Kyoto University

Elements in a noncommutative operator algebra can be regarded as noncommutative random variables from a probabilistic viewpoint. Such understanding has its origin in quantum theory. Theory of operator algebras focusing on the probabihstic aspect is called noncommutative probability theory. Noncommutative probability theory is divided into several directions. Some groups perform mathematical research, and others do physical research. The main focus of this article is free probability, a mathematical aspect of noncommutative probability. The name of free probability theory might sound strange for non-experts. This name was chosen because free probability fits in the analysis of the free product of groups or algebras. $\mathbb{R}ee $ probability has been developed in terms of operator algebras to solve prob-lems related to von Neumann algebras generated by free groups [HPOO, VDN92]. From a probabilistic aspect, when one considers random walks on free groups, free probability is useful to analyze the recurrence/transience of the random walks [W86].1 In addition, Voiculescu [V91] found that free probabihty has application to the anal-ysis of the eigenvalues of random matrices (see also [HPOO, VDN92]). Why eigenvalues of random matrices interest researchers? The original motivation is to model the energy levels of nucleons of nuclei. Then subsequent studies revealed many relations of random matrices to other mathematics as well as physics, e.g. integrable systems (such as Pein-lev\’e equations), the Riemann zeta function and representation theory [M04]. All these applications are based on the analysis of eigenvalue distributions of random matrices. In this article, we are going to present the basics of free probability, and then describe the summary of results obtained so far on freely infinitely divisible distributions, the author’s recent main subject. $A $ purpose of free probability is to analyze free convolution which describes the eigenvalue distribution of the sum of independent large random matrices. The set of freely infinitely divisible probability measures is the central subject associated to free convolution. *email