Geometry and probability on the noncommutative 2-torus in a magnetic field

TL;DR

This paper explores the geometric and probabilistic aspects of a noncommutative 2-torus under a magnetic field, focusing on invariance properties, scalar curvature, and stochastic processes influenced by the magnetic Laplacian.

Contribution

It introduces a novel analysis of geometric invariants and stochastic dynamics on the noncommutative 2-torus in a magnetic field using perturbation methods.

Findings

Volume invariance under magnetic perturbation

Characterization of scalar curvature in the noncommutative setting

Analysis of magnetic stochastic processes and their properties

Abstract

In this work, we describe the geometric and probabilistic properties of a noncommutative 2- torus in a magnetic field. We study the volume invariance, integrated scalar curvature and volume form by using the method of perturbation by inner derivation of the magnetic Laplacian in the noncommutative 2-torus. Then, we analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative 2-torus, derive and discuss the related main properties.