Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards

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.

Extensive numerical simulations of surface growth with temporally correlated noise

Surface growth processes can be significantly affected by long-range temporal correlations. In this work, we perform extensive numerical simulations of a (1+1)- and (2+1)-dimensional ballistic deposition (BD) model driven by temporally correlated noise, which is regarded as the temporal correlated Kardar-Parisi-Zhang universality class. Our results are compared with the existing theoretical predictions and numerical simulations. When the temporal correlation exponent is above a certain threshold, BD surfaces develop gradually faceted patterns. We find that the temporal correlated BD system displays nontrivial dynamic properties, and the characteristic roughness exponents satisfy α≃α_{loc}<α_{s} in (1+1) dimensions, which is beyond the existing dynamic scaling classifications.

Scaling of roughness and porosity in thin film deposition with mixed transport mechanisms and adsorption barriers

Thin film deposition with particle transport mixing collimated and diffusive components and with barriers for adsorption are studied using numerical simulations and scaling approaches. Biased random walks on lattices are used to model the particle flux and the analogy with advective-diffusive transport is used to define a Peclet number P that represents the effect of the bias towards the substrate. An aggregation probability that relates the rates of adsorption and of the dominant transport mechanism plays the role of a Damkohler number D, where D≲1 is set to describe moderate to low adsorption rates. Very porous deposits with sparse branches are obtained with P≪1, whereas low porosity deposits with large height fluctuations at short scales are obtained with P≫1. For P≳1 in which the field bias is intense, an initial random deposition is followed by Kardar-Parisi-Zhang (KPZ) roughening. As the transport is displaced from those limiting conditions, the interplay of the transport and adsorption mechanisms establishes a condition to produce films with the smoothest surfaces for a constant deposited mass: with low adsorption barriers, a balance of random and collimated flux is required, whereas for high barriers the smoothest surfaces are obtained with P∼D^{1/2}. For intense bias, the roughness is shown to be a power law of P/D, whose exponent depends on the growth exponent β of the KPZ class, and the porosity has a superuniversal scaling as (P/D)^{-1/3}. We also study a generalized ballistic deposition model with slippery particle aggregation that shows the universality of these relations in growth with dominant collimated flux, particle adsorption at any point of the deposit, and negligible adsorbate diffusion, in contrast with the models where aggregation is restricted to the outer surface.

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.

Simulations of Co-GISAXS during kinetic roughening of growth surfaces

The recent development of surface growth studies using X-ray photon correlation spectroscopy in a grazing-incidence small-angle X-ray scattering (Co-GISAXS) geometry enables the investigation of dynamical processes during kinetic roughening in greater detail than was previously possible. In order to investigate the Co-GISAXS behavior expected from existing growth models, calculations and (2+1)-dimension simulations of linear Kuramoto-Sivashinsky and non-linear Kardar-Parisi-Zhang surface growth equations are presented which analyze the temporal correlation functions of the height-height structure factor. Calculations of the GISAXS intensity auto-correlation functions are also performed within the Born/distorted-wave Born approximation for comparison with the scaling behavior of the height-height structure factor and its correlation functions.

Nonuniversality in the erosion of tilted landscapes

The anisotropic model for landscapes erosion proposed by Pastor-Satorras and Rothman [R. Pastor-Satorras and D. H. Rothman, Phys. Rev. Lett. 80, 4349 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.4349] is believed to capture the physics of erosion at intermediate length scale (≲3 km), and to account for the large value of the roughness exponent α observed in real data at this scale. Our study of this model-conducted using the nonperturbative renormalization group-concludes on the nonuniversality of this exponent because of the existence of a line of fixed points. Thus the roughness exponent depends (weakly) on the details of the soil and the erosion mechanisms. We conjecture that this feature, while preserving the generic scaling observed in real data, could explain the wide spectrum of values of α measured for natural landscapes.

Effects of film growth kinetics on grain coarsening and grain shape

We study models of grain nucleation and coarsening during the deposition of a thin film using numerical simulations and scaling approaches. The incorporation of new particles in the film is determined by lattice growth models in three different universality classes, with no effect of the grain structure. The first model of grain coarsening is similar to that proposed by Saito and Omura [Phys. Rev. E 84, 021601 (2011)PLEEE81539-375510.1103/PhysRevE.84.021601], in which nucleation occurs only at the substrate, and the grain boundary evolution at the film surface is determined by a probabilistic competition of neighboring grains. The surface grain density has a power-law decay, with an exponent related to the dynamical exponent of the underlying growth kinetics, and the average radius of gyration scales with the film thickness with the same exponent. This model is extended by allowing nucleation of new grains during the deposition, with constant but small rates. The surface grain density crosses over from the initial power law decay to a saturation; at the crossover, the time, grain mass, and surface grain density are estimated as a function of the nucleation rate. The distributions of grain mass, height, and radius of gyration show remarkable power law decays, similar to other systems with coarsening and particle injection, with exponents also related to the dynamical exponent. The scaling of the radius of gyration with the height h relative to the base of the grain show clearly different exponents in growth dominated by surface tension and growth dominated by surface diffusion; thus it may be interesting for investigating the effects of kinetic roughening on grain morphology. In growth dominated by surface diffusion, the increase of grain size with temperature is observed.

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.

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.

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.

From single-file diffusion to two-dimensional cage diffusion and generalization of the totally asymmetric simple exclusion process to higher dimensions

A two-dimensional constrained diffusion model is presented and characterized by numerical simulations. The model generalizes the one-dimensional single-file diffusion model by considering a cage diffusion constraint induced by neighboring particles, which is a more stringent condition than volume exclusion. Using numerical simulations we characterize the diffusion process and we particularly show that asymmetric transition probabilities lead to the two-dimensional Kardar-Parisi-Zhang universality class. Therefore, this very simple model effectively generalizes the one-dimensional totally asymmetric simple exclusion process to higher dimensions.

Smoothening in thin-film deposition on rough substrates

The evolution of the surface roughness W of a thin film deposited on a rough substrate is studied with a model of temperature-activated adatom diffusion, irreversible lateral aggregation, and no step energy barrier, in which the main parameter is the ratio R of diffusion and deposition rates. At sufficiently low temperatures (R≲10), the average number of adatom steps after adsorption is very small, thus W monotonically increases with time t due to an approximately uncorrelated deposition at short times. If the temperature is not very low (R∼10(3) or larger), smoothening occurs at short times and the Villain-Lai-Das Sarma (VLDS) growth equation governs the long time roughening, which is attained after a crossover time t(c) that increases with the correlation length ξ(i) of the substrate. Scaling arguments predict the dependence of t(c) on temperature and on the substrate production time and the scaling relation for the difference between the roughness of films deposited on rough and flat substrates, in good agreement with numerical results. The effect of temperature is not a direct extension of previous results on flat substrates because the short wavelength fluctuations delay the formation of terraces. For this reason, the effective energy obtained from the dependence of t(c) on R is 40% of the energy of activated adatom diffusion. A scaling law for the initial smoothening is proposed as W/W(i)=Ψ(t/t(c1)), with a crossover time t(c1)≡R(-θ)ξ(i)(z), where W(i) is the substrate roughness, θ≈0.4, and z is the VLDS dynamical exponent. It provides good data collapse if W is not very small and is suggested to be tested experimentally.

Numerical estimate of the Kardar-Parisi-Zhang universality class in (2+1) dimensions

We study the restricted solid on solid model for surface growth in spatial dimension d=2 by means of a multisurface coding technique that allows one to produce a large number of samples in the stationary regime in a reasonable computational time. Thanks to (i) a careful finite-size scaling analysis of the critical exponents and (ii) the accurate estimate of the first three moments of the height fluctuations, we can quantify the wandering exponent with unprecedented precision: χ(d=2)=0.3869(4). This figure is incompatible with the long-standing conjecture due to Kim and Koesterlitz that hypothesized χ(d=2)=2/5.

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.

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.

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.

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.

Kardar-Parisi-Zhang universality class in (2+1) dimensions: universal geometry-dependent distributions and finite-time corrections

The dynamical regimes of models belonging to the Kardar-Parisi-Zhang (KPZ) universality class are investigated in d=2+1 by extensive simulations considering flat and curved geometries. Geometry-dependent universal distributions, different from their Tracy-Widom counterpart in one dimension, were found. Distributions exhibit finite-time corrections hallmarked by a shift in the mean decaying as t(-β), where β is the growth exponent. Our results support a generalization of the ansatz h=v(∞)t+(Γt)(β)χ+η+ζt(-β) to higher dimensions, where v(∞), Γ, ζ, and η are nonuniversal quantities whereas β and χ are universal and the last one depends on the surface geometry. Generalized Gumbel distributions provide very good fits of the distributions in at least four orders of magnitude around the peak, which can be used for comparisons with experiments. Our numerical results call for analytical approaches and experimental realizations of the KPZ class in two-dimensional systems.

Multisurface coding simulations of the restricted solid-on-solid model in four dimensions

We study the restricted solid-on-solid model for surface growth in spatial dimension d=4 by means of a multisurface coding technique that allows us to analyze samples of size up to 256(4) in the steady-state regime. For such large systems we are able to achieve a controlled asymptotic regime where the typical scale of the fluctuations are larger than the lattice spacing used in the simulations. A careful finite-size scaling analysis of the critical exponents clearly indicate that d=4 is not the upper critical dimension of the model.

Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1, and 3+1 dimensions

We investigate the strong-coupling regime of the stationary Kardar-Parisi-Zhang equation for interfaces growing on a substrate of dimension d = 1, 2, and 3 using a nonperturbative renormalization group (NPRG) approach. We compute critical exponents, correlation and response functions, extract the related scaling functions, and calculate universal amplitude ratios. We work with a simplified implementation of the second-order (in the response field) approximation proposed in a previous work [Phys. Rev. E 84, 061150 (2011) and Phys. Rev. E 86, 019904(E) (2012)], which greatly simplifies the frequency sector of the NPRG flow equations, while keeping a nontrivial frequency dependence for the two-point functions. The one-dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second-order NPRG equations and with the exact scaling function. Furthermore, the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d = 2 and d = 3. We argue that our ansatz is reliable up to d [Symbol: see text] 3.5.

(2+1)-Dimensional directed polymer in a random medium: scaling phenomena and universal distributions

We examine numerically the zero-temperature (2+1)-dimensional directed polymer in a random medium, along with several of its brethren via the Kardar-Parisi-Zhang (KPZ) equation. Using finite-size and KPZ scaling Ansätze, we extract the universal distributions controlling fluctuation phenomena in this canonical model of nonequilibrium statistical mechanics. Specifically, we study point-point, point-line, and point-plane directed polymer geometries, scenarios which yield higher-dimensional analogs of the Tracy-Widom distributions of random matrix theory. Our analysis represents a robust, multifaceted numerical characterization of the 2+1 KPZ universality class and its limit distributions.