## 5. Fixed pointsThe fixed points, collectively denoted by
, are the solutions to the system of
algebraic equations obtained by setting the right hand sides of
equations (24) equal to zero. They control the stability properties of
the system of DynRG equations and therefore they characterize the
asymptotic behavior of correlation functions. When the couplings
attain their critical values, the system becomes critical and the
correlation functions enter into power law regimes, characteristic of
physical systems in the critical state where they are scale invariant.
The flows and the fixed points of our set of RG equations are represented in Fig. 1, for white noise and Fig. 2 for colored noise, while the features of our fixed point analysis are conveniently summarized in the adjoining Table 1.
## 5.1. The value of the critical exponents at each critical point. White noiseIn the case of white noise () we
solve for the fixed points of the (reduced set) of RG equations. Since
in this case, these fixed points
will all lie in a two-dimensional coupling space spanned by
. There are three fixed points,
labelled as and
, whose coordinates are listed in
Table 1. Two of them, lie in
the line , corresponding to
in the RG-evolved KPZ equation,
while lies in the region bounded
between the and
lines. By substituting these fixed
point values back into the original set of RG equations (23), we solve
for the corresponding fixed point exponents
and ## 5.2. The value of the critical exponents at each critical point. Colored noiseFor colored or correlated noise
(), we now deal with a three
dimensional space of dimensionless couplings
and The next pair of fixed points also have coordinates and exponents depending on the noise exponents. As in the case of , there are values of (same as for ) leading to complex couplings, which we rule out. The allowed values of the noise exponents lie roughly in the respective intervals and . The exponents for are given by the simple functions and , and the stability properties do not depend on the particular values of the noise exponents within the above mentioned intervals. For the choice and , is IR attractive while is an unstable saddle point. We must nonetheless exclude from our consideration since it lies below the plane , i.e., it corresponds to a renormalized KPZ mass , which therefore exceeds the scale of the momentum cut-off imposed on our perturbative calculation. © European Southern Observatory (ESO) 1999 Online publication: March 10, 1999 |