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 $\mathbf{0}$ (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for coloring various classes of graphs. We give a general method for estimating a norm of a symmetric nonregular 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 nonregular graphs, trees, planar graphs and graphs with given tree-width and genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sampling Eulerian orientations of triangular lattice graphs

AbstractWe consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although Mihail and Winkler (1989) showed that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar graph which moves between orientations by reversing the edges of directed faces. Using path coupling and the comparison method we obtain a polynomial upper bound on the mixing time of this chain for any solid subgraph of the triangular lattice. By considering the conductance of the chain we show that there exist non-solid subgraphs (subgraphs with holes) for which the chain will always take an exponential amount of time to converge. Finally, we show that the problem of counting Eulerian orientations remains #P-complete when restricted to planar graphs (Mihail and Winkler had already established this for general graphs)

Markov Chains for Sampling Matchings

Markov Chain Monte Carlo algorithms are often used to sample combinatorial structures such as matchings and independent sets in graphs. A Markov chain is defined whose state space includes the desired sample space, and which has an appropriate stationary distribution. By simulating the chain for a sufficiently large number of steps, we can sample from a distribution arbitrarily close to the stationary distribution. The number of steps required to do this is known as the mixing time of the Markov chain. In this thesis, we consider a number of Markov chains for sampling matchings, both in general and more restricted classes of graphs, and also for sampling independent sets in claw-free graphs. We apply techniques for showing rapid mixing based on two main approaches: coupling and conductance. We consider chains using single-site moves, and also chains using large block moves. Perfect matchings of bipartite graphs are of particular interest in our community. We investigate the mixing time of a Markov chain for sampling perfect matchings in a restricted class of bipartite graphs, and show that its mixing time is exponential in some instances. For a further restricted class of graphs, however, we can show subexponential mixing time. One of the techniques for showing rapid mixing is coupling. The bound on the mixing time depends on a contraction ratio b. Ideally, b < 1, but in the case b = 1 it is still possible to obtain a bound on the mixing time, provided there is a sufficiently large probability of contraction for all pairs of states. We develop a lemma which obtains better bounds on the mixing time in this case than existing theorems, in the case where b = 1 and the probability of a change in distance is proportional to the distance between the two states. We apply this lemma to the Dyer-Greenhill chain for sampling independent sets, and to a Markov chain for sampling 2D-colourings

Path Coupling without contraction

Path coupling is a useful technique for simplifying the analysis of a coupling of a Markov chain. Rather than defining and analysing the coupling on every pair in Ω × Ω, whereΩ is the state space of the Markov chain, analysis is done on a smaller set S ⊆ Ω × Ω. If the coefficient of contraction β is strictly less than one, no further analysis is needed in order to show rapid mixing. However, if β = 1 then analysis (of the variance) is still required for all pairs in Ω × Ω. In this paper we present a new approach which shows rapid mixing in the case β = 1 with a further condition which only needs to be checked for pairs in S, greatly simplifying the work involved. We also present a technique applicable when β = 1 and our condition is not met