Randomly coloring planar graphs with fewer colors than the maximum degree
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Rapid mixing of Gibbs sampling on graphs that are sparse on average
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
Journal of Combinatorial Theory Series B
Spectrally degenerate graphs: Hereditary case
Journal of Combinatorial Theory Series B
A Note on Degenerate and Spectrally Degenerate Graphs
Journal of Graph Theory
Dobrushin conditions for systematic scan with block dynamics
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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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.