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Having a 2D map filled uniformly by random values (Figure:top-left), the next maps are kernel averaged with a kernel of sizes 3x3, 5x5, ..., 11x11.
The questions are:
What are the patterns appeared in the kernel averaged maps?
What is the statistical basis of creation of such clusters/pattern?
How is the pattern related to the original dispersion model i.e., uniform?

enter image description here

Note:
To generate the maps a kernel based averaging system kind C was applied on the original data.

An answer from a different aspect has been given at this link. If you can develop based on more statistical/spatial statistics please do it.

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Smoothing is in general an implementation of diffusion, so what you see are more less the frames of this noise evolving into constant value (plus an evidence of fixed boundary condition). – mbq♦ Oct 9 '11 at 18:07
@mbq Could you please provide me some links to the diffusion concept with emphasizing on the points you mentioned? – Developer Oct 10 '11 at 9:12
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You can look around with keywords like heat equation or diffusion equation -- it is not the same, but can lead you to some clues. About patterns, you could look into CG papers about making noise textures (Perlin noise and following stuff). Or also ask this on Physics.SE, people from statistical mechanics may know something about it. You are asking very hard questions though. – mbq♦ Oct 10 '11 at 11:23
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@mbq Actually, this is a simple GIS/time-series question with a straightforward answer. Because the kernel is square and not circular, it's really a pair of independent "diffusions" in 1D, where its analysis (as a moving average) is a fundamental part of time series methodology. – whuber♦ Oct 10 '11 at 15:02
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@mbq I don't know what you mean by "characteristics" or by "density patterns"! The illustration shows moving averages of uniform noise using a sequence of $k\times k$ square kernels; those are straightforward to characterize as spatial random processes. The question hints at a generalization whereby some generic spatial process is smoothed in this way; the relationships among the process and its smooths are again straightforward to describe, especially in terms of the multivariate distributions to which the process gives rise. – whuber♦ Oct 10 '11 at 18:28
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