ncts

Free probability, arising from the theory of operator algebras, is a mathematical theory that studies random variables in non-commutative probability spaces. This theory was initiated by D. Voiculescu around 1985 in order to study an important unsolved problem in operator algebra, and is often regarded as a non-commutative parallelism of classical probability theory. Independence in non-commutative probability spaces (or in quantum physics) is called freeness, the analogue of the classical notion of independence. Surprisingly, the random matrix models play a major role in the study of free probability theory. Connections to other theories, such as classical probability theory, combinatorics, wireless communication, large deviations, quantum information theory, etc. were established later on. In this talk, we will briefly introduce fundamental results in free probability and random matrix theories, and their connections to classical probability theory. More precisely, we will talk about the free counterpart of limit theorems, infinitely divisible distributions along with their Levy-Hincin representations, Levy processes, stale laws, etc. If time permits, we will also introduce bi-free probability, the newly developed theory in free probability, and its recent developments.