Nearly accurate solutions for Ising-like models using Maximal Entropy Random Walk

Abstract

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 relatively costly, uncontrollable and inaccurate Monte-Carlo methods. There is discussed analytical approach for e.g. $width\times \infty$ approximation of lattice, also exploiting Hammersley-Clifford theorem to generate random Gibbs/Markov field through scanning line-by-line using local statistical model as in lossless image compression. While its conditional distributions could be found with Monte-Carlo methods, there is discussed use of Maximal Entropy Random Walk (MERW) to calculate them from approximation of lattice as infinite in one direction and finite in the remaining. Specifically, in the finite directions there is built alphabet of all patterns, then transition matrix containing energy for all pairs of such patterns is built, from its dominant eigenvector getting probability distribution of pairs of patterns in Boltzmann distribution of their infinite sequences, which can be translated into local 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 per node, getting maximal $\approx 0.02$ error from analytical solution near critical point, which quickly improves to extremely accurate e.g. $\approx 10^{-10}$ error for $J\approx 0.2$.