Pennies on a carpet problem

One variation that was solved by Baxter is the hard hexagon model, a discrete version where the pennies are hexagons and they are constrained to have their centers on the vertices of a triangular lattice. This example is rather famous because the solution involves the Rogers-Ramanujan identities. See Baxter's book for more details. As far as I know, the analogous "hard square model" has not been solved.

This is the two-dimensional hard spheres model, sometimes called hard discs in a box.

See Section 4 of Persi Diaconis's recent survey article, The Markov Chain Monte Carlo Revolution. The point here is that even though it very hard to sample a random configuration of nonoverlapping discs by dropping them on the carpet (because the probability of success is far too small for any reasonable number of discs), but it is nevertheless possible to sample a random configuration via Monte Carlo.

Rota (on page $viii$ of his introduction, page 10 of the pdf file) is talking about the difficulty of having an analytic solution for statistical mechanics in the 2-dimensional and 3-dimensional cases, while it is possible to attack the problem somewhat in the one-dimensional case.

He also mentions how stochastic methods and simulation can be used to come up with a quick-and-dirty approximation of the answers by modeling the physical sysyem and using Monte Carlo methods: iterating the system with random steps.

Topics to research would be Monte Carlo methods, stochastic models, random walks, etc. Can you say a little more about exactly what it is that you wish to study or examine?