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We study the optimization of nonperturbative renormalization group equations truncated both in fields and derivatives. On the example of the Ising model in three dimensions, we show that the Principle of Minimal Sensitivity can be unambiguously implemented at order ∂ 2 of the derivative expansion. This approach allows us to select optimized cutoff functions and to improve the accuracy of the critical exponents ν and η. The convergence of the field expansion is also analyzed. We show in particular that its optimization does not coincide with optimization of the accuracy of the critical exponents.

On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order ∂ 4 of the derivative expansion leads to ν = 0.632 and to an anomalous dimension η = 0.033 which is significantly improved compared with lower orders calculations.Many problems in high-energy as well as in statistical physics call for nonperturbative methods. On the one hand, several physical systems are described by field theories in their strong coupling regime so that the usual perturbative techniques become troublesome. They fail either because only the first orders of perturbation are computed and do not suffice, or because, even when high orders are known, standard resummation techniques do not provide converged results. On the other hand, some phenomena such as confinement in QCD or phase transitions induced by topological defects are genuinely nonperturbative.Apart from some methods restricted to specific dimensions or situations, very few nonperturbative techniques are available. During the last years, the Wilson approach 1 to the renormalization group (RG) has been turned into an efficient tool 2,3,4 . This nonperturbative RG can be implemented in very general situations and, in particular, in any dimension, so that it has allowed one to study several issues difficult to tackle within a perturbative framework among which the three-dimensional Gross-Neveu model 5 , frustrated magnets 6 , the randomly dilute Ising model 7 , and the Abelian Higgs model 8 .This method relies on a nonperturbative renormalization of the effective action Γ, i.e. the Gibbs free energy. It consists in building an effective action Γ k at the running scale k by integrating out only fluctuations greater than k. At the scale k = Λ, Λ −1 denoting the spacing of the underlying lattice, Γ k coincides with the Hamiltonian H since no fluctuation has yet been taken into account while, at k = 0, it coincides with the standard effective action Γ since all fluctuations have been integrated out. Thus, Γ k continuously interpolates between the microscopic Hamiltonian H and the free energy Γ. The running effective action Γ k follows an exact equation which controls its evolution with the running scale k 2 :where t = ln(k/Λ) and Γ(2) k [φ] is the second functional derivative of Γ k with respect to the field φ(q). In Eq. (1), R k (q) is an infrared cutoff function which suppresses the propagation of the low-energy modes without affecting the high-energy ones.Although exact, Eq. (1) is a functional partial integrodifferential equation which cannot be solved exactly. To handle it, one has to truncate Γ k . A natural and widely used truncation is the derivative expansion, which consists in expanding Γ k in powers of ∂φ, keeping only the lowest order terms. Physically, this truncation rests on the assumption that the long-distance physics of a given model is well described by the lowest derivative terms, the higher ones corresponding to less relevant operators. U...

We present a simple approximation of the nonperturbative renormalization group designed for the Kardar-Parisi-Zhang equation and show that it yields the correct phase diagram, including the strong-coupling phase with reasonable scaling exponent values in physical dimensions. We find indications of a possible qualitative change of behavior around d=4. We discuss how our approach can be systematically improved.

We generalize nonperturbative renormalization group methods to nonequilibrium critical phenomena. Within this formalism, reaction-diffusion processes are described by a scale-dependent effective action, the flow of which is derived. We investigate branching and annihilating random walks with an odd number of offspring. Along with recovering their universal physics (described by the directed percolation universality class), we determine their phase diagrams and predict that a transition occurs even in three dimensions, contrarily to what perturbation theory suggests.

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 Itō's discretization. We analyze the consequences of Itō'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. We stress, though, that the framework presented here also applies to genuinely out-of-equilibrium problems.

We apply the nonperturbative renormalization group method to a class of out-of-equilibrium phase transitions (usually called "parity-conserving" or, more properly, "generalized voter" class) which is out of the reach of perturbative approaches. We show the existence of a genuinely nonperturbative fixed point, i.e., a critical point that does not seem to be Gaussian in any dimension.

We demonstrate the full power of nonperturbative renormalization 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 [see text] 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--> infinity.