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Astron. Astrophys. 351, 405-412 (1999)

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2. Generalized dimension

A fractal point distribution is usually characterized by its dimension and there exists a large variety of ways in which the dimension can be defined and measured. Of these possibilities two which are particularly simple and can be easily applied to a finite distribution of points are the box-counting dimension and the correlation dimension. In this section we discuss the "working definitions" of these two quantities that we have adopted for analyzing a distribution of a finite number of points. For more formal definitions of these dimensions the reader is referred to Borgani (1995) and references therein. The formal definitions usually involve the limit where the number of particles tends to infinity and they cannot be directly applied to galaxy distributions.

We first consider the box-counting dimension. In calculating the box-counting dimension for a distribution of points, the space is divided into identical boxes and we count the number of boxes which contain at least one point inside them. We then progressively reduce the size of the boxes while counting the number of boxes with at least one point inside them at every stage of this process. This gives the number of non-empty boxes [FORMULA] as a function of the size of one edge of the box r at every stage of the procedure. If the number of non-empty boxes exhibits a power-law scaling as a function of the size of the box i.e.

[EQUATION]

we then define D to be the box-counting dimension. In practice the nature of the scaling may be different on different length scales and we look for a sufficiently large range of r over which [FORMULA] exhibits a particular scaling behaviour and we then use Eq. (2) to obtain the box-counting dimension valid over those scales. So finally we may get more than one value of box-counting dimension for the distribution, each value of the box-counting dimension being valid over a limited range of length scales.

To compute the correlation dimension for a point distribution with N points we proceed by first labeling the points using an index j which runs from 1 to N. We then randomly select M of the N points and the index i is used to refer to these M randomly chosen points.

For every point i, we count the total number of points which are within a distance r from the [FORMULA] point and this quantity [FORMULA] can be written as

[EQUATION]

where [FORMULA] is the position vector of the [FORMULA] point and [FORMULA] is the Heaviside function. [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]. We next divide [FORMULA] by the total number of points N to calculate [FORMULA], the probability of finding a point within a distance r from the [FORMULA] point. We then average the quantity, [FORMULA], over all the M randomly selected centers to determine the probability of finding a point within a distance r of another point and we denote this by [FORMULA] which is given by,

[EQUATION]

If the probability [FORMULA] exhibits a scaling relation

[EQUATION]

we then define [FORMULA] to be the correlation dimension.

As with the box-counting dimension, the nature of the scaling behaviour may be different on different length scales and we may then get more than one value for the correlation dimension, each different value being valid over a range of scales.

It is very clear that [FORMULA] - which is the probability of finding a point within a sphere of radius r centered on another point, is closely related to the volume integral of the two point correlation function. In a situation where the two point correlation function exhibits a power-law behaviour [FORMULA] on scales [FORMULA], we expect the correlation dimension to have a value [FORMULA] over these scales.

For a mono-fractal the box-counting dimension and the correlation dimension will be the same, and for a homogeneous, space filling point distribution they are both equal to the dimension of the ambient space in which the points are embedded.

The box-counting dimension and the correlation dimension quantify different aspects of the scaling behaviour of a point distribution and they will have different values in a generic situation. The concept of a generalized dimension connects these two definitions and provides a continuous spectrum of dimensions [FORMULA] for a range of the parameter q. The definition of the Minkowski-Bouligand dimension [FORMULA] (Falconer 1990 , Feder 1989) closely follows the definition of the correlation dimension. The only difference is that we use the [FORMULA] moment of the galaxy distribution [FORMULA] (Eq. 3) around any point. Eq. (4) can then be generalized to define

[EQUATION]

which is used to define the generalized dimension

[EQUATION]

The quantity [FORMULA] may exhibit different scaling behaviour over different ranges of length scales and we will then get more than one spectrum of generalized dimensions each being valid over a different range of length scales.

From Eqs. (6) and (7) it is clear that the generalized dimension [FORMULA] corresponds to the correlation dimension at [FORMULA]. In addition [FORMULA] corresponds to the box-counting dimension at [FORMULA].

For a mono-fractal the generalized dimension is a constant i.e. [FORMULA] which reflects the fact that for a mono-fractal the point distribution is characterized by a unique scaling behaviour. For a generic multi-fractal the values of [FORMULA] will be different for different values of q. The positive values of q give more weightage to the over-dense regions. Thus, for [FORMULA], [FORMULA] probes the scaling behaviour of the distribution of points in the over-dense regions like inside clusters etc. The negative values of q, on the other hand, give more weight-age to the under-dense regions and, hence, for negative q, [FORMULA] probes the scaling behaviour of the distribution of points in the under-dense regions like voids.

Finally it should be pointed out that the Minkowski-Bouligand generalized dimension [FORMULA] is one of the many possible definitions of a generalized dimension. The minimal spanning tree used by van der Weygaert & Jones (van der Weygaert & Jones, 1992) is another possible method which can be used. The Minkowski-Bouligand generalized dimension has the advantage of being easy to compute. In addition the various selection effects which have to be taken into account when analyzing redshift surveys can be easily accounted for when determining the Minkowski-Bouligand generalized dimension and hence we have chosen this particular method for the multi-fractal characterization of the galaxy distribution in LCRS,

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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