Random Fields and Their Geometry

This self-contained monograph contains a number of topics that focus on the many developments in the area of mathematical probability/statistics known as random fields. Robert Adler has been one of the major contributors to this subject. In an earlier work (1981) he dealt with a wide variety of geometrical problems that are unique to stochastic processes in dimensions higher than one, fashioning an interesting blend of geometry and probability. Some old "classical" topics ("non"-geometric in nature) are presented in the first few chapters, but the material has been recast in the post-1980 setting of entropy. This is followed by a treatment of random geometry which takes into account the recent work by Worsley et al, and a significant update on our understanding of the fractal nature of erratic random fields. The geometric theme of the book is carried throughout in a user- friendly style. As in his previous work, Adler engages scientists outside of pure and applied mathematics, in particular, researchers involved in brain imaging, and astrophysicists. Applications of excursion set theory in astrophysics and brain imaging are detailed in a short chapter.
It is hope that the approach employed in these two quite different areas will direct the reader to finding applications elsewhere. In recent years, the geometry of random fields has also found its way into the literature of materials (surface) science, oceanography, and geology. For applications in the medical community, a chapter discussing the "tube formula" approach involves the geometry on manifolds and an examination of the distribution of the maximum of Gaussian random