Long-range classical spin systems
My main line of work concerns phase transitions in long-range lattice models. A recurring problem is that interactions couple distant regions strongly enough that standard nearest-neighbour contour arguments no longer localize cleanly. I use multiscale contours, inspired by the Fröhlich–Spencer method, to recover the geometric and energetic control needed for Peierls arguments and cluster expansions.
This program includes long-range Ising, Potts, and clock models; random fields and decaying external fields; and the low-temperature decay of truncated correlations. I am also interested in probabilistic representations that can extend these methods beyond the regimes accessible to direct contour estimates.
Quantum statistical mechanics
My second research direction concerns equilibrium states of quantum spin systems. In classical statistical mechanics, the Dobrushin–Lanford–Ruelle equations describe equilibrium through conditional distributions and boundary conditions. I study quantum analogues of this viewpoint and their relation with the KMS condition.
This work uses operator algebras, groupoids, and path or Poisson representations to formulate boundary conditions for quantum systems. The broader aim is to develop a framework in which classical and quantum equilibrium can be compared through a common specification-theoretic language.
Methods and related interests
- Contour methods and Pirogov–Sinai theory
- Cluster expansions and correlation decay
- Random fields and disordered systems
- Gibbs specifications, DLR equations, and KMS states
- Probabilistic and groupoid representations of quantum systems