Measuring dependence in metric abstract elementary classes with perturbations

Abstract

We define and study a metric independence notion in a homogeneous metric abstract elementary class with perturbations that is $d^p$-superstable (superstable wrt. the perturbation topology), weakly simple and has complete type spaces and we give a new example of such a class based on B. Zilber's approximations of Weyl algebras. We introduce a way to measure the dependence of a tuple $a$ from a set $B$ over another set $A$. We prove basic properties of the notion, e.g. that $a$ is independent of $B$ over $A$ in the usual sense of homogeneous model theory if and only if the measure of dependence is $<\varepsilon$ for all $\varepsilon >0$. As an example of our measure of dependence we show a connection between the measure and entropy in models from quantum mechanics in which the spectrum of the observable is discrete. As an application, we show that weak simplicity implies a very strong form of simplicity and study the question of when the dependence inside a set of all realisations of some type can be seen to arise from a pregeometry in cases when the type is not regular. In the end of the paper, we demonstrate our notions and results in one more example: a class built from the $p$-adic integers.