Lecture 14 - Products of free random variables

Summary

In the previous lectures we treated the sum of freely independent variables. In particular, we showed how one can understand and solve from a combinatorial point of view the problem of describing the distribution of a+ b in terms of the distributions of a and of b if these variables are freely independent. Now we want to turn to the corresponding problem for the product. Thus we want to understand how we get the distribution of ab out of the distribution of a and of b if a and b are freely independent.

Note that for the corresponding classical problem no new considerations are required, since this can be reduced to the additive problem. Namely, if a and b commute, we have ab = exp(log a + log b) and thus we only need to apply the additive theory to log a and log b. In the non-commutative situation, however, the functional equation for the exponential function no longer holds, so there is no clear way to reduce the multiplicative problem to the additive one and some new considerations are needed. In our combinatorial treatment it will turn out that the description of the multiplication of freely independent variables is intimately connected with the complementation map K in the lattice of non-crossing partitions. Since there is no counterpart of the complementation map for all partitions, statements concerning the multiplication of freely independent variables might be quite different from what one expects classically.