Randomly distribute objects over a surface with some clusters

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I want to randomly distribute some(in thousands) objects over a surface. This I can achieve with a function say x,y = rand().

This will evenly distribute objects over the surface, but is it possible to have some control over the randomness like having some groupings (some objects close together).

I can try to achieve this with multiple iterations of distribution, but that might run into objects overlapping.

So was looking for some kind of parameterized algorithm that might help in this area. I do not have much knowledge in mathematics so would appreciate some help.

Thanks!