My research concerns topics which present challenges and opportunities for Analysis and Probability Theory, which are however motivated by their relevance for Physics.

The work has benefitted from the mix of perspectives which the different fields provide and an appreciation of the links between their basic concepts and problems.

Specific areas of current research include:

• Spectra and Dynamics of Random Operators
• Critical behavior in critical dimensions    (with possible implications for specific field theories)
• Disorder effects in Statistical Mechanics 

Somewhat more specifically:

• A recent work, done in collaboration with S. Warzel, has shown that a relevant phenomenon for the transition from localization (and pure point spectrum) to continuous spectrum of tree graph operators is the appearance of `resonant tunneling' between would-be localized states. This is different from the `continuity' reasons for the persistence of extended states under weak disorder, which is the mechanism behind the previously obtained proofs of extended states in a perturbative regime. Such resonant delocalization may play a significant role in conduction within systems of many particles.
  
  The resonance mechanism is currently being explored further both within the context of graphs of exponential volume growth and of systems of many particles.

Typically, many of the interesting phenomena encountered in statistical mechanics are beyond the reach of exact solutions and perturbative methods. My goal has been to develop rigorous methods which can shed light on the critical behavior even in such situations. A recurring theme has been the appearance of stochastic geometric effects which play important roles in the behavior of critical system. The work has contributed to the nascence of the current field of random fractal structures which capture the scaling limits of many critical two dimensional systems, and which bear close relations with conformal field theories. In previous works, some with done with collaborators, we have rigorously established the existence of model-dependent upper critical dimensions, above which the critical behavior simplifies.