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There has been a significant increase recently in activities on the interface between applied analysis and probability theory. With the potential of a combined approach to the study of various physical systems in view, this book is a collection of topical survey articles by leading researchers in both fields, working on the mathematical description of growth phenomena in the broadest sense. The main aim of the book is to foster interaction between researchers in probability and analysis, and to inspire joint efforts to attack important physical problems. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic analysis, multi-scale techniques, and homogenization of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose–Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe.

This text provides a broad account of the theory of traces and determinants on geometric algebras of differential and pseudodifferential operators over compact manifolds. Trace and determinant functionals on geometric operator algebras provide a means of constructing refined invariants in analysis, topology, differential geometry, analytic number theory and QFT. The consequent interactions around such invariants have led to significant advances both in pure mathematics and theoretical physics. As the fundamental tools of trace theory have become well understood and clear general structures have emerged, so the need for specialist texts which explain the basic theoretical principles and the computational techniques has become increasingly exigent. This text is the first to deal with the general theory of traces and determinants of operators on manifolds in a broad context, encompassing a number of the principle applications and backed up by specific computations which set out in detail to newcomers the nuts-and-bolts of the basic theory. Both the microanalytic approach to traces and determinants via pseudodifferential operator theory and the more computational approach directed by applications in geometric analysis, are developed in a general framework that will be of interest to mathematicians and physicists in a number of different fields.