Role in Free Probability Theory
The lattice of noncrossing partitions plays the same role in defining "free cumulants" in free probability theory that is played by the lattice of all partitions in defining joint cumulants in classical probability theory. To be more precise, let be a non-commutative probability space, a non-commutative random variable with free cumulants . (See free probability for terminology.) Then
where denotes the number of blocks of length in the non-crossing partition . That is, the moments of a non-commutative random variable can be expressed as a sum of free cumulants over the sum non-crossing partitions. This is the free analogue of the moment-cumulant formula in classical probability. See also Wigner semicircle distribution.
Read more about this topic: Noncrossing Partition
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