# Percolation networks in three dimensions by shapes of different sizes

Percolation is something which I feel I understand somewhat intuitively, but it is quite complex as far as I've read, and therefore my expectations may be wrong. It's related to my research in materials science, so I'd like to grasp it better if at all possible. It's likely that general aspects of the solution are enough to answer my question, so I don't have to force anyone to go into the gory details. My mathematical education also stopped somewhere around intermediate-level calculus, so go easy on me please!

I am interested in percolation pathways between two parallel flat sheets, separated by a certain distance $d$ in empty 3D space. Imagine one were to start adding small non-intersecting flat planes of area $A$ (with largest "diameter" $\ll d$) in the empty space, with rotational freedom in any axis. Would there be a difference in the "speed" at which the distance would be percolated, if the size of the added planes were varied (though always smaller than the distance between the limiting parallel sheets)? For example, considering adding planes of a constant shape (such as squares of differing sizes), what would the relation between the average "occupation density" for a successful percolation to happen be, if we were to compare adding squares of area $A$ with squares of area $A/10$?

Equivalently (or at least I think it is), if we were to compare adding $x$ square planes of area $A$, and $10x$ square planes of area $A/10$, in which case would the probability of percolating the distance $d$ be higher? My gut feeling is that using a lower amount of larger planes makes it easier to percolate than a higher amount of smaller planes.

A detailed analysis of the problem in its full form is likely infeasible, so I assume that relaxed constraints are required, such as adding square planes to a 3D cubic grid. Is this approximation of the problem sufficient to provide reliable answers that can be transferred to the general version (i.e. adding arbitrary 1D/2D/3D shapes, arbitrary placement, arbitrary shape sizes, including largest "diameters" $\simeq d$ or even $>d$ )? Are these conclusions generally in line with those obtained for percolation of lines in 2D space, or do they fundamentally differ (such as with 2D vs. 3D random walks)?

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