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Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting
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Memoirs of the American Mathematical Society
1994; 100 pp; softcover
Volume: 112
ISBN-10: 0-8218-2597-6
ISBN-13: 978-0-8218-2597-6
List Price: US$40 Individual Members: US$24
Institutional Members: US\$32
Order Code: MEMO/112/537

Various notions of the Markov property relative to a partial ordering have been proposed by both physicists and mathematicians. For the most part, the analysis has focused on the study of some important, but special, examples. This work develops general techniques for studying Markov fields on partially ordered sets. The authors introduce random transformations of the index set which preserve the Markov property of the field. These transformations yield new classes of Markov fields starting from relatively simple ones. Examples include a model for crack formation and a model for the distribution of fibres in a composite material. Given the burst of popularity of random fields, this self-contained and accessible book will prove useful in the many scientific areas where random field models are appearing.

• Splitting lemma
• Honest random sets
• Splitting random sets. Constructions of splitting random elements
• Markov systems and their transformations
• Shift invariance
• Some special partially ordered sets and Markov fields
• Ferry problem
• The linearly ordered case
• Fibres
• Cracks and splitting random elements
• Transformations of fields indexed by contours
• Bibliographical notes
• References