28
votes
Accepted
Is there a noncommutative Gaussian?
The theory of classical independence and classical convolution can be generalised to noncommutative settings in several ways. The most famous one is that of free independence and free convolution (...
- 114k
24
votes
Accepted
Why is free probability a generalization of probability theory?
Quite a lot of questions here!
It is perhaps worth making a distinction between scalar classical probability theory - the study of scalar classical random variables - and more general classical ...
- 114k
13
votes
Why is free probability a generalization of probability theory?
There have been many good answers to this question, but it might be that
the main point gets lost in too many details. So, as kind of expert on free
probability theory, let me try to give a short ...
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13
votes
Why did Voiculescu develop free probability?
Here is a more detailed account on the history by Dan Voiculescu himself. This is from his article "Background and Outlook" in the Lectures Notes
"Free Probability and Operator Algebras", see
http://...
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11
votes
What are applications of asymptotic freeness of random matrices?
Here are some applications of free probability of random matrices:
Neural networks: The asymptotic freeness assumption plays a fundamental role in the study of the propagation of spectral ...
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9
votes
Why is free probability a generalization of probability theory?
First of all, you are mixing many questions into one post...
It depends on what do you mean by "generalization". And I am not sure what you mean by talking about "commutative" without mentioning ...
- 8,071
9
votes
Accepted
Is $L(\mathbb{Z}*\mathbb{Z}_{2})$ a free group factor?
This should just be a comment- but for some reason I couldn't add a comment.
It seems to me that using Corollary 5.3 of this paper by Dykema, we indeed get a positive answer to your question.
...
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7
votes
Quick derivation of classical probability theory from von Neumann algebraic framework
I am not sure how far you want to go, but some basics are explained in this answer:
Is there an introduction to probability theory from a structuralist/categorical perspective?
In particular, you have ...
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7
votes
Why impossible events have some drawbacks or pathologies in probability theory?
What Halmos is referring to is the development of probability theory on the basis of measure algebras. This development is spelled out in some detail by I. E. Segal in Abstract Probability Spaces and ...
- 82.6k
6
votes
Independence of two noncommutative observables
Free probability theory was developed precisely to deal with noncommuting operators, represented by matrices $X$, $Y$. If $XY\neq YX$, the eigenvalues $\sigma$ of the sum $X+Y$ are not given by the ...
- 188k
6
votes
Accepted
Combinatorial formula to compute the moments of the product of two free random variables
The formula as stated in your reference seems to be wrong. The Kreweras complement of the partitions should also show up. You can find the correct version in my original paper https://link.springer....
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6
votes
Is there a noncommutative Gaussian?
If one does not insist on having a notion of "independence" in the background, but takes, as asked in the question, the Wick/Isserlis formula (which expresses general moments in terms of ...
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6
votes
Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
One can get a certain way towards this goal via a sort of "dimensional analysis". This isn't a completely satisfying heuristic argument - in particular, it only partially specifies what ...
- 114k
6
votes
Accepted
Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
I have a suggestion that seems to bring the two sides to be compared much closer together.
Take a block-diagonal matrix $B$ with $k$ blocks each a copy of $A$. This has the same eigenvalue measure as $...
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6
votes
What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?
I am not quite sure which generalization Rota had in mind, but this is a generalization that has found applications in the literature on stochastic processes:
Given a probability distribution function ...
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6
votes
What are applications of asymptotic freeness of random matrices?
Here are a few additional references, in the same directions as in Carlo's answer.
Wireless communication:
Channel Estimation and Robust Detection for IQ Imbalanced Uplink Massive MIMO-OFDM With ...
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5
votes
Why a random variable is better described by its cumulants than by its characteristic function?
The quote actually belongs to C.G.Rota.
Because cumulant sequences are closed under addition while moment sequences are not. That makes cumulant a more tractable algebraic structure altogether. ...
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5
votes
Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
I am not sure whether this counts as heuristics, as it goes even deeper into free probability results, but it might give some high-level kind of idea why this result should be true. The main fact ...
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4
votes
Does free probability have anything to say about the eigenvalue correlations of random matrices?
Eigenvalue correlations on the macroscopic level (fluctuations of linear statistics) are in free probability theory captured by the notion of "second order freeness"; a treatment of this can be found ...
- 1,210
4
votes
Accepted
Brown measure of left shift operator
Since the two-sided shift is a normal operator this is not really a question about the Brown measure, but about the spectral measure, so it suffices to compute the $*$-moments $\tau(L^nL^{*m})$. For ...
- 1,210
4
votes
The definition of amalgamated free product for general von Neumann algebras
Let's just consider the case of amalgamation over $\mathbb C$ as the general case is similar. Almost all definitions of free product of von Neumann algebras call for faithful normal states. This ...
- 3,984
4
votes
Independence of two noncommutative observables
I think some of the confusion in this question comes from the use of the phrase "joint distribution"; the slogan is that freeness of two variables allows to determine the joint distribution of the two ...
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4
votes
Accepted
Relating the R-transform in free probability to noncommutative group representations
The $R$-transform is related to harmonic analysis around free products of groups.
Actually, the computational machinery for the $R$-transform was also found independently from Voiculescu and about the ...
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4
votes
Accepted
Quick derivation of classical probability theory from von Neumann algebraic framework
At the beginning of his talk What actually is free probability theory? Tobias Mai explains how classical probability theory fits into the context of non-commutative probability theory.
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4
votes
Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$
Such "concrete" formulas are addressed in Section 3.4. of my memoir Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory. This is in the ...
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4
votes
Accepted
Does free multiplicative convolution become free additive convolution under logarithm?
No, this is not true. Let's cast the question as follows: Is the eigenvalue distribution of $\log [ (1+\epsilon A)(1+\epsilon B)]$ the same as the one of $\log (1+\epsilon A) + \log (1+\epsilon B)$, ...
- 6,696
3
votes
Accepted
For fixed $\lambda \ge 0$, Integrate the function $f_\lambda(x):=x/(x + \lambda)^2$ w.r.t. Marchenko-Pastur density
$$I(\lambda)=\int_{t_-}^{t_+}\frac{\sqrt{\left(t_+-t\right) \left(t-t_-\right)}}{2 \pi {\gamma} (t+\lambda)^2}\,dt
=\frac{-\sqrt{ {\gamma}^2+2 {\gamma} ( {\lambda}-1)+( {\lambda}+1)^2}+ {\gamma}+ {\...
- 188k
3
votes
Relationship between free probability and deterministic graphs?
The relation between free probability and graphs you are looking for comes from the free product of (rooted) graphs, see for example the paper of Accardi, Lenczewski and Salapata https://arxiv.org/abs/...
3
votes
Why a random variable is better described by its cumulants than by its characteristic function?
I think it is a bit misleading to contrast cumulants as combinatorial quantities with the characteristic function as analytic object. The characteristic function is an analytical device containing ...
- 1,210
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