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Methods of free probability

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arxiv 2208.07515 v5 pith:PO4XVN3A submitted 2022-08-16 math.PR math-phmath.COmath.MPmath.QA

classification math.PRmath-phmath.COmath.MPmath.QA
keywords freeprobabilityclassicalcomingexamplesfoundationsgroupslimiting
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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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  1. Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions

    cond-mat.stat-mech 2025-01 accept novelty 8.0 of 10

    For switching diffusion, the long-time position cumulants equal free cumulants of the diffusion-coefficient distribution, and the rate function shows dynamical transitions depending on the tail of that distribution.

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