**Translations of Mathematical Monographs**

1992;
204 pp;
Hardcover

MSC: Primary 82;
Secondary 60; 85

**Print ISBN: 978-0-8218-4563-9
Product Code: MMONO/104**

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

**Electronic ISBN: 978-1-4704-4515-7
Product Code: MMONO/104.E**

List Price: $94.00

AMS Member Price: $75.20

MAA Member Price: $84.60

# Wulff Construction: A Global Shape from Local Interaction

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*Roland Dobrushin; R. Kotecký; Senya Shlosman*

A theory of the equilibrium shape of crystal assuming minimal surface free energy was formulated at the beginning of the century by Wulff. Assuming that the anisotropic interfacial free energy (depending on the orientation of the interface with respect to the crystal axes) is known, the Wulff construction yields the shape of crystal in equilibrium and allows one to understand its main features. This research monograph considers the Wulff construction in the case of a two-dimensional Ising ferromagnet with periodic boundary conditions and at sufficiently low temperatures. Namely, the authors investigate the phenomenon of phase separation in a (small) canonical ensemble characterized by a fixed total spin in a finite volume. Its value is chosen to lie in the interval between the spontaneous magnetizations of pure phases. Heuristically, the main result can be stated this way: a droplet of one phase immersed in the opposite one will be formed with the separation line following with high accuracy the shape yielded by the Wulff construction. The book brings the reader through the entire development of the proof of this result.

#### Readership

Research mathematicians.

#### Table of Contents

# Table of Contents

## Wulff Construction: A Global Shape from Local Interaction

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Chapter I. Introduction 112
- Chapter II. Extremal properties of the Wulff functional 2334
- Chapter III. Limit theorems 4960
- Chapter IV. Surface tension 103114
- Chapter V. Large contours 151162
- Chapter VI. Proof of the main results 187198
- Bibliography 201212
- Back Cover Back Cover1218