Nearly accurate solutions for Ising-like models using Maximal Entropy Random Walk
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While one-dimensional Markov processes are well understood, going to higher dimensions there are only a few analytically solved Ising-like models, in practice requiring to use costly, uncontrollable and inaccurate Monte-Carlo methods. There is discussed analytical approach for approximated problem exploiting Hammersley-Clifford theorem, which allows to realize generation of random Markov field through scanning line-by-line using local statistical model as in lossless image compression. While conditional distributions for such statistical model could be found with Monte-Carlo methods, there is discussed use of Maximal Entropy Random Walk (MERW) to obtain it from approximation of lattice as infinite in one direction and finite in the remaining. Specifically, in the finite directions we build alphabet of all patterns, find transition matrix containing energy for all pairs of such patterns, from its dominant eigenvector getting probability distribution of pairs in their infinite 1D Markov process, which can be translated to statistical model for line-by-line scan. Such inexpensive models, requiring seconds on a laptop for attached implementation and directly providing probability distributions of patterns, were tested for mean entropy and energy, getting maximal ≈ 0.02 error from analytical solution near critical point, which quickly improves to extremely accurate e.g. ≈ 10^-10 error for coupling constant J≈ 0.2.
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