Lecture 21 - The full Fock space model for the R-transform

Summary

A convenient fact which was observed and then repeatedly used in the preceding lectures is that one can construct a family of noncommutative random variables with a prescribed joint distribution (where quite often the joint distribution is indicated in the guise of a prescribed joint R-transform). Sometimes there may be more than one way of doing such a construction; an example of such a situation is the one of free semicircular families, which can be obtained by an abstract free product construction, but can also be “concretely” put into evidence by using operators of creation and annihilation on the full Fock space (cf. Lecture 7).

Of course, all the different methods for constructing a family of elements with a prescribed joint distribution are ultimately equivalent, in the respect that the calculations with moments and with cumulants performed on the family give the same results, no matter how the family was constructed. Nevertheless, there can be a substantial difference in the transparency of the calculations – it may happen that the solution to the problem we are trying to solve shows up more easily if one method of construction is used over another.

So this is, in a nut-shell, the idea of “modeling”: find a good way of constructing a family of non-commutative random variables with a given joint distribution, so that we are at an advantage when computing moments and cumulants of that family.