By Daniel J. Velleman
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Additional info for American Mathematical Monthly, volume 117, May 2010
Kindermann and J. L. Snell, Markov Random Fields and Their Applications, American Mathematical Society, Providence, RI, 1980. 45. S. Kirkpatrick, C. D. Gelett, and M. P. Vecchi, Optimization by simulated annealing, Science 220 (1983) 671–680. 671 46. N. Metropolis, The beginning of the Monte Carlo method, Los Alamos Science 15 (1987) 125–130. 47. N. Metropolis and F. H. Harlow, Computing and computers: Weapons simulation leads to the computer era, Los Alamos Science 12 (1983) 132–141. 412 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 48.
M. Jerrum, A. Sinclair, and E. Vigoda, A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries, J. ACM 51 (2004) 671–697. 1008738 41. D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon, Optimization by simulated annealing: An experimental evaluation; Part I, Graph partitioning, Oper. Res. 37 (1989) 865–892. 865 42. , Optimization by simulated annealing: An experimental evaluation; Part II, Graph coloring and number partitioning, Oper. Res. 39 (1991) 378–406.
Around the same time, Rosenthal  described a method (called minorization) which gives explicit a priori polynomial bounds on the number of iterations needed to ensure satisfactory convergence. There has been much work within the computer science community to understand the convergence properties of MCMC methods. Of particular note is Jerrum and Sinclair’s results on polynomial-time bounds for mixing times for the Metropolis algorithm applied to the Ising model , counting , and permanents of matrices .