{'fraction_non_alphanumeric': 0.02845887387081437, 'fraction_numerical': 0.011206955104258643, 'mean_word_length': 4.760701150324752, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Statistical characteristics of simulated walls

The large scale matter distribution in three different simulations of CDM models is investigated and compared with corresponding results of the Zel'dovich theory of nonlinear gravitational instability. We show that the basic characteristics of wall-like structure elements are well described by this theory, and that they can be expressed by the cosmological parameters and a few spectral moments of the perturbation spectrum. Therefore the characteristics of such elements provide reasonable estimates of these parameters. We show that the compressed matter is relaxed and gravitationally confined, what manifests itself in the existence of walls as (quasi)stationary structure elements with life time restricted by their disruption into high density clouds. The matter distribution is investigated both in the real and redshift spaces. In both cases almost the same particles form the walls, and we estimate differences in corresponding wall characteristics. The same methods are applied to several mock catalogues of 'galaxies' what allows us to characterize a large scale bias between the spatial distribution of dark matter and of simulated `galaxies'.


INTRODUCTION
Over the past decade immense progress was achieved in the investigation of the large scale matter distribution. Now the galaxy distribution is studied up to the redshift z ∼ 3 (Steidel at al. 1996). At smaller redshifts the analysis of rich galaxy surveys with an effective depth ∼ (200 -400)h −1 Mpc, such as the Durham/UKST Galaxy Redshift Survey (Ratcliffe et al. 1996), and the Las Campanas Redshift Survey , have established the existence of wall-like structure elements as a typical phenomenon in the visible galaxy distribution incorporating ∼ (40 -50)% of galaxies (Doroshkevich et al. 1996, hereafter LCRS1;Doroshkevich et al. 1999a, hereafter LCRS2;Doroshkevich et al. 2000, hereafter DURS). The wall-like structure elements with a typical diameter ∼ (30 -50)h −1 Mpc surround low-density regions with a similar typical diameter ∼ (50 -70)h −1 Mpc. Within the wall-like structures, the observed galaxy distribution is also inhomogeneous (see, e.g., Fig. 5 of Ramella et al. 1992), and galaxies are concentrated in high density clumps and filaments.
The galaxies occupying low density regions are concentrated within a random network of filaments. Filaments incorporate ∼ 50% of galaxies and are clearly seen in many redshift surveys (see, e.g., de Lapparent, Geller & Huchra 1988). These results extend the range of investigated scales in the galaxy distribution up to ∼ 100h −1 Mpc. Further progress in the study of the observed large scale galaxy distribution could be reached with the 2dF redshift survey (Colless 1998;Cannon 1998) and the Sloan Digital Sky Survey (Loveday & Pier 1998;Maddox 1998).
The formation and evolution of structure on large scales are investigated in numerous simulations (see, e.g., Cole et al., 1997Jenkins et al. 1998;Governato et al. 1998;Müller et al. 1998;Doroshkevich et al. 1999b, hereafter DMRT). These simulations are performed in large boxes (∼ 350 -500h −1 Mpc) and reproduce the main properties of the observed large scale matter distribution. In particular, they confirm formation of large wall-like matter condensations due to a nonlinear anisotropic matter compression on a typical scale ∼ (20 -30)h −1 Mpc that is about one half of the typical wall separation.
The statistical characteristics of wall formation are described by an approximate theoretical model (Lee & Shandarin 1998;Demiański & Doroshkevich 1999a, b, hereafter DD99) based on the Zel'dovich nonlinear theory of gravitational instability (Zel'dovich 1970(Zel'dovich , 1978Shandarin & Zel'dovich 1989). This approach relates the structure parameters with the main parameters of the underlying cosmological scenario and the initial power spectrum of density perturbations. The impact of large scale perturbations is found to be important throughout all evolutionary stages and some statistical characteristics of structure elements -filaments and walls formed in the course of nonlinear evolution -are directly connected with the parameters of these perturbations. Another theoretical model of large scale structure formation was discussed in Bond, Kofman & Pogosyan (1996).
The simulated large scale matter distribution does not exactly reproduce the theoretical expectations due to the influence of some essential factors, the most important ones are the small scale clustering and relaxation of compressed matter, and the large scale matter flow within sheet-like structure elements. Thus, compression of matter along one of the transversal directions transforms sheet-like elements into filaments, while expansion of matter in both transversal directions results in the erosion of pancakes. The disruption of walls and the small scale clustering of compressed matter substantially accelerate the relaxation and are responsible for strong matter concentration within walls. This is apparent from the isotropy of velocity dispersion within walls noticed in DMRT.
The combined influence of these (and other) factors complicates the statistical description of the large scale matter distribution at late evolutionary stages, what is typical for the final evolutionary stages of the standard COBEnormalized CDM (SCDM) model with Ωm = 1. For low density models, such as the open CDM (OCDM) model and the ΛCDM model with ΩΛ > Ωm, the situation is not so complex, and some statistical characteristics of structure can be successfully compared with the approximate theoretical expectations.
The investigation of wall-like massive structure elements is more promising in this respect because walls represent the first step in the process of structure formation and, so, hold more information about characteristics of the initial matter flow. Such walls are observed as superclusters of galaxies similar to the Great Wall (de Lapparent et al. 1988) and the Pisces-Perseus supercluster (Giovanelli & Haynes 1993). In simulations such wall-like structure elements are also easily identified because of their relatively high overdensity. Samples of such elements were investigated in DMRT and LCRS2. The connection between properties of walls and the amplitude and the spectrum of initial perturbations was discussed in DD99, and some of these results can be compared with measured properties of simulated wall-like structure elements. Examples considered in DD99 and DMRT had rather illustrative character, but they seem to be quite promising.
Here we will compare more accurately some of the expected and measured characteristics of wall-like matter condensations. We concentrate our attention on the physical aspects of the formation and evolution of the large scale matter distribution in order to better understand these processes and the phenomenon of wall-like matter condensations. Both theoretical and numerical estimates are inevitably approximate, but nevertheless, such comparison allows us to test the theoretical conclusions, to reveal and illustrate the influence of essential factors mentioned above, and to examine the abilities of statistical methods used to describe the large scale matter distribution.
These methods allow us to reveal, in particular, some differences in characteristics of the large scale matter distribution in the real and redshift spaces. Various aspects of this problem were widely discussed during the past decade (see, e.g., Kaiser 1987;Davis, Miller & White 1997;Hamilton 1998;Melott et al. 1998;Hui, Kofman & Shandarin 1999, Tadros et al. 1999). Here we show that the differences between characteristics of walls in the real and redshift spaces depend on the basic cosmological model and increase during the cosmic evolution. Characteristics of walls in the real and redshift spaces are almost identical for the low-density models, but they differ more strongly for the SCDM model.
We do not discuss the application of these methods to the observed galaxy catalogues, what is a much harder problem, due to the strong influence of selection effects and other factors. We will consider this problem in the future.
This paper is organized as follows: In Sec. 2 the basic notations are introduced. In Sec. 3 the statistical characteristics of wall-like structure elements in the Zel'dovich theory are presented. In Sec. 4 we consider the methods used to measure the required characteristics of matter distribution. Our results are presented in Secs. 5 & 6 where they are also compared with the theoretical expectations. Sec. 7 contains summary and a short discussion of our main results. Some technical details are given in Appendixes A.

STATISTICAL CHARACTERISTICS OF LARGE SCALE STRUCTURE
It is generally recognized that the formation of observed large scale structure is driven by the middle part of the power spectrum, p(k), with 0.2hMpc −1 ≥ k ≥ 0.01hMpc −1 (k is the comoving wave number), and it is weakly sensitive to the small and large scale perturbations. In many publications authors use an artificial smoothing of the spectrum to describe this process (see, e.g., Bardeen et al. 1986, hereafter BBKS, Coles et al. 1993). However, as was shown in DD99 it is possible to avoid this artificial smoothing if the process of structure formation is described in terms of the displacement, Si(q), and velocity rather than density field. Indeed, in contrast with the density field, the statistical characteristics of displacements are weakly sensitive to the small and large scale perturbations and are reasonably well described by the middle part of the initial power spectrum. Even the strong nonlinear matter clustering does not significantly influence the main characteristics of displacements and, so, such (approximate) description of structure holds during long period of structure evolution. Of course, this approach cannot describe the formation of gravitationally confined walls and their disruption into a system of high density clouds.
Bearing in mind these comments we will describe the structure parameters using characteristics directly connected with the displacement. One of them is the large scale amplitude of perturbations measured by the dispersion of displacements, Other convenient parameter is the coherent length of the displacement and velocity fields, lv, expressed through the moment m−2 of the initial power spectrum, p(k). Suitably defined coherent length lv provides simple expressions for the correlation functions of these fields and the basic characteristics of the large scale structure (DD99 and Sec. 3).

The Zel'dovich approximation
The Zel'dovich theory connects the Eulerian, ri, and the Lagrangian, qi, coordinates of fluid elements (particles) by the expression where z denotes the redshift, B(z) describes growth of perturbations in the linear theory, and the random vector Si or the random potential Φ characterize the spatial distribution of perturbations. The Lagrangian coordinates of a particle, qi, are its unperturbed comoving coordinates. The velocity of a particle can be found from (2.2) as where H is the Hubble constant (H0 = 100 · h km/s/Mpc). Analytical fits for the functions B(z) and β(z) were given in DD99. Approximately, at z ≪ 1, we have .
(2.4) 2.2 Main structure characteristics for the CDM-like power spectrum The standard CDM-like power spectrum with a Harrison -Zel'dovich large scale asymptote can be taken as a reasonable approximation of the initial power spectrum used in Zel'dovich' theory. Here A(z) is the amplitude of perturbations, T (x) is a transfer function and ργ & ρ rel are the densities of CMB photons and relativistic particles (photons, neutrinos etc.). For this spectrum the parameters lv and σs are expressed through the spectral moments, mj, as follows: For the CDM transfer function (BBKS) m−2 = 0.023, and the expressions for the scale lv and the characteristic masses of DM and baryonic components associated with the scale lv can be written more explicitly as Here Ω b is the dimensionless mean density of the baryonic component. The same characteristic scale, lv, as given by (2.7) can be used for the structure description as long as the Zel'dovich theory can be applied. More details can be found in DD99. The same approach can be used for other power spectra as well.

The amplitude of large scale perturbations
The large scale amplitude of perturbation as measured by A(z) in (2.5) and σs (2.1) can be successfully used to describe the structure evolution in the framework of the Zel'dovich theory. As was shown in DD99 it is convenient to use -together with σs -an effective dimensionless 'time', τ (z, Ωm, h), which is proportional to the large scale amplitude of perturbations and describes suitably the evolutionary stage reached in the model. This 'time' is similar to that used in the adhesion model (Shandarin & Zel'dovich 1989). As was noticed in DD99 the structure evolution shows strong features of self-similarity and is described by universal expressions depending on the dimensionless variables q/lv and τ . This is a direct consequence of the Zel'dovich approximation.
The 'time' τ can be measured by different methods, some of which are discussed below. It is sensitive to the sample under investigation and to the method of measurement. It can be used to quantify bias between spatial distributions of different objects, such as, for example, large scale bias between distributions of galaxies and the DM component.
The quadrupole component of the CMB anisotropy, TQ, the variance of density in a sphere with radius 8h −1 Mpc, σ8, and the velocity dispersion, σ vel are the more often used characteristics of the large scale amplitude. All these characteristics are proportional to each other, but their dependence on Ωm and h is different, and they are sensitive to matter distribution in different scales. Thus, the quadrupole component of CMB anisotropy characterizes the perturbations on scales comparable with the horizon, while the values σ vel and σ8 are more sensitive to the matter distribution in moderate and small scales.
The connection of these characteristics with σs and τ can be summarized as follows: (i) Using the fits for the CMB anisotropy proposed by Bunn & White (1997)  , ΩΛ = 0, where τT denotes the amplitude of large scale perturbations, τ , measured by the CMB anisotropy. These estimates depend on the spectral moment m−2 only which is very stable and does not change during the considered period of evolution. But the estimates should be corrected if a possible contribution of gravitational waves is taken into account.
(ii) The amplitude of perturbations, σs and τ , can be directly expressed through the two point autocorrelation function as follows: and for the autocorrelation function ξ(r) approximated by the power law Here r ξ is the first zero-point of the autocorrelation function and τ ξ denotes the amplitude τ measured by this function. The parameter r ξ is usually found with small precision, but for γ ≈1.5 -1.7, 1 − γ/2 ≈0.25 -0.15 even some variations of r ξ do not change significantly the final estimate of τ .
(iii) The parameter σ8 can be also expressed through the two point autocorrelation function, ξ(r), (Peebles 1993), and for ξ(r) approximated by a power law (2.11) we have (2.13) (2.14) (iv) The dispersion of the peculiar velocity of particles at small redshifts, z=0, can be written as in the linear theory (DD99) and for τ we obtain the independent estimate (2.16) Here τ vel denotes the amplitude τ measured by the velocity dispersion. σ vel takes into account also the high velocities generated by the gravitational compression of matter (in particular, within clusters of galaxies) and, so, it gives actually an upper limit of the amplitude.

STATISTICAL CHARACTERISTICS OF WALLS IN THE ZEL'DOVICH THEORY
In both observed and simulated catalogues, at small redshifts, the wall-like structure elements accumulate ∼ 50% of galaxies and form the skeleton of large scale structure. So, investigation of the characteristics of these elements is important in itself. It allows us also to obtain information about processes of nonlinear structure evolution. In particular, we can find two independent measures of the large scale amplitude, τ . As walls represent the first step of the large scale nonlinear matter compression their characteristics can be compared with predictions of the Zel'dovich theory. In this Section we will consider five characteristics of walls, namely, the surface density of walls, mw, defined as the amount of matter per unit of wall surface, for example, per h −2 Mpc 2 , the thickness of walls, hw, the wall separation, Dsep, the velocity dispersion of matter compressed within walls, ww, and the dispersion of wall velocities, σv. All these characteristics can be derived from the Zel'dovich theory (DD99) and can be found for simulated point distributions as well.

Formation of walls
Following DD99, we will consider the intersection of two fluid particles with Lagrangian coordinates q1 and q2 as the formation of a wall (Zel'dovich pancake) with the surface density mw = np |q1 − q2| where np is the mean particles density in the sample. In Zel'dovich theory statistical characteristics of such walls are described by the initial power spectrum (2.5) and can be expressed through the characteristic scale, lv, the surface density of wall, mw, or dimensionless surface density, qw = mw/lv/ np , and the 'time', τ , introduced in Secs. 2. To do this, the structure functions of the initial power spectrum can be used. For the standard SDM-like power spectrum (2.5) with the BBKS transfer function these functions were introduced in DD99.
Naturally, the theoretical considerations describe the idealized model of structure evolution. Thus, it uses the rigid wall boundary though in reality such boundaries are always blured. Other important factor is the compression and expansion of pancakes in transversal directions. These motions transform pancakes into filaments and/or lead to the dissipation of poor pancakes. They are not so important for rich walls but can change the wall surface density by a factor of 1.3 -1.5. The small scale clustering and relaxation of matter distorts also the measured characteristics of walls with respect to theoretical expectations.
These factors distort the actual power spectrum with respect to the used one and introduce differences between the expected and actually measured parameters of walls which however cannot be evaluated a priori. The actual power and limitations of this approach must be tested at first with Nbody simulations.

Surface density of walls
The most fundamental characteristic of walls is the surface density, mw. The approximate expression for the probability distribution function (PDF) of the pancakes surface density, mw, defined as above has been obtained in DD99 in the same manner as the well known Press-Schechter mass function. It characterizes the process of one dimensional matter compression and formation of wall-like pancakes as described by the Zel'dovich theory.
For Gaussian initial perturbations and the standard CDM-like power spectrum with the BBKS transfer function, it can be written as follows: where np is the mean particle density in the sample, lv is defined by (2.7), τm characterizes the amplitude of perturbations and the evolution stage of structures, τ , as measured by the surface density of walls, and q1 & q2 are Lagrangian coordinates of wall boundaries. This relation was corrected for the merging of neighboring walls and this process is described by the erf -function in (3.1). These expressions connect τm with the mean surface density of walls and allow us to estimate τm from measurements of qw . For other models and/or other distributions of initial perturbations the PDFs similar to (3.1) could be obtained using the technique described in DD99.

The wall separation
We have not been able to find a simple theoretical description of the wall separation. Nonetheless, taking into account the mainly one dimensional character of wall formation, we can roughly link the mean measured wall separation, Dsep , to the mean surface density of walls, qw .
Indeed, the matter conservation law along the direction of wall compression can be approximately written as follows: where fw is the matter fraction assigned to walls. It implies that on average a fraction fw of particles situated at the distance ±0.5Dsep from the center of wall will be collected by the wall. For simulations when the mean wall separation is comparable to the box size, L box , we will use the more accurate relation The averaging can be performed analytically assuming the exponential distribution function for the wall separation. The factor f dq defined by Eq. (3.2) characterizes the matter fraction assigned to walls as it is determined by comparison of independently measured characteristics qw and Dsep . In turn, difference between f dq and fw characterizes the robustness and degree of self consistency of the model and the measurements. These estimates are only approximations because the wall formation is actually a three dimensional process.

Velocity of structure elements
For the pancakes defined in Sec. 3.1 the 1D velocity of walls, vw, can be found from relations (2.2) and (2.3) as follows: where n is a unit vector normal to the wall. The small scale clustering and relaxation of compressed matter does not influence the velocities of walls and, so, they are the most stable characteristics of the evolutionary stage reached. As was shown in DD99 the mean velocity of walls, vw , is expected to be negligible as compared with its dispersions, σv, and the expected PDF of this velocity, Nv, is Gaussian for Gaussian initial perturbations. For the standard spectrum (2.5) with the BBKS transfer function, and for qw ≪ 1, the velocity dispersion is related to the amplitude of initial perturbations as follows: what is similar to (2.16) and also is identical to expectations of the linear theory. Here τv denotes the amplitude τ as measured by the dispersion of wall velocity and u0 was introduced by (2.15).

Velocity dispersion of matter compressed within walls
The variance of velocity of matter accumulated by walls, can be found in the framework of the Zel'dovich theory using the structure functions described in DD99. As is shown in Appendix A, it can be written as follows: In fact, this function characterizes the mean kinetic energy of particles compressed into a wall of a given size qw. After averaging over a sample of walls with the PDF Nm (3.1), in the Zel'dovich theory, we obtain The comparison of the expected mean kinetic energy of the compressed particles with the kinetic energy measured in simulations characterizes the mean degree of relaxation of compressed matter at a given τ . For richer walls, with qw ≫ qw , the relation (3.6) is transformed into and for such walls, the PDF is similar to (3.1). For a rich sample of walls, this relation can be also used for the direct measurement of the amplitude τ (DMRT; DD99).

Wall thickness
The methods discussed in DD99 allow us also, in the framework of the Zel'dovich theory, to obtain the expected thickness of walls along the direction of maximal compression, hw. It can be characterized by the thickness of a homogeneous slice with the same surface density. The corresponding expression (Appendix A) is This relation shows that the wall thickness is strongly correlated with its surface density. After averaging with the PDF (3.1) we obtain for the mean thickness of walls The degree of matter compression in the Zel'dovich theory, δz(q, τ ), is characterized by the ratio After averaging with the PDF (3.1) we have for the mean degree of matter compression So, in the Zel'dovich theory the averaged degree of matter compression is small.

Wall properties in the redshift space
In observed catalogues only the redshift position of galaxies along the line-of-sight is known, and therefore the parameters of observed structures with respect to those found above can differ due to the influence of the velocity field. The statistical characteristics of walls in redshift space predicted by the Zel'dovich theory can be found with the methods described above. This information is not so rich as in the real space because in the redshift space, positions of particles are determined by their velocities, and, for example, such a useful characteristic as the wall velocity cannot be found.

Surface density of walls
In the real space (Sec. 3.1) the pancake formation was defined as an intersection of particles with coordinates q1 & q2.
In the redshift space the velocity (2.3) along the line-of-sight must be used instead of the coordinate. In Zel'dovich theory the velocity dispersion exceeds the dispersion of displacement by a factor of (1+β). Hence, this substitution increases the wall surface density in the redshift space in respect to that in the real space, and now we must use Here the factor f rd ≥1 describes the more effective matter compression in the redshift space predicted by the Zel'dovich theory, β was introduced in (2.3), (2.4), and φ is a random angle between the direction of wall compression and the line-of-sight (0 ≤ φ ≤ π/2). Evidently, τ rd = τ for The PDF of wall surface densities in the redshift space is identical to (3.1) with a substitution of τ rd for τ , and now for the mean surface density of walls we have where τ characterizes the evolutionary stage as before. Probably, these relations can be used for the description of poorer pancakes and earlier evolutionary stages when the influence of other factors is less important. At small redshifts we must take into account the influence of the high velocity dispersion of compressed matter generated by the small scale matter clustering and relaxation. The influence of this factor, well known as the 'finger of God' effect, is opposite to that discussed above. It changes the observed particle position within walls along the line-of-sight what blurs the wall boundary and increases the thickness of observed walls. It artificially removes the high velocity particles from the selected wall and effectively decreases the surface density of walls selected in redshift space with respect to the estimates (3.13).
The impact of this factor can be approximately described by a modification of PDF of wall surface density, and a new normalization of distribution N rd m . The second term in the square brackets describes the artificial rejection of high velocity particles from the wall with a surface density qw bounded by a threshold density δ thr . In this term the exponent gives the fraction of matter accumulated by the wall in real space for some qw & τ , whereas W (qw, τ, δ thr ) is the fraction of high velocity particles which are removed from the wall in the redshift space. The function W (qw, τ, δ thr ) cannot be found in the Zel'dovich theory as it depends on the distributions of particles positions and velocities arising due to the small scale clustering and relaxation of matter compressed within walls.
An other factor which can suppress the expected difference of wall characteristics, measured in the real and redshift spaces at small redshifts, is the strong matter condensation within structure elements with various richnesses. The strong matter rearrangement transforms the continuous matter infall on walls into a discontinuous one, increases the separation of infalling structure elements, even in the redshift space and, so, at least partly, prevents the erosion of wall boundaries.
These comments show that in the redshift space the Zel'dovich theory with the factor f rd given by (3.12) & (3.13) overestimates the matter concentration within walls. Therefore, instead of the factor f rd in (3.13) a factor κ rd (Γ, τ, l thr ) should be used and the more realistic relation connects the amplitude τm with the wall richness qw in the redshift space.
The actual value of κ rd depends on the parameters of the cosmological model and on the method of identification of walls. The analysis performed below shows that for the walls selected in 3D space as described in Sec. 6.1, no growth of qw was found, and the parameters qw & τm are connected by the relation (3.1) as in the real space.

Wall separation
The separations of richer walls is not sensitive to relatively small shifts of particle positions introduced by the random velocities, but these shifts can result in an artificial merging of poorer walls. The influence of this factor can be tested with the relation (3.2) as before.

Velocity dispersion of matter compressed within walls and the wall thickness
In the redshift space the expression for the velocity dispersion of matter compressed within walls in the Zel'dovich theory is identical to (3.6) with a substitution of τ rd = τ κ rd instead of τ , but now it characterizes also the observed thickness of walls. For walls selected from the 3D sample of particles, as is described in Sec. 6.1, we have This value exceeds the corresponding real thickness of walls given by (3.9). The expected overdensity of compressed matter is given by

Core-sampling approach
The core-sampling approach was proposed by Buryak et al. (1994) for the analysis of the galaxy distribution in deep pencil beam redshift surveys. In the original form it allows to obtain the mean free-path between the filaments and walls. It was improved and described in detail in LCRS1 where some characteristics of the large scale galaxy distribution were found for the Las Campanas Redshift Survey. For simulated matter distributions as considered here these characteristics were discussed in DMRT. The potential of the core-sampling approach is not exhausted by these applications, and it could be used to measure parameters of the large scale matter distribution discussed in the previous Sections. Here we will use this approach to obtain the characteristics of the wall-like structure component.
The core-sampling method deals with a sample of points (galaxies) lying within relatively narrow cores -rectangular and/or cylindrical in simulations, and conical in observations -and it studies the point distribution along these cores. For some applications the transversal coordinates of points can be used as well. To take into account the selection effects, which are important for observed catalogues, appropriate corrections can be incorporated. The sampling core is characterized by the size, Dcore, that is the side of a rectangular core or the angular diameter of a conical core.

Measured characteristics of walls
Here we will apply the core sampling technique to the sample of wall-like structure elements selected by a 3D-cluster analysis (DMRT, Sec. 6.1). This means, the sampling cores contain only the particles assigned to walls. Further on, all particles are projected onto the core axes and are collected into a set of clusters with a linking length l link . Clusters with richness larger than a threshold richness, Nmin, are identified with walls within the sampling core.
The measured wall parameters are sensitive to the influence of small scale clustering of matter within walls. For strongly disrupted walls and a narrow core, the results depend on the random position of high density clumps, what strongly increases the scatter of measured wall properties. The influence of this factor is partly suppressed for larger sizes of the sampling core, Dcore.
However, the random intersection of the core with a wall boundary generates artificially poor clusters. The number of such intersections increases proportionally to Dcore what restricts the maximal Dcore. To suppress the influence of this factor a threshold richness of cluster, Nmin, was used. If however Nmin becomes too large, the statistical estimates become unreliable. For large Dcore the overlapping of projections of neighboring walls becomes also important what distorts the measured wall characteristics.
It is also important to choose an optimal linking length, l link , because for small l link , only the high density part of walls is measured, whereas for larger l link , again the impact of the random overlapping of wall projections becomes important.
The influence of these factors cannot be eliminated completely, and our final estimates of properties of walls are always distorted to some degree. These distortions can be minimized for an optimal range of parameters Dcore, Nmin & l link . Practically, these factors do not distort the velocity dispersion of walls, σv, which therefore provides the best characteristic of the actual evolutionary stage of the wall formation. On the other hand, the comparison of results obtained for different l link and Dcore allows to characterize the inner structure of walls.

Measurement and correction of wall parameters
The richness of clusters in the core measures the surface density of walls, where Nm is the number of particles in a cluster. The velocity of walls, vsim, the velocity dispersion of particles accumulated within walls, wsim, and the proper sizes of walls, hsim, are found as follows: Here ri, rw and ui are the coordinates of a particle, of a wall, and the velocity of a particle along the sampling core, respectively. The wall separation, Dsim, is measured by the distance between neighboring clusters.
The parameters msim, vsim, wsim and hsim as given by (4.1) and (4.2) are found along the sampling core and, so, are not identical to the parameters discussed in Sec. 3. These parameters must be corrected for the random orientation of walls with respect to the sampling core. The impact of this factor increases the measured surface density, and the corrected wall surface density, mc, is connected with the measured one by where φ is a random polar angle between the core and the vector orthogonal to the surface of the wall, and the averaging is performed in a spherical coordinate system. Corrected values of the wall velocity and the walls thickness are as follows: In the redshift space the wall thickness is connected with the velocity dispersion by Eq. (3.16). The velocity dispersion within walls was found to be almost isotropic (DMRT), and, so, we will use the measured wsim as the actual velocity dispersion across walls. The measured PDF of the wall surface density, Nm(mc), and the mean wall surface density, mc , are distorted due to the small statistics of rich walls and rejection of poor walls with a richness Nm ≤ Nmin. The correction for these distortions can be estimated by comparing the simulated PDF with the expected PDF (3.1).
To do this we will fit the measured PDF to the function (4.5) The parameter bm describes deviations of measured and expected mean surface density of walls msim , and am is a normalization factor. If the measured PDF is well fitted to the function (4.5) then the value mt = mc /bm (4.6) can be taken as a measure of the 'true' mean surface density of walls. Finally, the mean dimensionless surface density of walls, qw and the amplitudes of perturbations, τm & τv, measured by the surface density and velocity of wall-like structure elements, can be estimated as follows: The small statistics of rich and poor walls distorts also the measured wall separation, Dsep. The expected distribution of wall separations is exponential, and therefore it is possible to correct the mean separation using the fit of the measured PDF, Nsep(Dsim), to the function Nsep = asep exp(−bsepDsim/ Dsim ). (4.8) As before, the parameter bsep describes deviations of the measured and expected mean separation of walls, and asep is a normalization factor. If the measured PDF is well fitted to the function (4.8) then the value can be taken as a measure of the 'true' mean separation of walls.

Basic simulations
The theoretical model discussed above describes the evolution of the DM distribution and, so, should be tested with the simulated DM distribution as well. Here we use three simulations as a basis for our analysis -the COBE normalized standard CDM model (SCDM), a ΛCDM with ΩΛ > Ωm, and an open CDM (OCDM) model. These models were described and investigated with 3D cluster analysis and Minimal Spanning Tree technique in DMRT. It was found that the ΛCDM and OCDM models successfully reproduce the main observed characteristics of large scale matter distribution while the SCDM model demonstrates strong signatures of overevolution. Here we study these three models bearing in mind that only the ΛCDM and OCDM models can be considered as realistic models of the observed large scale matter distribution. The SCDM model represents the matter distribution typical for a late evolutionary stage. The simulations were performed with a PM code in a box of (500h −1 Mpc) 3 with (300) 3 particles for the Harrison-Zel'dovich primordial power spectrum and the BBKS transfer function. The force and mass resolutions are ∼ 0.9h −1 Mpc and ∼ 10 11 M⊙, respectively. The point distribution in redshift space was produced by adding an apparent shift to one coordinate due to the peculiar velocity of particles.
Four mock catalogues were prepared on the basis of the OCDM model with various degrees of large scale bias between the spatial DM distribution and the 'galaxies'. These mock catalogues were constructed by identifying randomly 'galaxies' with DM particles, but with a probability depending on the environmental density, thereby identifying more particles as 'galaxies' in high density regions (walls). These catalogues were investigated also both in real and redshift spaces.
The main characteristics of the simulations are listed in Table 1. A more detailed description can be found in DMRT.

Large scale amplitude of perturbations
The evolutionary stages reached in the models under discussion can be suitably characterized using the methods described in Sec. 2. The value τT listed in Table 1 characterizes the large scale amplitude used for the normalization of simulated perturbations. Other measures of the amplitude, such as σ8, τ ξ & τ vel , are sensitive to both the actually realized sample of random perturbations (cosmic variance) and to the nonlinear distortions of power spectrum produced during the evolution. For the considered mock catalogues these measures are also sensitive to the large scale bias between the spatial DM and 'galaxies' distributions what allows us to characterize it quantitatively. The spatial matter distribution and the bias between spatial distributions of DM component and 'galaxies' can be characterized by the correlation length, r0, and the slop of the correlation function, γ, introduced in (2.11). These parameters are listed in Table 1 for all samples. Using relations (2.12) and (2.13), these values allows to calculate σ8 and τ ξ , which are also listed in Table 1.
The characteristics of correlation function, r0 and γ, are sensitive to the perturbations in scales k ∼0.5 -0.1h Mpc −1 . As is seen from (2.12), estimates τ ξ are very sensitive to the value of 2−γ, and, so, to small scale perturbations. The first zero-point of autocorrelation function, r ξ ≈ 40h −1 Mpc, can be usually found with a large uncertainty (∼20 -30 per cent) but its impact is reduced by the small exponent 1−γ/2 ≤ 0.3 in (2.12).
For OCDM and ΛCDM models the impact of small scale matter clustering is moderate, and differences between τ ξ and τ vel are found to be ∼10 per cent. The differences between the same parameters and τT can be considered as a reasonable measure of simulated 'cosmic variance'. For these models differences between the parameter τ ξ calculated for the real and redshift spaces also do not exceed ∼ 10%. For the SCDM model both τ ξ and τ vel are distorted by the strong small scale clustering. This divergence indicates that for the SCDM model the successful application of methods discussed in Sec. 3 is also in question.
The progressive growth of τ ξ and σ8 for mock catalogues characterizes the degree of the large scale bias between the spatial distribution of DM component and 'galaxies'.

PROPERTIES OF WALL -LIKE STRUCTURE ELEMENTS
The main basic characteristics of walls were discussed in DMRT for three DM and four mock catalogues mentioned above both in the real and redshift spaces. In this Sec. the wall characteristics discussed in Sec. 3 are found with the core-sampling technique for the same simulations and the same samples of walls.

Selection of wall-like structure elements
The sample of wall-like structure elements was selected with the two-parameter method described and exploited in DMRT. It identifies the wall-like structure elements with clusters found using a threshold linking length, l thr , and a threshold richness, N thr . As usual, the boundary of the clusters is defined by the threshold overdensity, δ thr , which is connected with the threshold linking length by δ thr = n thr n = 3 4π n l 3 thr . (6.1) The threshold richness, N thr , restricts the matter fraction, fw, associated with walls. The main characteristics of these samples both in real and redshift spaces are listed in Table 2. The values of fw ≈ 0.4 -0.45 are consistent with the theoretically expected and observed matter fraction accumulated by walls (DD99, LCRS1, LCRS2). The analysis performed in DMRT shows that for the low density models the main characteristics of such wall-like elements are similar to the observed characteristics of superclusters of galaxies (Oort 1983 a,b; LCRS2; DURS).

DM walls in the real space
The analysis of DM catalogues in the real space is most interesting as in this case we can study the clear signal from the gravitational interaction of compressed matter and can reveal and characterize statistically the matter relaxation. Five basic characteristics of DM walls discussed in Sec. 3, namely, the wall thickness, hw, the dispersions of wall velocities, σv, the velocity dispersion of matter compressed within walls, ww, the dimensionless surface density, qw, and mean separation of walls, Dsep, can be found with the coresampling method and can be compared with those found in DMRT. The surface density of walls is closely connected with the size of proto-walls as discussed in DMRT.
Comparison of such characteristics of matter distribution as τ vel listed in Table 1 and τv and τm related to the wall properties allows us to test the influence of small scale matter clustering and other random factors discussed in Sec. 4.3, and to find the optimal ranges of core size, Dcore, and of threshold richness, Nmin, as well as the optimal linking length, l link . The results listed in Table 2 are obtained with the linking length l link = 5h −1 Mpc, and are averaged over 7 core sizes, 6h −1 Mpc≤ Dcore ≤ 9h −1 Mpc, and over 7 threshold richness, 10≤ Nmin ≤35 .

Basic characteristics of DM walls
For all models, the dispersion of wall velocities, σv, is found to be the best and most stable characteristic of the evolutionary stage reached. This is the direct consequence of the discrimination between the wall velocity and the velocity dispersion of particles compressed within walls. The PDFs, Nv, plotted in Fig. 1, are well fitted to Gaussian functions with the measured dispersion.
For the OCDM and ΛCDM models the mean dimensionless surface density of walls, qw , and the amplitudes, τm ≈ τv ≈ τ vel , are found with scatters ∼10 -15% for the used Nmin, Dcore, and l link . This scatter characterizes the moderate action of random factors discussed in Sec. 3.2 and the procedure of measurement. The values of lv qw are consistent with estimates of the size of proto walls obtained in DMRT. The PDFs of the surface density plotted in Fig. 2 are consistent with that expected form (3.1). These results demonstrate that for lower density cosmological models the Zel'dovich approximation successfully describes these basic characteristics of rich walls. For the SCDM model, the results listed in Table 2 are more sensitive to the method of measurement and the surface density of walls is underestimated, τm < τv ≈ τ vel . This difference can be mainly ascribed to the strong dis- ruption of walls occurring at late evolutionary stages in this model. Other important factors are the faster compression and/or expansion of walls in transversal directions and the existence of richer halos of evaporated particles around the walls mixed with infalling particles. Such a halo becomes richer for larger τ , i.e. for the ΛCDM, and especially, for the SCDM models.
The distribution function of wall separation, Nsep, plotted in Fig. 3 is well fitted to (truncated) exponential distribution. The mean wall separation Dsep is sensitive to the threshold richness Nmin and to the core size Dcore. The separation Dsep ∼ 40h −1 Mpc, found for the lower threshold richness, Nmin=5, and larger core sizes, Dcore = 9h −1 Mpc, coincides with the results obtained in DMRT. It increases with Nmin as the number of rich walls progressively decreases. For smaller Dcore and larger Nmin some of highly disrupted walls are lost due to their small covering factor. This parameter can be found with relatively large scatter.
Using relation (3.2) we can compare our estimates of Dsep and qw . For all models we have f dq ≈ (0.75 − 0.9)fw and the mean wall separation is probably overestimated.
For all models under consideration, the mean wall thickness, hw , is similar to that found in DMRT with the inertia tensor technique, where a wall is represented by a homogeneous ellipsoid. It is about of 2 -4 times smaller than that expected in the Zel'dovich approximation (3.10) what bears a sign of the relaxation of gravitationally bounded DM particles within walls. For the OCDM model the velocity dispersion of matter compressed within walls is found to be similar to the mean velocity of walls and of all particles, ww ∼ σv ≈ σ vel . In contrast, for the SCDM and ΛCDM models the dispersion ww is about 30% smaller than that obtained for the complete walls in DMRT and the dispersions σ vel and σv discussed above. This divergence characterizes statistically the evaporation of high energy particles in course of the relaxation of compressed matter and is reinforced by the procedures of measurement and wall selection. The relatively small value of ww demonstrates that in contrast to the clusters of galaxies the moderate degree of one dimensional matter compression within walls is not accompanied by an essential growth of velocity dispersion.

Relaxation of compressed matter
For ΛCDM and SCDM models the wall thickness, hw ∼ (3 -4)h −1 Mpc, is 2 -3 times smaller than that expected in the Zel'dovich theory (3.10). So large compression of matter within walls means that the selected particles are strongly confined and, probably, relaxed. For 1D matter compression the relaxation is expected to be weak, but in reality it is reinforced due to the small scale clustering and disruption of walls.
The degree of relaxation reached can be characterized by the parameters δ & ǫ , Table 2. Here δ measures the mean degree of matter compression, and ǫ is the mean kinetic energy of compressed particles with respect to the expectations of the Zel'dovich theory. The function wz(τ ) given by (3.7) is evaluated at τ = τv.
The divergence between the expectations of Zel'dovich theory and simulations is moderate for the OCDM model and becomes strong for the ΛCDM model as the evolution progresses. For the SCDM model the estimate of δ is artificially decreased together with qw . The small value of ǫ ∼ 0.1 -0.2 confirms an essential deficit of energy of compressed particles in comparison to that expected in the Zel'dovich theory. This deficit is partly enhanced by the procedure of wall selection, as the wall boundaries are blured, and particles placed far from the wall center are not included into walls.
In Zel'dovich theory the strong correlation of ww and hw with the wall richness, mw, is described by expressions (3.6) & (3.9). In simulations the measured linear correlation coefficients of qw, ww and hw are also ∼ 0.4 -0.5 what indicates that the essential mass dependence of these parameters remains also after relaxation. To discriminate the regular and random variations of functions ww and hw we will consider the reduced wall thickness, ζ, and the reduced velocity dispersion, ω, which can be defined as follows: In all considered cases the PDFs of the reduced velocity dispersion within walls, Nω, and of the reduced wall thick- Here δ thr is threshold parameters of clusters, fw is the fractions of all particles forming the selected walls, and fcr is the fraction of particles belonging to walls in both real and redshift spaces. Parameters τm, τv are the amplitude of perturbations as given by (3.1) & (3.4). The other quantities are explained in the text. Averaging was performed over 7 core sizes, 6h −1 Mpc≤ Dcore ≤ 9h −1 Mpc, and over 7 threshold richness, 10≤ N min ≤35 for the linking length l link = 5h −1 Mpc.
ness, N ζ , can be roughly fitted to Gaussian functions. The PDFs Nω are plotted in Fig. 4 for all three models. These results show that due to the strong relaxation of compressed matter the correlations between the considered characteristics of walls predicted by the Zel'dovich theory in Eqs. (3.6), (3.8) & (3.10) are replaced by relations (6.4) which are also universal.

DM walls in the redshift space
If the analysis of wall characteristics in the real space allows to reveal the influence of gravitational interaction of the compressed matter, then a similar analysis performed in the redshift space reveals the influence of random velocities on the observed characteristics of the large scale matter distribution.
In the redshift space the analysis of wall characteristics was performed for samples of walls selected as described in Sec. 6.1 . As was shown in DMRT, in low density cosmological models the main characteristics of these walls are similar to the observed characteristics of superclusters of galaxies. The determination of wall characteristics and their corrections are discussed in Sec. 4.3. The wall parameters were found in the same ranges of Dcore and Nmin as in the real space for l link = 5h −1 Mpc.
The main results are listed in Table 2 and are plotted in Figs. 2 -4.

Walls in the real and redshift spaces
The samples of walls selected in the real and redshift spaces are not identical with each other due to influence of random velocities of particles. This difference can be suitably characterized by the fraction of the same particles assigned to walls in both spaces. Here this fraction was defined as a ratio of number of the particles, Ncom, to the number of particles assigned to the selected walls, Nw. For all models under consideration this fraction, listed in Table 2, is Small variations of number of particles, Nw, assigned to walls in the real and redshift spaces lead to these variations.
These results indicate that the influence of high random velocities generated by the small scale wall disruption and the matter relaxation moderately distorts the sample of walls selected in the redshift space. More strong deviations between such wall parameters as the wall thickness and degree of matter compression, measured in the real and redshift spaces, are caused by the redistribution of matter within walls and the procedure of measurement rather than by the incorrect wall identification. The impact of these factors rapidly increases with τm and becomes extreme for the SCDM model.
These deviations can be sensitive to the code used for simulation (see, e.g., discussion in Splinter et al. 1998). For example, in the P 3 M code, these variations may increase due to larger velocities of compressed matter generated there.

Basic characteristics of DM walls
For all three models the mean surface density of selected walls listed in Table 2 is similar to that found in the real space. This fact shows that the artificial growth of matter concentration within walls discussed in Sec. 3.2 is effectively suppressed by the influence of the velocity dispersion and the procedure of wall selection, and the relation (3.1), as before, connects the mean surface density of selected walls, qw , with the amplitude, τm. Variations of qw and τm with Dcore and Nmin are shown in Table 2 as a scatter of these parameters. The PDFs Nm plotted in Fig. 2 are also similar to those found in the real space.
The mean wall separation is consistent with the estimate found in real space and, as before, for all models f dq ≈ (0.75 − 0.9)fw . The PDFs Nsep plotted in Fig. 3 are also similar to those found in the real space.
In the redshift space the used method of wall identification selects mainly particles with a small relative velocity what essentially restricts the measured velocity dispersion within walls and the wall thickness. Results listed in Table 2 show that only for the OCDM model, the velocity dispersion of compressed matter is consistent with the values found in the real space and in DMRT. For ΛCDM and SCDM models they are even smaller than those found in the real space. The measured wall thickness is now linked with the velocity dispersion by the relation (3.16).

Characteristics of matter relaxation
In the redshift space walls are less conspicuous than in the real space but, even so, for all three models the mean overdensity, δ , listed in Table 2, differs from the estimates based on the Zel'dovich theory (3.7). As in the real space, the velocity dispersion in the redshift space is strongly correlated with the surface density of walls, what is described by the relation (6.4) with an exponent pw ≈ 0.5. The PDFs of the reduced velocity dispersions, Nω, plotted in Fig. 4, demonstrate some excess of particles with lower ω, but it can also be roughly fitted to a Gaussian function with ω ≈1 and dispersion σω ≈ 0.4. This dispersion is about two times larger than that in the real space.
These results show that, although in the redshift space walls are not so conspicuous as in the real space, in the range of 'time' ∼ 0.2 ≤ τ ≤ ∼ 0.5, the relaxation of compressed dark matter can be directly recognized with these methods.

Walls in mock catalogues
The analysis of mock catalogues characterizes how the considered simple model of large scale bias influences the measured wall properties. These catalogues were investigated also both in the real and redshift spaces. The analysis was performed for 10 values Nmin (15≤ Nmin ≤60) and for 7 values of the core radius Dcore (7h −1 Mpc≤ Dcore ≤ 10h −1 Mpc) using a linking length l link = 5h −1 Mpc. The main results averaged over these Nmin and Dcore are listed in Table 2.

'Galaxy' walls in the real space
In the real space for all mock catalogues the parameters τv, hw and ǫw are similar to those found for the basic OCDM model. The velocity dispersion of 'galaxies' within walls, ww ≈ σv ≈ σ vel , exceeds that found for the basic model by about of 20 -30%. These variations can be attributed to the preferential identification of 'galaxies' in the central high density regions of walls, where the relative velocities of DM particles are also larger than the mean values. The wall thickness and the velocity dispersion of 'galaxies' can be reduced and turned into dimensionless quantities in the same manner as in Eq. (6.4), and the PDFs for the reduced wall thickness and velocity dispersion within walls, N ζ and Nω, are also similar to Gaussian functions. The PDFs Nω are shown in Fig. 5 for the mock4 catalogue.
As was expected, the mean surface density of walls, qw , exceeds that found for the basic OCDM model, and this excess progressively increases together with the biasing factor used. This excess can be considered as a suitable measure of the bias. This means that to characterize this bias the difference between τm and τv and/or between τm and other amplitudes measured for the same catalogues can be used together with the autocorrelation function. The growth of qw leads to a proportional growth of δ, as the wall thickness is only weakly distorted.
The large scale bias increases the contrast between richer and poorer walls what is seen as an essential growth of the mean wall separation. In all mock catalogues Dsep is about two times larger than in the basic OCDM model. The growth of both qw and Dsep do not distort the relation between fw and f dq .

'Galaxy' walls in the redshift space
In the redshift space the fraction of the same particles assigned to walls both in the real and redshift spaces becomes fcr ∼ 80 -85% (Table 2), what explains the similarity of parameters qw and τm listed in Tables 2 for both cases. The expected growth of wall richness in the redshift space according to (3.13) is not found, and the surface densities of walls, qw , are, in the range of errors, the same both in the real and redshift spaces. This fact shows that for 'galaxies' the expected growth of wall richness in the redshift space is suppressed even more strongly than for DM component due to the relaxation of compressed matter. Then Eq. (3.1) describes correctly the time dependence of the mean wall surface density.
The parameters hw & ww for 'galaxy' walls are similar to those found for the underlying OCDM model. The difference of ww found for the same samples of walls in the real and redshift spaces and a slow decrease of ww for stronger biased models can be assigned to the loss of a small fraction of particles with large velocities, what demonstrates the sensitivity of these functions to the method of wall identification.
In the redshift space we have not so a reliable independent estimator of the amplitude as τv. There the bias is seen as a relation of the amplitudes τm ≥ τ ξ . This makes it difficult to quantitatively estimate the relatively moderate large scale bias in observed catalogues because both τm & τ ξ are sensitive to the bias.
This discussion shows that the simple algorithm used in DMRT for the 'galaxy' identification does not essentially distort the basic characteristics of simulated walls, and a stronger bias can be seen as an excess of the surface density of 'galaxies' relative to that found for the DM component in the basic model. At the same time the mean velocity dispersions of both the DM component and the 'galaxies' assigned to walls, ww , tends to be smaller than σv & τv and other characteristics of the amplitude of perturbations.

SUMMARY AND DISCUSSION
In this paper we continue the investigation of large scale matter distribution and processes of large scale structure formation and evolution. Some aspects of these problems were discussed in our previous papers (LCRS-1, LCRS-2, DURS, DMRT, Müller et al. 1998) where the 3-dimensional analysis of the observed and simulated large scale structure was performed with the core-sampling and the Minimal Spanning Tree techniques. Another approach to this problem, based on the percolation technique, was discussed in Sahni et al. (1994), Shandarin & Yess (1998) and Sathyaprakash et al. (1998). The statistical description of structure formation and evolution based on the Zel'dovich theory of nonlinear gravitational instability can be found in Lee & Shandarin (1998) and DD99.
Here we direct our attention to the physical aspects of the process of wall formation, what implies a more detailed discussion of the properties of DM walls in real space. The simulations described and investigated in DMRT are used to test the theoretical expectations, to estimate the influence of small scale clustering and relaxation of compressed matter and other random factors, and to examine the power of the statistical methods used to describe the large scale matter distribution. Three cosmological models, at different evolutionary stages, were analyzed in the same manner, and the comparison of results obtained for these models allows us to estimate the properties of walls at various τ .
In the redshift space the influence of small scale clustering and large velocity dispersion of compressed matter noticeably distorts some characteristics of the walls. These distortions appear also in the considered mock catalogues, and can even be enhanced by the possible large scale bias between the spatial distribution of DM and galaxies.
Some of these results may depend on the code used for the simulations (see, e.g., the discussion in Splinter et al. 1998), and they should be checked with simulations employing a code with higher spatial resolution.

Identification of walls
The core-sampling approach described in Sec. 5 allows us to characterize, in more details, the matter distribution along the sampling core and to estimate the uncertainty in measured properties of wall-like condensations introduced by the influence of velocity dispersion and small scale clustering. The influence of these random factors is demonstrated by comparing results obtained with various Dcore, Nmin & l link .
Results presented in Sec. 6 show that some fraction of the early compressed matter has subsequently evaporated due to relaxation processes. These DM particles together with the infalling matter form an extended halo around the walls and, therefore, it is difficult to separate the walls from the background. The same problem is met by the correct definition of boundaries of galaxies and clusters of galaxies. It was also discussed in the DMRT, LCRS-2 and DURS, where the methods of wall selection, described in Sec. 6, were applied to simulated DM and observed galaxy distributions.
The central high density part of walls is reliably selected in all the cases, but various definitions of the wall boundaries can noticeably change the measured characteristics of walls. To provide more objective comparisons of wall characteristics the same dimensionless parameters fw and δ thr should be used for identification of walls in different catalogues and simulations.

Measured characteristics of walls
The results presented in Sec. 6 show that the core-sampling approach can be successfully used for the investigation and description of the large scale matter distribution and the wall-like matter condensations. It allows to estimate the surface density, thickness, velocity dispersion and other basic parameters of DM walls corrected for the influence of random curvature and shape of walls. These parameters differ from those obtained in 3D space with the Minimal Spanning Tree and inertia tensor methods, and these methods suitably complement each other.
The measured wall characteristics can be compared with predictions of the Zel'dovich theory what reveals the influence of relaxation of compressed matter on the properties of walls and allows to correct the theoretical expectations. The small scale clustering of compressed matter and the wall disruption lead to noticeable variations of measured wall characteristics for different parameters of the sampling core. These variations are not so large for the low density models, but they increase rapidly with τ .
The dimensionless surface density of walls, qw, is closely connected with the size of proto-walls as discussed in DMRT, LCRS2 and DURS. The high surface density of walls, qw ≥ 0.6, found above even for the low density models, demonstrates that processes of strong nonlinear matter evolution occur at a typical scale of ∼ qwlv ∼ (15 -25)h −1 Mpc. This evolution is correctly described by the Zel'dovich theory.
This characteristic is sensitive to the basic cosmological parameters, Ωm & h, what allows us to select the class of more perspective models for further investigation.

Relaxation of compressed matter
The problem of relaxation of compressed matter is now in the forefront, and the obtained results allow to begin discussion of the statistical characteristics of this relaxation. The analysis performed in the real space is more important for the discussion of the basic physical processes which occurred during the formation of wall-like matter condensations, such as the small scale matter clustering and the relaxation of the compressed matter. These processes generate the large velocity dispersion within walls and lead to the evaporation of high velocity particles. Thus, in all these cases an essential deficit of energy in DM walls as compared with the expectations of the Zel'dovich theory -∼ (50 -80)% and more -was found. The growth of this deficit with τ from the OCDM to SCDM models demonstrates that the DM relaxation becomes more and more important for later evolutionary stages, and its influence on the observed parameters of the large scale matter distribution becomes crucial for τ ≥ 0.5.
The relaxation is seen in rich superclusters of galaxies such as the Perseus-Pisces (Saslaw & Haque-Copilah 1998). It is essentially accelerated and amplified by the small scale clustering of compressed matter. This clustering is clearly seen in observations as, for example, a strongly inhomogeneous galaxy distribution within the Great Wall (Ramella et al. 1992). The clusters of galaxies situated within wall-like superclusters similar to the Great Wall and the Perseus-Pisces can be considered as extreme examples of this process.
The merging of earlier formed structure elements is very important for the formation of large walls (DD99). This means that actually the relaxation occurs step by step during all the evolutionary history beginning with the formation of first low mass, high density pancakes which later are successively integrated and merged to larger structure elements. This means also that the finally reached degree of relaxation and the properties of compressed matter depend on the (unknown) evolutionary history of the considered walls and, therefore, can be characterized only statistically.
The relaxation of compressed matter destroys the tight correlation between the surface density and velocity dispersion predicted by the Zel'dovich theory (3.6), but it generates other correlations between the same characteristics described by the relations (6.4). This fact indicates that the properties of compressed matter are sufficiently general, and these characteristics can be used to improve the methods of wall selection and the description of wall properties.
The velocity dispersion within walls increases gradually with τ from the OCDM to the SCDM model. As was discussed in Sec. 7, the particles with high velocity are gravitationally confined and occupy preferentially the high density central regions of walls. This fact confirms that, probably, these particles are relaxed and have a (quasi)stationary distribution. This distribution is not so stationary as, for example, in clusters of galaxies, and it is slowly evolving due to the large scale matter flow along the walls and the persisting merging of neighboring structure elements, but presumably, this evolution does not significantly distort the formed matter distribution.

DM walls in the redshift space
The matter condensation seen in the redshift space can be partly artificially enhanced by the influence of streaming velocities. The possible influence of this effect was widely discussed over the past decade (see, e.g., Kaiser 1987Davis, Miller & White 1997;Hamilton 1998;Hui, Kofman & Shandarin 1999) and, as applied to properties of absorption lines in the spectra of high redshift quasars, by McGill (1990b) and more recently by Levshakov & Kegel (1996. These tendencies are also clearly seen from the direct application of the Zel'dovich approximation to the wall formation in the redshift space as was discussed in Sec. 3.2. Of course, it is impossible to decide which particles belong to walls, but we can estimate statistically the properties of DM walls identified in the redshift space. However, the influence of this uncertainty cannot be separated from the influence of the relaxation and other factors discussed above. For all models the comparison of DM walls selected in the real and redshift spaces demonstrates, that they are composed mainly from the same particles -this fraction is about fcr ∼ (70 -80)% (Table 2). This means that in both cases we find the same walls, and the fraction of randomly added or lost particles is indeed small. In spite of this, some properties of walls in the redshift space are quite sensitive to the velocity dispersion and to the methods of wall identification. Thus, the strong growth of wall thickness -about a factor of 2 -confirms results obtained by Melott et al. (1998). This effect is quite similar to the well known 'finger of God' effect observed in clusters of galaxies.
The wall surface density, qw, is most interesting, as it is directly connected with the basic cosmological parameters, Ωm & h. Our analysis shows that for low density models -ΛCDM and OCDM -the measured value of qw is similar both in the real and redshift spaces. This means that the growth of the matter condensation within walls due to streaming velocities as predicted by the Zel'dovich theory is strongly suppressed by the influence of the matter relaxation and the transformation of a continuous matter infall to a discontinuous one. Actually similar relations connect the fundamental wall characteristics such as qw and τm.
The velocity dispersion within walls selected in the 3D redshift space can be noticeably underestimated what is a direct consequence of the method of wall selection. In the redshift space, particles with large velocities are artificially shifted to the periphery of selected walls and, so, can be omitted from the analysis.

Walls in mock catalogues
For the considered mock catalogues the influence of velocity dispersion is enhanced by the methods used for 'galaxy' selection. The large scale bias increases the 'galaxy' concentration within walls and, so, increases the density gradient near the wall boundary. When the 'galaxies' are identified preferentially in the high density central parts of the walls (in the real space), than their velocity dispersion exceeds that for the DM particles, and this excess may be as large as ∼ (20 -30)%. In the redshift space, the parameters of 'galaxy' walls such as hw and ww are similar to those in the underlying DM distribution.
The bias is clearly seen both in the real and redshift spaces as an excess of the mean surface density of walls. The comparison of parameters qw & τm found for observed wall-like galaxy condensations with possible independent estimates of the same parameters gives us a chance to obtain a reasonable observational estimates of the large scale bias.

The amplitude of large scale perturbations
These results demonstrate again that all characteristics of the amplitude and evolutionary stage of large scale structure considered in Secs. 2 & 3 are similar, but not identical to each other, as they are sensitive to different properties of perturbations. The best and most stable measure, τv, comes from measurements of the velocity of structure elements. It is insensitive to the nonlinear evolution of perturbations, large scale bias and small scale clustering or relaxation of the compressed matter.
The comparison of other estimates for the same parameter τ , namely, τ vel , τ ξ , & τm obtained in the same simulations demonstrates their sensitivity to various natural and artificial factors. For the low density models -ΛCDM and OCDM -the parameters τv and τm are usually sufficiently close to each other, what is a direct consequence of the close connection of the process of wall formation with the large scale perturbations. The parameter τm is sensitive to a possible large scale bias, but to reveal this factor, we need to have independent unbiased estimates of the same amplitude.
The most interesting independent estimate of the amplitude is τ ξ which is however more sensitive to small scale matter clustering. Thus, for the SCDM model where this clustering is stronger it significantly overestimates the large scale amplitude. It is less sensitive to the large scale bias than τm.
Independent estimates of the large scale amplitude come from measurements of the CMB anisotropy. The COBE data are consistent with other available estimates of cosmological parameters and of the large scale amplitude, and therefore, τT can be considered as the best estimate of the combination (2.9) of Γ and the amplitude. It can be connected with estimates of cosmological parameters Ωm ≈ 0.3, ΩΛ ≈ 0.7 obtained from observations of high-redshift supernovae (Perlmutter et al. 1998). Nonetheless, τT should be corrected for a possible contribution of gravitational waves.
The investigation of the space density of clusters of galaxies and its redshift evolution (see, e.g., Bahcall & Fan 1998;Eke et al. 1998;Wang & Steinhardt 1998) seems also to be promising and can give the required independent measure of the large scale amplitude. The formation and evolution of galaxy clusters is caused by large scale perturbations, and their characteristics can be connected with these perturbations. But they are sensitive to the thermal evolution of clusters and, moreover, are related to only ∼ (10 -15)% of matter accumulated by the clusters. This means that they are not free from random variations what is seen, in particular, as the well known variations of the autocorrelation function with the cluster sample.
The critical discussion of available measurements of cosmological parameters (Wang et al. 1999;Efstathiou 1999) shows that in spite of a large progress reached during last years, we do not have yet a reliable unbiased estimate of these parameters, and these data should be tested with respect to possible random large scale variations. The application of the discussed methods to large observed redshift surveys can help to achieve this goal.