Statistical Fluctuation of Infinitesimal Spaces

This paper is a follow-up on the noncommutative differential geometry on infinitesimal spaces [19]. In the present work, we extend the algebraic convergence from [19] 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 brief introduction of the Von Mises-Fisher distribution on manifolds, we show that when the Dirac operators are interpreted as stochastic matrices, the sequence $(D_n)_{n\in \mathbb{N}}$ converges in average to the usual Dirac operator on a spin manifold. The same conclusion can be drawn for the Laplace operator.