Quantitative Phase Diagrams of Branching and Annihilating Random Walks

Abstract

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.

What can be learnt from the nonperturbative renormalization group?

“…Using the NPRG method, we have re-examined the physics of this system. We have been able to compute the critical exponents in all dimensions and have shown that the phase diagram obtained perturbatively is wrong [42]. More precisely, above dimension two, the system is indeed always in the active phase at small σ/D and λ/D as predicted by the perturbative analysis.…”

“…But there is a threshold value of λ/D above which an absorbing phase can exist. By numerically integrating the RG flow from dimensions 3 to 6 we have determined these threshold values as well as the complete phase diagram [42]. In figure 1 there are displayed both the NPRG results and those obtained from numerical simulations.…”

“…Computing it requires to keep track all along the RG flow of both relevant and irrelevant couplings since they all contribute to the non-universal behavior of the model. While this is in general a formidable task in perturbation theory, that anyway is useless if these couplings are large, it just consists in the NPRG approach in integrating the flow equations for the potential with the bare potential of equation (27) as initial condition at scale Λ [42,44,45]. …”

“…However, all these differences do not prevent us from constructing an analog of the effective average action method for out of equilibrium systems [41,42]. The nontrivial point is to choose a R k function.…”

“…The NPRG method seems to be a promising way to tackle the strong coupling regime of (at least some) field theories while avoiding problematic resummation methods used in perturbative renormalization. It also enables us to compute non-universal quantities such as phase diagrams because it can handle the functional nature of the effective potential with both its relevant and irrelevant couplings, that are most of the time infinitely numerous [42,45]. For the reaction-diffusion problem we have discussed in this article, the NPRG has allowed us to find that a phase transition exists in all dimensions while perturbative renormalization failed, probably even in a all-order analysis.…”