I recently read the following "open problem" titled "Pennies on a carpet" in "An Introduction To Probability and Random Processes" by Baclawski and Rota (page viii of book, page 10 of following pdf), found here: <http://www.ellerman.org/Davids-Stuff/Maths/Rota-Baclawski-Prob-Theory-79.pdf>


"We have a rectangular carpet and an indefinite supply of perfect pennies. What is the probability that if we drop the pennies on the carpet at random no two of them will overlap? This problem is one of the most important problems in statistical mechanics. If we could answer it we would know, for example, why water boils at 100C, on the basis of purely atomic computations. Nothing is known about this problem.”


I was wondering if this problem goes by a more popular name and whether or not some form of progress has been made on it. In particular, references would be highly appreciated!



Pennies on a carpet problem
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](http://www-stat.stanford.edu/~cgates/PERSI/papers/MCMCRev.pdf). 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.


