Scaling behaviour in random non-commutative geometries

TL;DR

This paper investigates phase transitions in random non-commutative geometries, analyzing how observables scale with system size and exploring critical behavior and exponents for different geometry types.

Contribution

It provides a detailed analysis of phase transitions and critical scaling in specific types of random non-commutative geometries, advancing understanding of their non-perturbative properties.

Findings

Identification of pseudo critical points for different geometries

Scaling behavior of observables with system size

Initial insights into critical exponents and correlations

Abstract

Random non-commutative geometries are a novel approach to taking a non-perturbative path integral over geometries. They were introduced in arxiv.org/abs/1510.01377, where a first examination was performed. During this examination we found that some geometries show indications of a phase transition. In this article we explore this phase transition further for geometries of type $(1,1)$, $(2,0)$, and $(1,3)$. We determine the pseudo critical points of these geometries and explore how some of the observables scale with the system size. We also undertake first steps towards understanding the critical behaviour through correlations and in determining critical exponents of the system.