Gibbs and quantum discrete spaces

The Gibbs field is one of the central objects of modern probability theory, mathematical statistical physics, and Euclidean field theory. In this paper we introduce and study a natural generalization of this field to the case in which the background space (a lattice, a graph) on which the random field is defined is itself a random object. Moreover, this randomness is given neither a priori nor independent of the configuration; on the contrary, the space and the configuration on it depend on each other, and both objects are given by a Gibbs construction. We refer to the resulting distribution as a Gibbs family because it parametrizes Gibbs fields on different graphs belonging to the support of the distribution. We also study the quantum analogue of Gibbs families and discuss relationships with modern string theory and quantum gravity.