Universality classes of the Kardar-Parisi-Zhang equation

We re-examine mode-coupling theory for the Kardar-Parisi-Zhang (KPZ) equation in the strong coupling limit and show that there exists two branches of solutions. One branch (or universality class) only exists for dimensionalities $d<d_c=2$ and is similar to that found by a variety of analytic approaches, including replica symmetry breaking and Flory-Imry-Ma arguments. The second branch exists up to $d_c=4$ and gives values for the dynamical exponent $z$ similar to those of numerical studies for $d\ge2$.Comment: 4 pages, 1 figure, published versio

Non-perturbative Approach to Critical Dynamics

This paper is devoted to a non-perturbative renormalization group (NPRG) analysis of Model A, which stands as a paradigm for the study of critical dynamics. The NPRG formalism has appeared as a valuable theoretical tool to investigate non-equilibrium critical phenomena, yet the simplest -- and nontrivial -- models for critical dynamics have never been studied using NPRG techniques. In this paper we focus on Model A taking this opportunity to provide a pedagological introduction to NPRG methods for dynamical problems in statistical physics. The dynamical exponent $z$ is computed in $d=3$ and $d=2$ and is found in close agreement with results from other methods.Comment: 13 page

Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence

In this paper, we present theoretical results on the statistical properties of stationary, homogeneous and isotropic turbulence in incompressible flows in three dimensions. Within the framework of the Non-Perturbative Renormalization Group, we derive a closed renormalization flow equation for a generic $n$-point correlation (and response) function for large wave-numbers with respect to the inverse integral scale. The closure is obtained from a controlled expansion and relies on extended symmetries of the Navier-Stokes field theory. It yields the exact leading behavior of the flow equation at large wave-numbers $|\vec p_i|$, and for arbitrary time differences $t_i$ in the stationary state. Furthermore, we obtain the form of the general solution of the corresponding fixed point equation, which yields the analytical form of the leading wave-number and time dependence of $n$-point correlation functions, for large wave-numbers and both for small $t_i$ and in the limit $t_i\to \infty$. At small $t_i$, the leading contribution at large wave-number is logarithmically equivalent to $-\alpha (\epsilon L)^{2/3}|\sum t_i \vec p_i|^2$, where $\alpha$ is a nonuniversal constant, $L$ the integral scale and $\varepsilon$ the mean energy injection rate. For the 2-point function, the $(t p)^2$ dependence is known to originate from the sweeping effect. The derived formula embodies the generalization of the effect of sweeping to $n-$point correlation functions. At large wave-number and large $t_i$, we show that the $t_i^2$ dependence in the leading order contribution crosses over to a $|t_i|$ dependence. The expression of the correlation functions in this regime was not derived before, even for the 2-point function. Both predictions can be tested in direct numerical simulations and in experiments.Comment: 23 pages, minor typos correcte

General framework of the non-perturbative renormalization group for non-equilibrium steady states

This paper is devoted to presenting in detail the non-perturbative renormalization group (NPRG) formalism to investigate out-of-equilibrium systems and critical dynamics in statistical physics. The general NPRG framework for studying non-equilibrium steady states in stochastic models is expounded and fundamental technicalities are stressed, mainly regarding the role of causality and of Ito's discretization. We analyze the consequences of Ito's prescription in the NPRG framework and eventually provide an adequate regularization to encode them automatically. Besides, we show how to build a supersymmetric NPRG formalism with emphasis on time-reversal symmetric problems, whose supersymmetric structure allows for a particularly simple implementation of NPRG in which causality issues are transparent. We illustrate the two approaches on the example of Model A within the derivative expansion approximation at order two, and check that they yield identical results.Comment: 28 pages, 1 figure, minor corrections prior to publicatio

Quantitative Phase Diagrams of Branching and Annihilating Random Walks

We demonstrate the full power of nonperturbative renormalisation group methods for nonequilibrium situations by calculating the quantitative phase diagrams of simple branching and annihilating random walks and checking these results against careful numerical simulations. Specifically, we show, for the 2A->0, A -> 2A case, that an absorbing phase transition exists in dimensions d=1 to 6, and argue that mean field theory is restored not in d=3, as suggested by previous analyses, but only in the limit d -> $\infty$.Comment: 4 pages, 3 figures, published version (some typos corrected

Reaction-diffusion processes and non-perturbative renormalisation group

This paper is devoted to investigating non-equilibrium phase transitions to an absorbing state, which are generically encountered in reaction-diffusion processes. It is a review, based on [Phys. Rev. Lett. 92, 195703; Phys. Rev. Lett. 92, 255703; Phys. Rev. Lett. 95, 100601], of recent progress in this field that has been allowed by a non-perturbative renormalisation group approach. We mainly focus on branching and annihilating random walks and show that their critical properties strongly rely on non-perturbative features and that hence the use of a non-perturbative method turns out to be crucial to get a correct picture of the physics of these models.Comment: 14 pages, submitted to J. Phys. A for the proceedings of the conference 'Renormalization Group 2005', Helsink

Single-site approximation for reaction-diffusion processes

We consider the branching and annihilating random walk $A\to 2A$ and $2A\to 0$ with reaction rates $\sigma$ and $\lambda$, respectively, and hopping rate $D$, and study the phase diagram in the $(\lambda/D,\sigma/D)$ plane. According to standard mean-field theory, this system is in an active state for all $\sigma/D>0$, and perturbative renormalization suggests that this mean-field result is valid for $d >2$; however, nonperturbative renormalization predicts that for all $d$ there is a phase transition line to an absorbing state in the $(\lambda/D,\sigma/D)$ plane. We show here that a simple single-site approximation reproduces with minimal effort the nonperturbative phase diagram both qualitatively and quantitatively for all dimensions $d>2$. We expect the approach to be useful for other reaction-diffusion processes involving absorbing state transitions.Comment: 15 pages, 2 figures, published versio

Non-perturbative renormalisation group for the Kardar-Parisi-Zhang equation: general framework and first applications

We present an analytical method, rooted in the non-perturbative renormalization group, that allows one to calculate the critical exponents and the correlation and response functions of the Kardar-Parisi-Zhang (KPZ) growth equation in all its different regimes, including the strong-coupling one. We analyze the symmetries of the KPZ problem and derive an approximation scheme that satisfies the linearly realized ones. We implement this scheme at the minimal order in the response field, and show that it yields a complete, qualitatively correct phase diagram in all dimensions, with reasonable values for the critical exponents in physical dimensions. We also compute in one dimension the full (momentum and frequency dependent) correlation function, and the associated universal scaling functions. We find an excellent quantitative agreement with the exact results from Praehofer and Spohn (J. Stat. Phys. 115 (2004)). We emphasize that all these results, which can be systematically improved, are obtained with sole input the bare action and its symmetries, without further assumptions on the existence of scaling or on the form of the scaling function.Comment: 21 pages, 6 figures, revised version, including the correction of an inconsistency and accordingly updated figures 5 and 6 and table 2, as published in an Erratum (see Ref. below). The results are improve