Monte Carlo without chains

Monte Carlo without Chains Alexandre J. Chorin Department of Mathematics, University of California and Lawrence Berkeley National Laboratory Berkeley, CA 94720 Abstract A sampling method for spin systems is presented. The spin lattice is written as the union of a nested sequence of sublattices, all but the last with conditionally independent spins, which are sampled in succession using their marginals. The marginals are computed concurrently by a fast algorithm; errors in the evaluation of the marginals are offset by weights. There are no Markov chains and each sample is independent of the previous ones; the cost of a sample is proportional to the number of spins (but the number of samples needed for good statistics may grow with array size). The examples include the Edwards-Anderson spin glass in three dimensions. Keywords: Monte Carlo, no Markov chains, marginals, spin glass Introduction. Monte Carlo sampling in physics is synonymous with Markov chain Monte Carlo (MCMC) for good reasons which are too well-known to need repeating (see e.g. [2],[17]). Yet there are problems where the free energy landscape exhibits multi- ple minima and the number of MCMC steps needed to produce an independent sample is huge; this happens for example in spin glass models (see e.g. [13],[16]). It is therefore worthwhile to consider alternatives, and the purpose of the present paper is to propose one. An overview of the proposal is as follows: Consider a set of variables (“spins”) located at the nodes of a lattice L with a probability density P that one wishes to sample. Suppose one can construct a nested sequences of subsets L 0 ⊃ L 1 ⊃ · · · ⊃ L n with the following properties: L 0 = L; L n contains few points; the marginal density of the variables in each L i is known, and given values of the spins in L i+1 , the remaining variables in L i are independent. Then the following is an effective sampling strategy for the spins in L: First sample the spins in L n so that each configuration is sampled with a frequency equal to its probability by first listing all the states of the spins in L n and calculating their probabilities. Then sample the variables in each L i−1 as i decreases from n − 1 to zero using the independence of these variables, making sure that each state is visited with a frequency equal to its probability. Each state of L = L 0 is then

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