A simple condition implying rapid mixing of single-site dynamics on spin systems

Spin systems are a general way to describe local interactions between nodes in a graph. In statistical mechanics, spin systems are often used as a model for physical systems. In computer science, they comprise an important class of families of combinatorial objects, for which approximate counting and sampling algorithms remain an elusive goal. The Dobrushin condition states that every row sum of the “influence matrix” for a spin system is less than 1 − ε, where ε > 0. This criterion implies rapid convergence (O(n log n) mixing time) of the single-site (Glauber) dynamics for a spin system, as well as uniqueness of the Gibbs measure. The dual criterion that every column sum of the influence matrix is less than 1 − ε has also been shown to imply the same conclusions. We examine a common generalization of these conditions, namely that the maximum eigenvalue of the influence matrix is less than 1 − ε. Our main result is that this criterion implies O(n log n) mixing time...

• Type: Conference
• Year: 2006
• Where: FOCS
• Authors: Thomas P. Hayes
• Keywords: FOCS 2006, Influence Matrix, Mixing Time, Spin System, Theoretical Computer Science