Gibbs and Quantum Discrete Spaces

TL;DR

This paper introduces a generalized Gibbs measure framework for random spaces dependent on configurations, including quantum analogs, with applications to discrete quantum gravity.

Contribution

It defines and studies Gibbs families where the underlying space is random and dependent on the configuration, extending classical Gibbs measures.

Findings

Defined Gibbs families for random spaces dependent on configurations

Explored quantum (KMS) analogs of Gibbs families

Applied the framework to discrete quantum gravity

Abstract

Gibbs measure 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. Various applications to discrete quantum gravity are given.