scholarly journals On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions

2010 ◽  
Vol 46 (3) ◽  
pp. 644-652 ◽  
Author(s):  
Florent Benaych-Georges
Keyword(s):  
Free Convolutions

scholarly journals Local limit theorems for multiplicative free convolutions

2014 ◽  
Vol 267 (9) ◽  
pp. 3469-3499 ◽  
Author(s):  
Michael Anshelevich ◽  
Jiun-Chau Wang ◽  
Ping Zhong
Keyword(s):  
Limit Theorems ◽  
Local Limit ◽  
Free Convolutions ◽  
Local Limit Theorems

scholarly journals Finite free convolutions via Weingarten calculus

2020 ◽  
pp. 2150038
Author(s):  
Jacob Campbell ◽  
Zhi Yin
Keyword(s):  
Unitary Groups ◽  
Orthogonal Groups ◽  
Signed Permutation ◽  
Permutation Matrices ◽  
Explicit Formulae ◽  
Free Convolutions

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain quadrature property, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.

scholarly journals Boolean cumulants and subordination in free probability

2020 ◽  
pp. 2150036
Author(s):  
Franz Lehner ◽  
Kamil Szpojankowski
Keyword(s):  
Free Probability ◽  
Borel Function ◽  
General Setting ◽  
Main Tool ◽  
Explicit Calculation ◽  
Conditional Expectations ◽  
Multiplicative Convolution ◽  
Noncommutative Polynomials ◽  
Free Random Variables ◽  
Free Convolutions

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

scholarly journals Approximation of free convolutions by free infinitely divisible laws

2021 ◽  
Vol 66 (4) ◽  
pp. 806-838
Author(s):  
Геннадий Петрович Чистяков ◽  
Gennadii Petrovich Chistyakov ◽  
Фридрих Гeтце ◽  
Friedrich Gotze
Keyword(s):  
Infinitely Divisible ◽  
Infinitely Divisible Laws ◽  
Free Convolutions

Основываясь на методе сyбoрдинационных функций, мы получаем оценки минимальных ошибок аппроксимации $n$-кратных свободных сверток вероятностных мер безгранично делимыми свободными вероятностными мерами.

scholarly journals Rank one HCIZ at high temperature: interpolating between classical and free convolutions

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Pierre Mergny ◽  
Marc Potters
Keyword(s):  
High Temperature ◽  
Limit Theorem ◽  
Simple Interpretation ◽  
Numerical Examples ◽  
Free Convolution ◽  
Rank One ◽  
Beta 2 ◽  
Poisson Limit ◽  
Associated Family ◽  
Free Convolutions

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, with parameter c between the classical and the free convolution. This c-convolution has a simple interpretation in terms of another associated family of distribution indexed by c, called the Markov-Krein transform: the c-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulant-moment relations, a central limit theorem, a Poisson limit theorem and show several numerical examples of c-convoluted distributions.

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