The running time of such an algorithm depends on the rate of convergence to equilibrium of this Markov chain, as formalised in the notion of. .Series: Lectures in Mathematics. Paperback: 112 pages.

The running time of such an algorithm depends on the rate of convergence to equilibrium of this Markov chain, as formalised in the notion of "mixing time" of the Markov chain. A significant proportion of the volume is given over to an investigation of techniques for bounding the mixing time in cases of computational interest.

F. Martinelli, Lectures on Glauber dynamics for discrete spin models, Lectures on probability theory and statistics (Saint-Flour, 1997), Lecture Notes in Math

Levin, . Y. Peres, and E. Wilmer, Markov chains and mixing times, American Mathematical Society, 2008. F. Martinelli, Lectures on Glauber dynamics for discrete spin models, Lectures on probability theory and statistics (Saint-Flour, 1997), Lecture Notes in Math. vol. 1717, Springer, Berlin, 1999, pp. 123– 136. Martinelli, E. Olivieri, and R. Schonmann, For 2-D lattice spin systems weak mixing implies strong mixing, Comm.

The subject of the lecture series was counting (of combinatorial structures) and related . Sampling and counting. 26. Coupling and colourings.

The subject of the lecture series was counting (of combinatorial structures) and related topics, viewed from a computational perspective. As we shall see, "related topics" include sampling combinatorial structures (being computationally equivalent to approximate counting via efficient reductions), evaluating partition functions (being weighted counting) and calculating the volume of bodies (being counting in the limit). We shall be inhabiting a different world to the one conjured up by books with titles like Combinatorial Enumeration or Graphical Enumeration. 34. Canonical paths and matchings 49. 50.

Text comprised of notes originating in a postgraduate lecture series given by the author at the ETH in Zurich in the Spring of 2000. Covers the counting of combinatorial structures and related topics, viewed from a computational perspective. Download (djvu, . 2 Mb) Donate Read.

1 Two good counting algorithms. Perfect matchings in a planar graph. The class . A primal problem. Computing the permanent is hard on average. 3 Sampling and counting. Reducing approximate countingto almost uniform sampling. 4 Coupling and colourings. Colourings of a low-degree graph. Bounding mixing time using coupling.

Polynomial-time approximation algorithms for the Ising model Mark Jerrum and Alistair Sinclair, SIAM Journal on Computing 22.

Polynomial-time approximation algorithms for the Ising model Mark Jerrum and Alistair Sinclair, SIAM Journal on Computing 22 (1993), 1087-1116. Chapter 1 (with Zsuzsanna Lipták), Two good counting algorithms. Chapter 2 (with Uli Wagner)

Lectures in Mathematics ETH ZÃ rich. Birkhäuser Verlag, Basel, 2003. xii+112 pp. ISBN: 3-7643-6946-9.

Lectures in Mathematics ETH ZÃ rich. Mitzenmacher, Michael; Upfal, Eli. Probability and computing. Randomized algorithms and probabilistic analysis. Cambridge University Press, Cambridge, 2005. xvi+352 pp. ISBN: 0-521-83540-2.

Extension of Fill’s perfect rejection sampling algorithm to general chains (Extended abstract). One approach is to run an ergodic (. irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately, it can be difficult to determine how large M needs to be.