Thomas C. Spencer

My research is focused on problems in analysis arising from mathematical physics. I am particularly interested in phase transitions in statistical mechanics and in the spectral properties of random band and random Schrödinger matrices. My colleagues and I have established the existence of phase transitions for many three dimensional lattice field theories for which the energy of a field configuration is invariant under a continuous symmetry. As temperature is lowered, the field configurations change from a spatially disordered to an ordered pattern. In two dimensions, J. Fröhlich and I proved that certain O(2) spin models and Coulomb gases exhibit the Kosterlitz-Thouless transition. My most recent research (with Disertori and Zirnbauer) proves a phase transition for a statistical mechanics model with a hyperbolic supersymmetry in three dimensions. This work is motivated by a simplified model of random band matrices. The phase transition for this model roughly corresponds a qualitative change of the spectral properties of random band matrices indexed by a three dimensional lattice as the band width is varied. Our proof relies on multi-scale analysis and numerous identities arising from symmetry.