PreprintCC BY-NC-ND 4.0HAL
Methods of free probability
Abstract
This is an introduction to free probability, with all the needed preliminaries included. We first discuss the most basic noncommuting variables, namely the usual matrices, followed by the basic random matrices, and with a look into Lie groups too. Then we present, following Voiculescu, the main concepts and tools of free probability, notably with the various limiting theorems that can be obtained. We then go on a more technical discussion, featuring cumulants, functional transforms and group theory, on the correspondence between classical and free. Finally, we discuss matrix modeling questions for the various more specialized free laws and objects that can be imagined.
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BibTeX
@misc{banica2026methods,
title = {{Methods of free probability}},
author = {Teo Banica},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-03751880v6},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Methods of free probability
AU - Teo Banica
PY - 2026
DA - 2026/09/26
PB - HAL
UR - https://hal.science/hal-03751880v6
AB - This is an introduction to free probability, with all the needed preliminaries included. We first discuss the most basic noncommuting variables, namely the usual matrices, followed by the basic random matrices, and with a look into Lie groups too. Then we present, following Voiculescu, the main concepts and tools of free probability, notably with the various limiting theorems that can be obtained. We then go on a more technical discussion, featuring cumulants, functional transforms and group theory, on the correspondence between classical and free. Finally, we discuss matrix modeling questions for the various more specialized free laws and objects that can be imagined.
ER -