Random surfaces / Scott Sheffield

This treatise, which is the author's thesis, gives an authorative description of lattice models of random surfaces modelled by “gradient Gibbs measures”. These gradient Gibbs measures are defined on observables in which the height of the surface is divided out. This height can be either discrete-valued or have values in the continuum. In cases where the surface fluctuates too strongly and cannot be described by a Gibbs measure, gradient Gibbs measure, which then are “rough” (as opposed to the “smooth” case, where one has the restriction of proper Gibbs measures), still may exist. In some cases uniqueness results for surfaces of a given slope may be obtained. Although these gradient Gibbs measures had been introduced before, Sheffield's work is a very valuable addition to the literature, both as a careful review of the general theory, describing gradient Gibbs measures, large deviation principles and variational principles, and at the same time containing various new results of interest for more specific classes of models. (Zentralblatt)

Bibliogr. p.169-175