# Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting

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*I. V. Evstigneev; P. E. Greenwood*

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.

#### Table of Contents

# Table of Contents

## Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting

- Contents v6 free
- Introduction 18 free
- 1. Splitting lemma 714 free
- 2. Honest random sets 815
- 3. Splitting random sets. Constructions of splitting random elements 1421
- 4. Markov systems and their transformations 2330
- 5. Shift invariance 3037
- 6. Some special partially ordered sets and Markov fields 3542
- 7. Ferry problem 4350
- 8. The linearly ordered case 4653
- 9. Fibres 5461
- 10. Cracks and splitting random elements 6976
- 11. Transformations of fields indexed by contours 8491
- 12. Bibliographical notes 93100
- References 97104