Matrix norms and rapid mixing for spin systems

We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is 0 (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for colouring various classes of graphs. We give a general method for estimating a norm of a symmetric non-regular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as non-regular graphs, trees, planar graphs and graphs with given tree-width and genus.