Certain questions in physics can best be addressed with a theorem-oriented approach, especially if heuristic methods give conflicting indications. One example is the Ising model in a random magnetic field, which had conflicting heuristic descriptions with different predictions for the existence of long-range order in three dimensions. By proving that the ground state of the system did possess long-range order, I was able to settle this question definitively.

The idea of dimensional reduction, in which certain d-dimensional models are connected with related (d-2) dimensional models, is an attractive concept which originates from a supersymmetry of the problem. Although my work on the random-field Ising model showed that dimensional reduction does not work there, my recent work with Brydges shows that dimensional reduction works for branched polymers, and this leads to exact results on that problem, including critical exponents in 2, 3, and 4 dimensions.