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Ising models and multiresolution quad-trees

Part of: Equilibrium statistical mechanics Inference from stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

W. S. Kendall  and
R. G. Wilson
W. S. Kendall*
Affiliation:
University of Warwick
R. G. Wilson*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: [email protected]
∗∗ Postal address: Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

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.

Keywords

Gibbs stateIsing modelfinite island propertyimage analysisMarkov random fieldmultiresolution Markov random fieldpercolationquad-treerandom-cluster modelsegmentationunique infinite cluster

MSC classification

Primary: 62M40: Random fields; image analysis
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 96 - 122
Copyright
Copyright © Applied Probability Trust 2003 

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