Pennies on a carpet problem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:40:55Z http://mathoverflow.net/feeds/question/37438 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37438/pennies-on-a-carpet-problem Pennies on a carpet problem Alex R. 2010-09-01T22:42:15Z 2010-09-02T07:02:32Z <p>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: <a href="http://www.ellerman.org/Davids-Stuff/Maths/Rota-Baclawski-Prob-Theory-79.pdf" rel="nofollow">http://www.ellerman.org/Davids-Stuff/Maths/Rota-Baclawski-Prob-Theory-79.pdf</a></p> <p>"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.”</p> <p>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!</p> http://mathoverflow.net/questions/37438/pennies-on-a-carpet-problem/37440#37440 Answer by hannah-abel for Pennies on a carpet problem hannah-abel 2010-09-01T23:27:03Z 2010-09-01T23:27:03Z <p>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. </p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/37438/pennies-on-a-carpet-problem/37442#37442 Answer by Gerry Myerson for Pennies on a carpet problem Gerry Myerson 2010-09-01T23:54:26Z 2010-09-01T23:54:26Z <p>I suppose one could phrase the question this way: given $r$, $s$, $n$, and $\epsilon$, what is the probability that of $n$ points, selected uniformly and independently at random from a rectangle of dimensions $r$ by $s$, no two will be within $\epsilon$ of each other. </p> <p>In the theory of uniform distribution modulo one, there is a concept of discrepancy which can be brought to bear on this problem. I hope that gives you a few keywords to look for. </p> http://mathoverflow.net/questions/37438/pennies-on-a-carpet-problem/37452#37452 Answer by Richard Borcherds for Pennies on a carpet problem Richard Borcherds 2010-09-02T02:20:32Z 2010-09-02T02:20:32Z <p>One variation that was solved by Baxter is the <a href="http://en.wikipedia.org/wiki/Hard_hexagon_model" rel="nofollow"> hard hexagon model</a>, 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 <a href="http://en.wikipedia.org/wiki/Rogers-Ramanujan_identities" rel="nofollow">Rogers-Ramanujan identities</a>. See <a href="http://tpsrv.anu.edu.au/Members/baxter/book" rel="nofollow">Baxter's book</a> for more details. As far as I know, the analogous "hard square model" has not been solved.</p> http://mathoverflow.net/questions/37438/pennies-on-a-carpet-problem/37465#37465 Answer by Matthew Kahle for Pennies on a carpet problem Matthew Kahle 2010-09-02T07:02:32Z 2010-09-02T07:02:32Z <p>This is the two-dimensional hard spheres model, sometimes called hard discs in a box.</p> <p>See Section 4 of Persi Diaconis's recent survey article, <a href="http://www-stat.stanford.edu/~cgates/PERSI/papers/MCMCRev.pdf" rel="nofollow">The Markov Chain Monte Carlo Revolution</a>. 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.</p>