical dimension at d=6. The extrapolation of the data to dimensions d≥7 predicts that the upper critical dimension of the KPZ class is infinite.

Relaxation after a change in the interface growth dynamics

The global effects of sudden changes in the interface growth dynamics are studied using models of the Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) classes during their growth regimes in dimensions d=1 and d=2. Scaling arguments and simulation results are combined to predict the relaxation of the difference in the roughness of the perturbed and the unperturbed interfaces, ΔW^{2}∼s{c}t{-γ}, where s is the time of the change and t>s is the observation time after that event. The previous analytical solution for the EW-EW changes is reviewed and numerically discussed in the context of lattice models, with possible decays with γ=3/2 and γ=1/2. Assuming the dominant contribution to ΔW{2} to be predicted from a time shift in the final growth dynamics, the scaling of KPZ-KPZ changes with γ=1-2β and c=2β is predicted, where β is the growth exponent. Good agreement with simulation results in d=1 and d=2 is observed. A relation with the relaxation of a local autoresponse function in d=1 cannot be discarded, but very different exponents are shown in d=2. We also consider changes between different dynamics, with the KPZ-EW as a special case in which a faster growth, with dynamical exponent z_{i}, changes to a slower one, with exponent z. A scaling approach predicts a crossover time t_{c}∼s{z/z_{i}}≫s and ΔW{2}∼s{c}F(t/t_{c}), with the decay exponent γ=1/2 of the EW class. This rules out the simplified time shift hypothesis in d=2 dimensions. These results help to understand the remarkable differences in EW smoothing of correlated and uncorrelated surfaces, and the approach may be extended to sudden changes between other growth dynamics.

Aging of the (2+1)-dimensional Kardar-Parisi-Zhang model

Extended dynamical simulations have been performed on a (2+1)-dimensional driven dimer lattice-gas model to estimate aging properties. The autocorrelation and the autoresponse functions are determined and the corresponding scaling exponents are tabulated. Since this model can be mapped onto the (2+1)-dimensional Kardar-Parisi-Zhang surface growth model, our results contribute to the understanding of the universality class of that basic system.

Explore or exploit? A generic model and an exactly solvable case

Finding a good compromise between the exploitation of known resources and the exploration of unknown, but potentially more profitable choices, is a general problem, which arises in many different scientific disciplines. We propose a stylized model for these exploration-exploitation situations, including population or economic growth, portfolio optimization, evolutionary dynamics, or the problem of optimal pinning of vortices or dislocations in disordered materials. We find the exact growth rate of this model for treelike geometries and prove the existence of an optimal migration rate in this case. Numerical simulations in the one-dimensional case confirm the generic existence of an optimum.

Kardar-Parisi-Zhang equation with spatially correlated noise: a unified picture from nonperturbative renormalization group

We investigate the scaling regimes of the Kardar-Parisi-Zhang (KPZ) equation in the presence of spatially correlated noise with power-law decay D(p) ∼ p(-2ρ) in Fourier space, using a nonperturbative renormalization group approach. We determine the full phase diagram of the system as a function of ρ and the dimension d. In addition to the weak-coupling part of the diagram, which agrees with the results from Europhys. Lett. 47, 14 (1999) and Eur. Phys. J. B 9, 491 (1999), we find the two fixed points describing the short-range- (SR) and long-range- (LR) dominated strong-coupling phases. In contrast with a suggestion in the references cited above, we show that, for all values of ρ, there exists a unique strong-coupling SR fixed point that can be continuously followed as a function of d. We show in particular that the existence and the behavior of the LR fixed point do not provide any hint for 4 being the upper critical dimension of the KPZ equation with SR noise.

Universal aspects of curved, flat, and stationary-state Kardar-Parisi-Zhang statistics

Motivated by the recent exact solution of the stationary-state Kardar-Parisi-Zhang (KPZ) statistics by Imamura and Sasamoto [ Phys. Rev. Lett. 108 190603 (2012)], as well as a precursor experimental signature unearthed by Takeuchi [ Phys. Rev. Lett. 110 210604 (2013)], we establish here the universality of these phenomena, examining scaling behaviors of directed polymers in a random medium, the stochastic heat equation with multiplicative noise, and kinetically roughened KPZ growth models. We emphasize the value of cross KPZ-class universalities, revealing crossover effects of experimental relevance. Finally, we illustrate the great utility of KPZ scaling theory by an optimized numerical analysis of the Ulam problem of random permutations.