Statistical Fluctuation of Infinitesimal Spaces

Abstract

This paper is a follow-up on the \emph{noncommutative differential geometry on infinitesimal spaces} [15]. In the present work, we extend the algebraic convergence from [15] to the geometric setting. On the one hand, we reformulate the definition of finite dimensional compatible Dirac operators using Clifford algebras. This definition also leads to a new construction of a Laplace operator. On the other hand, after a well-chosen Green's function defined on a manifold, we show that when the Dirac operators can be interpreted as stochastic matrices. The sequence $(D_n)_{n\in \mathbb{N}}$ converges then in average to the usual Dirac operator on a spin manifold. The same conclusion can be drawn for the Laplace operator.