Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables. These operations have some interpretations in terms of empirical spectral measures of random matrices. The notion of free convolution was introduced by Voiculescu.
Free additive convolution
Let and be two probability measures on the real line, and assume that is a random variable in a non commutative probability space with law and is a random variable in the same non commutative probability space with law. Assume finally that and are freely independent. Then the free additive convolution is the law of. Random matrices interpretation: if and are some independent by Hermitian random matrices such that at least one of them is invariant, in law, under conjugation by any unitary matrix and such that the empirical spectral measures of and tend respectively to and as tends to infinity, then the empirical spectral measure of tends to. In many cases, it is possible to compute the probability measure explicitly by using complex-analytic techniques and the R-transform of the measures and.
Rectangular free additive convolution
The rectangular free additive convolution has also been defined in the non commutative probability framework by Benaych-Georges and admits the following random matrices interpretation. For, for and are some independent by complex random matrices such that at least one of them is invariant, in law, under multiplication on the left and on the right by any unitary matrix and such that the empirical singular values distribution of and tend respectively to and as and tend to infinity in such a way that tends to, then the empirical singular values distribution of tends to. In many cases, it is possible to compute the probability measure explicitly by using complex-analytic techniques and the rectangular R-transform with ratio of the measures and.
Let and be two probability measures on the interval, and assume that is a random variable in a non commutative probability space with law and is a random variable in the same non commutative probability space with law. Assume finally that and are freely independent. Then the free multiplicative convolution is the law of random matrices such that at least one of them is invariant, in law, under conjugation by any unitary matrix and such that the empirical spectral measures of and tend respectively to and as tends to infinity, then the empirical spectral measure of tends to. A similar definition can be made in the case of laws supported on the unit circle, with an orthogonal or unitary random matrices interpretation. Explicit computations of multiplicative free convolution can be carried out using complex-analytic techniques and the S-transform.
Free convolution can be used to compute the laws and spectra of sums or products of random variables which are free. Such examples include: random walk operators on free groups ; and asymptotic distribution of eigenvalues of sums or products of independent random matrices.
Through its applications to random matrices, free convolution has some strong connections with other works on G-estimation of Girko. The applications in wireless communications, finance and biology have provided a useful framework when the number of observations is of the same order as the dimensions of the system.
