Education.

In mathematics, problems with exact solutions are like gold nuggets. They often allow one to illustrate the fundamental behavior of a class of problems with elegance and simplicity, and at a level the neophyte can understand. One such nugget is described in this issue's Education paper, "Critical Percolation on a Bethe Lattice Revisited" by Braga, Sanchis, and Schieber. Percolation theory lives at the intersection of graph theory and probability. The following is a typical percolation theory question: Given a porous material (such as a sponge) with a set of pores connected by channels that are randomly chosen to be open or closed, what is the probability that a pore in the center is connected to the outside boundary? (That is, what is the probability that fluid can percolate from the outside boundary to the central pore?) Questions of this type arise in numerous applications, including modeling the spread of an epidemic or a forest fire, the flow of groundwater, and the formation of polymer networks. Our paper considers percolation theory on a Bethe lattice (of which a binary tree is perhaps the most familiar example). The authors build the theory from the ground up and describe a beautiful example of what physicists call a phase transition and the associated concepts of critical probability and critical exponents. Moreover, in this model the authors describe how to compute these quantities exactly. The paper also illustrates how percolation theory can become particularly subtle as the domain size becomes infinite. This material could naturally be introduced into the undergraduate curriculum as a case-study in a discrete mathematics or a graph theory course. The paper also may kindle interest in what is an active and applied area of graduate study and research. On a more personal note, this issue commences my term as Editor of the Education section. Fortunately, my predecessor, Bobby Schnabel, is not far away - I wish him every success in his new role as Editor-in-Chief of SIAM Review. [ABSTRACT FROM AUTHOR]