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Independence and Convolutions in Noncommutative Probability

Final Report Summary - ICNCP (Independence and Convolutions in Noncommutative Probability)

The research project is classified into three parts:

I Analysis of infinitely divisible distributions in classical and free probability. Papers, preprints or work in progress (J4), (P1), (P2), (P5), (P6), (P8), (P9); Talks (Talk1), (Talk2),(Talk3), (Talk4), (Talk7), (Talk8), (Talk9), (Talk10).

II Applications of monotone independence to free probability. (P4)

III Application and development of Lenczewski’s matricial free independence. (P3), (P7);(Talk5), (Talk6).

On all of these parts significant new results have been obtained, in particular concerning part I.

I Many distributions were shown to be FID, including beta distributions of the first and second kinds, gamma distributions, inverse gamma distributions and scale mixtures of Boolean stable laws. Some of them are HCM, and some of them are completely monotone. Since now we know examples, the next step is to find a general theory abstracting these examples and relating classical theory of ID distributions. During one year, talks and participations in conferences have been mostly devoted to Project I.

II We found that monotone increment processes are intimately connected to the theory of univalent functions. More direct contribution to univalent functions are expected.

III It is known that the study of cumulants in noncommutative probability often involves combinatorics and graph theory. This project also finds new connections to these fields, involving trees and symmetric groups. We found that matricial free cumulants give us a formula connecting monotone cumulants and free cumulants using trees. Looking at this, further study of matricial freeness may yield more combinatorial and graph theoretical consequences.

For more details see the enclosed PDF file