the free convolution
The free convolution is the operation that answers the central computational question of free probability: given the eigenvalue distributions of two large free random matrices A and B, what is the eigenvalue distribution of their sum A + B (or, for the multiplicative version, of a product)? In classical probability the law of a sum of independent variables is the ordinary convolution of their laws, computed easily because characteristic functions multiply. Free probability has its own convolution, written mu_A box-plus mu_B for addition, and its own transform that linearises it — but the operation itself is genuinely different from classical convolution and gives genuinely different answers.
The free additive convolution mu box-plus nu of two probability measures mu and nu is, by definition, the limiting eigenvalue distribution of A_N + B_N when A_N has limiting spectrum mu, B_N has limiting spectrum nu, and the two are asymptotically free. The whole point is that there is a transform that turns this complicated operation into simple addition, exactly as the logarithm of the characteristic function (the cumulant generating function) turns classical convolution into addition. That transform is the R-transform: R_{mu box-plus nu} = R_mu + R_nu. So to free-convolve two measures you compute each one's R-transform, add them, and invert. There is also a free multiplicative convolution mu box-times nu, the limiting spectrum of products like A_N^{1/2} B_N A_N^{1/2} (which has the same eigenvalues as A_N B_N when one factor is positive), and it is linearised by a different transform, the S-transform, which makes the relevant transforms multiply rather than add.
Free convolution is the practical engine of the theory. The Marchenko-Pastur law for a sample covariance matrix with a general (non-identity) population covariance Sigma is precisely the free multiplicative convolution of the standard MP law with the spectrum of Sigma — this is how you predict the eigenvalue spectrum of real, structured data. It also lets you 'deconvolve': given the observed (noisy) spectrum, the inverse free convolution recovers the true population spectrum, the basis of modern spectrum-estimation methods. A caveat that must not be dropped: free convolution computes the spectrum of A + B only when A and B are free, which for random matrices requires their eigenvectors to be in generic relative position. If you naively free-convolve two spectra of matrices that happen to share eigenvectors, you get the wrong answer; the right answer there is the classical convolution of the joint eigenvalues. The free convolution is the law of the sum precisely under the freeness hypothesis, not in general.
You have stock-return data whose true population covariance has spectrum nu (say a couple of large 'market' eigenvalues plus a flat noise floor). The observed sample covariance spectrum, in the high-dimensional limit, is the free multiplicative convolution of nu with the Marchenko-Pastur law of aspect ratio y = p/n. Running this convolution forward predicts exactly the smeared, edge-broadened spectrum you actually see; running it backward (free deconvolution) recovers the clean nu from the noisy observation.
The general sample-covariance spectrum is MP free-convolved with the population spectrum.
Free convolution gives the spectrum of A + B only under the freeness hypothesis (generic relative eigenvectors). For matrices that share eigenvectors the correct rule is classical convolution of the joint eigenvalues — do not free-convolve commuting matrices.