My research interests lie in rigorous methods in probability, quantum theory, and statistical mechanics, especially as relates to critical phenomena, where the length over which local effects are felt tends to infinity. Examples include the self-avoiding walk, branched polymers, spin systems, random Schroedinger operators, many-body quantum systems, disordered systems, and quantum field theory (theories of elementary particles).

The mathematics of functional integrals plays an essential role in the analysis of such systems. The renormalization group method organizes the analysis into a sequence of more tractible problems associated with an increasing sequence of length scales. 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 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. 

Recent research on quantum many-body systems indicates that for large disorder, there is a failure of thermalization, at least in one dimension -- a dramatic breakdown of ergodicity, one of the key postulates of statistical mechanics. Nonperturbative effects could potentially spoil MBL -- and in fact De Roeck and Huveneers argued that this is exactly what happens in two or more dimensions. But in one dimension, I ruled out this possibility by demonstrating in a particular model that for strong disorder all eigenstates are many-body localized. In effect, each particle comes with its own conservation law, which prevents it from straying very far from its position. The proof depends on an assumption that the spacing between eigenvalues of the Hamiltonian is not too tight -- that is, there is no more than a limited amount of level attraction (or clumping) of the energy levels. This is a physically realistic assumption, since all known systems exhibit either repulsive or neutral statistics with no clumping.