scholarly journals The chordal Loewner equation and monotone probability theory

2017 ◽  
Vol 20 (03) ◽  
pp. 1750016 ◽  
Author(s):  
Sebastian Schleißinger
Keyword(s):  
Probability Theory ◽  
Evolution Equations ◽  
Free Probability ◽  
Noncommutative Probability ◽  
Loewner Equation ◽  
Free Probability Theory ◽  
Chordal Loewner Equation ◽  
Monotone Probability ◽  
Noncommutative Probability Theory

In Ref. 5, O. Bauer interpreted the chordal Loewner equation in terms of noncommutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

Free Probability Theory

2019 ◽  
Vol 15 (4) ◽  
pp. 3147-3215
Author(s):  
Alice Guionnet ◽  
Roland Speicher ◽  
Dan-Virgil Voiculescu
Keyword(s):  
Probability Theory ◽  
Free Probability ◽  
Free Probability Theory

Free Probability Theory

2005 ◽  
pp. 827-880
Author(s):  
Philippe Biane ◽  
Roland Speicher ◽  
Dan-Virgil Voiculescu
Keyword(s):  
Probability Theory ◽  
Free Probability ◽  
Free Probability Theory

scholarly journals On the Outlying Eigenvalues of a Polynomial in Large Independent Random Matrices

2019 ◽  
Author(s):  
Serban T Belinschi ◽  
Hari Bercovici ◽  
Mireille Capitaine
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
Empirical Distribution ◽  
Free Probability ◽  
Probability Measures ◽  
Unitary Operators ◽  
Free Probability Theory ◽  
Wigner Matrix ◽  
Asymptotic Eigenvalue ◽  
Special Case

Abstract Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent Hermitian random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu$ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu ,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\eta \ (\textrm{sometimes denoted}\ P^\square(\mu,\nu))$ and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu ,\nu ,P$, and the spikes of $A_N$. We establish a similar result when $B_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

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