Gibbs and quantum discrete spaces

Abstract

Gibbs field is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields on different graphs in the support of the distribution. We study also quantum (KMS) analog of Gibbs families.

The Ising Model on the Random Planar Causal Triangulation: Bounds on the Critical Line and Magnetization Properties

“…We will show (Lemma 3.1) that the family of conditional distributions constructed from the introduced above Gibbs family is consistent. This gives a ground for the Dobrushin-Lanford-Ruelle construction of Gibbs measure on T ∞ (see [17]).…”

“…One can find in Malyshev [17] a general outline of the theory of Gibbs families on spin graphs. We follow his approach and develop it here for a class of planar triangulations.…”

“…Malyshev [17] gave a solid mathematical ground for such models, developing a general theory of Gibbs fields on random spaces where matter (represented by the configurations of spins) is naturally coupled with the gravity (represented by the graph).…”

Mathematics for some classes of networks

“…jointly with a field on it) is called random graph grammars [2, 5, 1] and [21,22]. It appears to be quite natural in connection with the emerging new physical theories [16,17,25] and social networks [20]. Namely, if eventually the local space-time appears to be discrete, then the most natural language for it is a graph with some physical fields on it.…”

“…There are many results-examples (by physicists and mathematicians) related to "quantum gravity", see for example [17,25] and references therein.…”