I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to noncrossing partitions.

As far as I understand it, the R-transform consists of the following steps:

- Given a probability distribution function $f(t)$ over some domain $D$ (which I usually take to be $\mathbb R$), find its Cauchy transform $$ g(s) = \int_D \frac {f(t)} {t-s} dt $$
- Calculate the functional inverse $ g^{-1}(w) $ and subtract $\frac 1 w$ to obtain the R-transform $$ r(w) = g^{-1}(w) - \frac 1 w $$

and that the free convolution of two pdfs $f_1 \boxplus f_2$ consists of:

- Adding the two R-transformed functions $$ r_s (w) = r_1(w) + r_2(w) $$
- Adding $\frac 1 w$ to the sum, then computing the functional inverse of $$ g_s^{-1}(w) = r_s(w)+\frac 1 w$$
- Computing the inverse Cauchy transform using the Plemelj relation $$ f_s(t) = \frac 1 \pi \Im g_s (s) $$

I am trying to understand the mechanics of R-transform (why does it work?) and relating it to calculating the free convolution using random matrices, i.e. that if we have matrices $A$ and $B$ with eigenvalue spectra $f_A$ and $f_B$, that by taking a random orthogonal matrix $Q$ of Haar measure you can form the sum $$A + Q B Q^T$$ that has spectrum $f_A \boxplus f_B $ in the large $N$ limit.

I can see that expanding the resolvent in the Cauchy transform produces a formal power series in $s$, but I don't really see how the coefficients come out to free cumulants. What is $\frac 1 w$ doing? What is the relationship with noncrossing partitions? My intuition is that they must play a role in counting the number of terms with the same value in this expansion, and that the noncrossing must arise from where the $Q$ and $Q^T$s show up in the series (effecting a change of basis), but I don't quite see it yet.

non-expertin random matrices, your question is a bit too unfocused than is ideal for a MathOverflow question, and has too many "moving parts". Maybe you could split this into two more focused questions, one of which might be "why does the R-transform linearize free convolution, and how do the noncrossing partitions come up?" – Yemon Choi Sep 25 '11 at 0:00