 # Hammersley-Clifford theorem and maximal entropy random walk for Ising-like models?

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It seems that condensed matter people usually just brute force use Monte-Carlo, but there are some subtle mathematical tools which might be worth considering, for example here is fresh paper with Mathematica implementation calculating in seconds parameters for Ising-like models with many digits of accuracy, also probability distribution of patterns or allowing to generate new uncorrelated field with single scan.

1) Hammersley-Clifford theorem ( https://en.wikipedia.org/wiki/Hammersley%E2%80%93Clifford_theorem ) saying that Gibbs fields are equivalent with Markov fields, which allow to simplify models, e.g. through local Markov condition:
Pr(value in node | values in remaining nodes) = Pr(value in node | values in its neighbors)

2) Maximal entropy random walk ( https://en.wikipedia.org/wiki/Maximal_Entropy_Random_Walk ) provides probability distribution of patterns for Boltzmann ensemble of infinite sequences. Applying it to transition matrix (M_uv = exp(-beta E_uv)), while there are usually used its eigenvalues, here from its dominant eigenvector we get probability distribution of patterns e.g. in 2D Ising model:

Pr(u) = (psi_u)^2

Pr(u,v) = psi_u (M_uv / lambda) psi_v

Is there a literature applying any of them for Ising-like models?
Beside Monte-Carlo and molecular dynamics simulations, what interesting mathematical tools are used in modern condensed matter physics?

Diagram from the paper with concepts and plots of errors: ## Create an account

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