CS 298-2
Theory Seminar
Dror Weitz
UC Berkeley
Spin systems are a class of models that originated in Statistical Physics, though interest in them has since expanded to many other areas, including Probability Theory, Statistics, Artificial Intelligence, Communication, and Theoretical Computer Science. A spin system consists of particles which interact locally, giving rise a global equilibrium distribution.
This dissertation studies relationships between fast convergence to equilibrium (mixing in time) of a natural local process (the Glauber dynamics), and decay of correlations with distance in the equilibrium distribution (mixing in space).
We generalize tools for establishing both types of mixing (Dobrushin type conditions), give combinatorial proofs of direct relationships between the two for systems on the integer lattice, and establish new relationships between mixing in time and space for systems on trees. The latter allows us to significantly extend the regime of parameters for which the dynamics is known to mix in $n\log n$ time for various models on trees including the Ising model, independent sets (hard-core model), and colorings.