Ising models and multiresolution quad-trees

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 2 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.