Algorithms and Computation

Volume 2906 of the series Lecture Notes in Computer Science pp 663-675

Rapid Mixing of Several Markov Chains for a Hard-Core Model

  • Ravi KannanAffiliated withDepartment of Computer Science, Yale University
  • , Michael W. MahoneyAffiliated withDepartment of Mathematics, Yale University
  • , Ravi MontenegroAffiliated withSchool of Mathematics, Georgia Institute of Technology

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The mixing properties of several Markov chains to sample from configurations of a hard-core model have been examined. The model is familiar in the statistical physics of the liquid state and consists of a set of n nonoverlapping particle balls of radius r * in a d-dimensional hypercube. Starting from an initial configuration, standard Markov chain monte carlo methods may be employed to generate a configuration according to a probability distribution of interest by choosing a trial state and accepting or rejecting the trial state as the next configuration of the Markov chain according to the Metropolis filter. Procedures to generate a trial state include moving a single particle globally within the hypercube, moving a single particle locally, and moving multiple particles at once. We prove that (i) in a d-dimensional system a single-particle global-move Markov chain is rapidly mixing as long as the density is sufficiently low, (ii) in a one-dimensional system a single-particle local-move Markov chain is rapidly mixing for arbitrary density as long as the local moves are in a sufficiently small neighborhood of the original particle, and (iii) the one-dimensional system can be related to a convex body, thus establishing that certain multiple-particle local-move Markov chains mix rapidly. Difficulties extending this work are also discussed.