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We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has 2d daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities and interaction strengths differ according to whether the relevant bond is between mother and daughter or between neighbours. Bounds are established which locate phase transitions and show the existence of a coexistence phase for the percolation model. Results are extended to the corresponding Ising model using the Fortuin-Kasteleyn random-cluster representation.

It is shown that the set of Markov random fields and Gibbs states with nearest neighbour potentials are the same for any finite graph. The set of Markov random fields is also shown to be the same as the equilibrium states of time-reversible birth/death processes with nearest neighbour interactions defined on the graph.

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Ising models and multiresolution quad-trees

Generalized Gibbs states and Markov random fields

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2 results

Ising models and multiresolution quad-trees

Generalized Gibbs states and Markov random fields