Random surfaces.

*(English)*Zbl 1104.60002
Astérisque 304. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-186-3/pbk). vi, 175 p. (2005).

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.

Reviewer: A. C. D. van Enter (Groningen)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60D05 | Geometric probability and stochastic geometry |

60F10 | Large deviations |

60G60 | Random fields |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |