Relationship between R-transform and free convolution of random matrices? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:05:47Z http://mathoverflow.net/feeds/question/76285 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76285/relationship-between-r-transform-and-free-convolution-of-random-matrices Relationship between R-transform and free convolution of random matrices? Jiahao Chen 2011-09-24T19:59:19Z 2011-09-25T16:36:29Z <p>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.</p> <hr> <p>As far as I understand it, the R-transform consists of the following steps:</p> <ol> <li>Given a probability distribution function $f(t)$ over some domain $D$ (which I usually take to be $\mathbb R$), find its Cauchy transform $$g(s) = \int_D \frac {f(t)} {t-s} dt$$</li> <li>Calculate the functional inverse $g^{-1}(w)$ and subtract $\frac 1 w$ to obtain the R-transform $$r(w) = g^{-1}(w) - \frac 1 w$$</li> </ol> <p>and that the free convolution of two pdfs $f_1 \boxplus f_2$ consists of:</p> <ol> <li>Adding the two R-transformed functions $$r_s (w) = r_1(w) + r_2(w)$$ </li> <li>Adding $\frac 1 w$ to the sum, then computing the functional inverse of $$g_s^{-1}(w) = r_s(w)+\frac 1 w$$</li> <li>Computing the inverse Cauchy transform using the Plemelj relation $$f_s(t) = \frac 1 \pi \Im g_s (s)$$</li> </ol> <hr> <p>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.</p> <p>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.</p>