My research tends to focus on the mathematical analysis of phenomena of interest in Statistical and Quantum Physics, and related developments in modern Probability Theory and Analysis. I find this mix of perspectives very stimulating, and enjoy seeing the different fields enriched through such contacts.
Areas of current and recent research include:
• Quantum spin chains: symmetry breaking, emergent features of their ground states,
Quantum <=> Classical relations which appear through functional integral representations.
• Critical phenomena in statistical mechnics, analized by uncovering the models' stochastic geometric scaffolding
(random currents, random clusters, loop models).
• The φ4d field theory, studied as the scaling limit of critical systems.
• Spectra and dynamics of random operators.
• Disorder effects in statistical mechanics.
Witht the notable exception of planar models, and models with mean-field interactions, the critical phenomena encountered in statistical mechanics lie beyond the reach of exact solutions and/or perturbative methods starting from the short-scale formulation of the theory. Yet there are rigorous methods which shed light on key aspects of the critical behavior even in such situations. A recurring theme in our work has been the appearance of random geometric structures which mediate the long distance correlations. The anlysis of their stochastic geometric properties yield insight on the essential differences in the critical behavior in basic models of statistical mechanics between low and high dimensions.
Another body of work has focused on the analysis of the effects of disorder on phase transitons, and on quantum spectra. For the latter, tools were developped for analyzing the phenomenon of Anderson localization. In the converse direction we found that resonant tunneling may, under certain conditions (in situations modeled by tree graphs), lead to unexpected persistence of delocalization