Paper Info
Title
A simple condition implying rapid mixing of single-site dynamics on spin systems
Abstract
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 - \in, where \in 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 - \in 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 - \in. Our main result is that this criterion implies O(n log n) mixing time for the Glauber dynamics. As applications, we consider the Ising model, hard-core lattice gas model, and graph colorings, relating the mixing time of the Glauber dynamics to the maximum eigenvalue for the adjacency matrix of the graph. For the special case of planar graphs, this leads to improved bounds on mixing time with quite simple proofs.
Year
DOI
Venue
2006
10.1109/FOCS.2006.6
FOCS
Keywords
Field
DocType
spin system,graph colorings,ising model,hard-core lattice gas model,maximum eigenvalue,single-site dynamic,glauber dynamic,dual criterion,n log n,simple condition,adjacency matrix,influence matrix,computational complexity,graph coloring,graph theory,mixing time,eigenvalues,quantum computing,planar graph,statistical mechanics,planar graphs,gibbs measure
Adjacency matrix,Gibbs measure,Graph theory,Glauber,Discrete mathematics,Combinatorics,Matrix (mathematics),Ising model,Time complexity,Mathematics,Planar graph
Conference
ISSN
ISBN
Citations 
0272-5428
0-7695-2720-5
26
PageRank 
References 
Authors
1.52
5
1
Name
Order
Citations
PageRank
Thomas P. Hayes165954.21