I am indebted to Terence Tao for this highly comprehensible Executive Summary:
When one heats an iron bar magnet above a certain special temperature the Curie temperature the iron bar will cease to be magnetised; when one cools the bar again below this temperature, the bar can once again spontaneously magnetise in the presence of an external magnetic field. This phenomenon is still not perfectly understood; for instance, it is difficult to predict the Curie temperature precisely from the fundamental laws of physics, although one can at least prove that this temperature exists. However, Chayes, McKellar, and Winn have shown that for a certain simplified model for magnetism (known as the Ashkin-Teller model), the Curie temperature is equal to the critical temperature below which percolation can occur; this means that even when the bar is unmagnetised, enough of the iron atoms in the bar spin in the same direction that they can create a connected path from one end of the bar to another. Percolation in the Ashkin-Teller model is not fully understood either, but it is a simpler phenomenon to deal with than spontaneous magnetisation, and so this result represents an advance in our understanding of how the latter phenomenon works.
The advantage of the Ashkin-Teller model is that its one-dimensional nature makes it possible to solve exactly, even if its correspondence to reality is occasionally a trifle askew.
The Chayes-McKellar-Winn paper is available in PDF form from McKellar’s Web site, and this is the McKellar in question:
I do not claim to understand all the mathematics involved. I am delighted, however, that she does: by now Danica McKellar might be better known for her several math books than for playing Winnie Cooper in the not-quite-forgotten TV series The Wonder Years. (She’s thirty-seven years old today.)