TL;DR
This paper introduces classical and free probability theories, reviewing their foundations, main theorems, and examples from random matrices, Lie groups, quantum groups, and subfactor theory, highlighting their interrelations and advanced aspects.
Contribution
It provides a comprehensive overview connecting classical and free probability, including foundational results, examples, and advanced topics like free geometry and subfactors.
Findings
Classical probability limit theorems (CLT, CCLT, PLT, CPLT) explained.
Free probability limit theorems and examples from quantum groups discussed.
Connections between free probability, free geometry, and subfactor theory explored.
Abstract
This is a joint introduction to classical and free probability, which are twin sisters. We first review the foundations of classical probability, notably with the main limiting theorems (CLT, CCLT, PLT, CPLT), and with a look into examples coming from Lie groups and random matrices. Then we present the foundations and main results of free probability, notably with free limiting theorems, and with a look into examples coming from quantum groups and random matrices. We discuss then a number of more advanced aspects, in relation with free geometry and with subfactor theory.
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Operator Algebra Research · Random Matrices and Applications