Free probability theory analyzes noncommutative distributions obtained by replacing independence in ordinary probability theory with free independence derived from the free product of groups.
The semicircle distribution is central to free probability theory like the normal distribution in probability theory.
In this talk, I would like to explain the free independent semicircle distribution, its characterization of moments using noncrossing partitioning, its representation in Fock space, and its relation to a random matrix called GUE.