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Astron. Astrophys. 344, 27-35 (1999)

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5. Fixed points

The fixed points, collectively denoted by [FORMULA], 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. Which fixed point the system is attracted to or repelled from depends on the basin of attraction where the initial conditions for the DynRG equations are located. As the couplings evolve under graining into their critical values, the n-point correlation functions approach power law (or scaling) forms because they are the solution to first order quasi-linear partial differential equations (the Callan-Symanzik (CS) equations (Amit 1984; Binney et al. 1992)) whose characteristics are the solutions to Eqns. (23). This CS equation simply expresses the fact that the correlation functions are invariant under the renormalization group transformation. In other words, the system is scale-invariant at its fixed points, hence, correlation functions must take the form of algebraic power laws, since these are the only mathematical functions that are scale-invariant. The physics of each of these fixed points is different, because it depends to a large extent on the magnitude and sign of the scaling exponent as well as on the attractive or repulsive nature of the fixed point.

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.

[FIGURE] Fig. 1. Flow diagram in ([FORMULA],V) space. Arrows indicate IR ([FORMULA]) flow.

[FIGURE] Fig. 2. Flow diagram in ([FORMULA], [FORMULA], V) space around P5+.


Table 1. Characteristics of the fixed points.
These fixed points are obtained as real solutions of a second degree algebraic equation, and therefore the set of pairs that give real fixed points are restricted by a discriminant inequality.
- Only P5+ has a basin of attraction, which is approximately in the region [FORMULA]. P2 has a large "basin of repulsion", and the other points are saddle points.
- The trajectories outside the P5+ basin of attraction reach infinity or fall to the singular [FORMULA] plane at a finite "time".
- Only with P5+ can we adjust the observed values [FORMULA], because P5- has [FORMULA] in this region, which implies smaller length scales than the one corresponding to the UV cut-off [FORMULA].

5.1. The value of the critical exponents at each critical point. White noise

In the case of white noise ([FORMULA]) we solve for the fixed points of the (reduced set) of RG equations. Since [FORMULA] in this case, these fixed points will all lie in a two-dimensional coupling space spanned by [FORMULA]. There are three fixed points, labelled as [FORMULA] and [FORMULA], whose coordinates are listed in Table 1. Two of them, [FORMULA] lie in the line [FORMULA], corresponding to [FORMULA] in the RG-evolved KPZ equation, while [FORMULA] lies in the region bounded between the [FORMULA] and [FORMULA] lines. By substituting these fixed point values back into the original set of RG equations (23), we solve for the corresponding fixed point exponents [FORMULA] and z. These values are listed in the third column of Table 1. Linearizing the RG equations about each one of their fixed points allows one to calculate the infrared (IR) stability properties of the fixed points and thus characterize their behavior with respect to the coarse-graining. This entails expanding the RG equations in [FORMULA] and [FORMULA] to first order in the perturbations and solving for [FORMULA] and [FORMULA]. In general, the eigenvectors of this linearized system will involve linear combinations of [FORMULA] and [FORMULA]. The associated eigenvalues, which determine the stability properties of the fixed point, are also listed in Table 1. A positive eigenvalue indicates that the coarse-graining induces a flow away from the point along the eigen-direction, while a negative eigenvalue indicates the point is stable in the infrared, since the flow will be into the fixed point. The type, or class, of fixed point, whether it be IR-attractive (all eigenvalues negative), IR-repulsive (all eigenvalues positive) or a saddle point (mixed sign eigenvalues) is listed in Table 1. The coarse-graining flow of the couplings in the white noise case is given in Fig. 1. These flow lines were calculated by integrating numerically the set of two coupled first order differential equations for [FORMULA] and V, using general choices for the RG equation initial conditions.

5.2. The value of the critical exponents at each critical point. Colored noise

For colored or correlated noise ([FORMULA]), we now deal with a three dimensional space of dimensionless couplings [FORMULA] and V. Using the same procedure as discussed above, we solve for the RG equation fixed points which lead to a total of seven fixed points, including the same three points [FORMULA] and [FORMULA] obtained in the white noise limit. Thus, allowing for colored noise yields four additional fixed points which we will denote by [FORMULA] and [FORMULA], since they arise in pairs. Their coordinates are listed in the table, together with their associated exponents [FORMULA] and z and IR eigenvalues and stability properties under coarse-graining. Both, the positions of these points and values of their exponents, depend in general upon the values of the noise exponents [FORMULA] and [FORMULA], which parametrize spatial and temporal correlations in the noise. The pair of fixed points labelled as [FORMULA] always lie in the plane [FORMULA], (i.e., [FORMULA]) but their location within this plane varies with the noise exponents. Moreover, any RG flow which starts off in this plane will always remain in this plane (this plane acts as a separatrix). It is important to point out that there are values of the noise exponents [FORMULA] which lead to complex values of the fixed point coordinates for [FORMULA]. This is because these fixed points arise as solutions of a quadratic algebraic equation whose discriminant can become negative for values of [FORMULA] and [FORMULA] in certain domains in parameter space. We must rule out such values of the noise exponents because of physical reasons. For the allowed values of [FORMULA] and [FORMULA] (i.e., those that lead to real fixed points), we find that [FORMULA] have exponents z and [FORMULA] that are complicated functions of these parameters, but we have checked that for all allowed values, either [FORMULA] or [FORMULA]. The IR eigenvalues and the nature of these fixed points depend on [FORMULA]. For the choice shown, i.e., [FORMULA] and [FORMULA], both [FORMULA] are saddle points. In fact, we have confirmed that the pair [FORMULA] will always be saddle points whenever [FORMULA] and [FORMULA]. The reason for choosing these particular exponent intervals will become clear in Sect. 6.

The next pair of fixed points [FORMULA] also have coordinates and exponents depending on the noise exponents. As in the case of [FORMULA], there are values of [FORMULA] (same as for [FORMULA]) leading to complex couplings, which we rule out. The allowed values of the noise exponents lie roughly in the respective intervals [FORMULA] and [FORMULA]. The exponents for [FORMULA] are given by the simple functions [FORMULA] and [FORMULA], and the stability properties do not depend on the particular values of the noise exponents within the above mentioned intervals. For the choice [FORMULA] and [FORMULA], [FORMULA] is IR attractive while [FORMULA] is an unstable saddle point. We must nonetheless exclude [FORMULA] from our consideration since it lies below the plane [FORMULA], i.e., it corresponds to a renormalized KPZ mass [FORMULA], which therefore exceeds the scale of the momentum cut-off imposed on our perturbative calculation.

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999