Statistical fluctuation of infinitesimal spaces

Abstract

This paper is a follow-up on the Noncommutative Differential Geometry on Infinitesimal Spaces [16]. In the present work, we extend the algebraic convergence established in [16] to the Riemannian 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 choosing a family of Green functions to define the coefficients of the Dirac matrices, we show that these matrices can be interpreted as stochastic matrices. The sequence of operators \((D_n)_{n\in \mathbb {N}}\) converges then in average to the local expression of the classical Dirac operator on an oriented compact Riemann manifold. The same result is drawn for the Laplace operator.