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.

Let ${\displaystyle \mu }$ and ${\displaystyle \nu }$ be two probability measures on the real line, and assume that ${\displaystyle X}$ is a random variable in a non commutative probability space with law ${\displaystyle \mu }$ and ${\displaystyle Y}$ is a random variable in the same non commutative probability space with law ${\displaystyle \nu }$. Assume finally that ${\displaystyle X}$ and ${\displaystyle Y}$ are freely independent. Then the free additive convolution ${\displaystyle \mu \boxplus \nu }$ is the law of ${\displaystyle X+Y}$. Random matrices interpretation: if ${\displaystyle A}$ and ${\displaystyle B}$ are some independent ${\displaystyle n}$ by ${\displaystyle n}$ 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 ${\displaystyle A}$ and ${\displaystyle B}$ tend respectively to ${\displaystyle \mu }$ and ${\displaystyle \nu }$ as ${\displaystyle n}$ tends to infinity, then the empirical spectral measure of ${\displaystyle A+B}$ tends to ${\displaystyle \mu \boxplus \nu }$.

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