Stochastic Geometry: An Introduction and Reading-List

Abstract

This informal review is aimed at nonspecialist readers in the hope of making the literature and methods of stochastic geometry more widely accessible. It develops around one elementary example and is combined with a selected reading list, including some references to early history. The review can serve as a complement to detailed surveys which have recently appeared (see ? 4 for references). Here I simply set down a few essential themes, stressing the continuity of methods applied to widely different problems. The selection is subjective, and viewpoints expressed are often rhetorical or unqualified. Stochastic geometry, or geometrical probability, is a rather broad heading for probability problems which incorporate geometrical figures as the 'outcomes' or 'data'. Examples are to be found in the earliest probability theories. Although the physical realities of coin-tossing, dice and roulette can usually be ignored, this is not true of certain games which depend on the final position of a thrown coin. Random positions of a geometrical figure are the centre of the classical theory. The recreational flavour (or stigma) once associated with these ideas is less noticeable in the presence of applications ranging from astronomy to cell biology and materials science. Additionally the probabilistic foundations are connected to other branches of pure mathematics. Scientific problems may have a geometrical content in the data, in sampling, in theoretical models, or in all three. Examples of geometrical data are: maps of trees, of meteorite finds, or of petty crimes; linear tracks made by subatomic particles; vegetation patterns; images of microscopic structure in minerals or ceramics; linear fractures in rock; thin sections of biological tissues. Here even the question of summarizing data is nontrivial. Secondly, many experimental sampling techniques can be described as geometrical. Spatial variation of vegetation patterns or bird-nest sites might be sampled along a linear transect, inside an arbitrary grid square or within a fixed radius of an arbitrary reference point. Solid structures like minerals and biological tissues are frequently studied by taking thin sections for microscopic examination. Here the deep problem of relating solid geometry to plane sections is compounded with questions of sampling variability, bias, choice of sampling design and inference. The multidisciplinary science of stereology