Free convolution

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 (see below; in the classical case, what would be the analog of free multiplicative convolution can be reduced to additive convolution by passing to logarithms of 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 formula_1 and formula_2 be two probability measures on the real line, and assume that formula_3 is a random variable in a non commutative probability space with law formula_1 and formula_5 is a random variable in the same non commutative probability space with law formula_2. Assume finally that formula_3 and formula_5 are freely independent. Then the free additive convolution formula_9 is the law of formula_10. Random matrices interpretation: if formula_11 and formula_12 are some independent formula_13 by formula_13 Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the empirical spectral measures of formula_11 and formula_12 tend respectively to formula_1 and formula_2 as formula_13 tends to infinity, then the empirical spectral measure of formula_20 tends to formula_9.
In many cases, it is possible to compute the probability measure formula_9 explicitly by using complex-analytic techniques and the R-transform of the measures formula_1 and formula_2.
Rectangular free additive convolution.
The rectangular free additive convolution (with ratio formula_25) formula_26 has also been defined in the non commutative probability framework by Benaych-Georges and admits the following random matrices interpretation. For formula_27, for formula_11 and formula_12 are some independent formula_13 by formula_31 complex (resp. real) 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 (resp. orthogonal) matrix and such that the empirical singular values distribution of formula_11 and formula_12 tend respectively to formula_1 and formula_2 as formula_13 and formula_31 tend to infinity in such a way that formula_38 tends to formula_25, then the empirical singular values distribution of formula_20 tends to formula_41.
In many cases, it is possible to compute the probability measure formula_41 explicitly by using complex-analytic techniques and the rectangular R-transform with ratio formula_25 of the measures formula_1 and formula_2.
Free multiplicative convolution.
Let formula_1 and formula_2 be two probability measures on the interval formula_48, and assume that formula_3 is a random variable in a non commutative probability space with law formula_1 and formula_5 is a random variable in the same non commutative probability space with law formula_2. Assume finally that formula_3 and formula_5 are freely independent. Then the free multiplicative convolution formula_55 is the law of formula_56 (or, equivalently, the law of formula_57. Random matrices interpretation: if formula_11 and formula_12 are some independent formula_13 by formula_13 non negative Hermitian (resp. real symmetric) random matrices such that at least one of them is invariant, in law, under conjugation by any unitary (resp. orthogonal) matrix and such that the empirical spectral measures of formula_11 and formula_12 tend respectively to formula_1 and formula_2 as formula_13 tends to infinity, then the empirical spectral measure of formula_67 tends to formula_55.
A similar definition can be made in the case of laws formula_69 supported on the unit circle formula_70, 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.
Applications of free convolution.
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.