Questions tagged [free-probability]

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combined experiments consisting sub-experiments which became dependent on each other

The following is stated in the Papoulis' book (p. 48) If we have two probability spaces ${(Ω_1, F_1, P_1)}$ and ${(Ω_2, F_2, P_2)}$, whether these two experiments are independent or dependent, the ...
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Explaning why the spectrum of a setting simple structure random matrix is always spiked ($d-1$ eigenvalues close to zero, and $1$ away from zero)

For concreteness, let $m=500$, $d=600$, $N=1000$. Let $W$ be and $d \times m$ matrix with unit-norm rows and let $u$ be a uni-norm vector of length $m$. Given a binary vector $b$ of length $m$, length ...
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4 votes
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105 views

Determinant bounded away from zero

Suppose $P(A_1,\dots,A_k,A_1^{-1},\dots,A_k^{-1})$ is a noncommutative polynomial with positive coefficients. We may then consider the map $g:U(n)^k\rightarrow\mathbb{C}$ from the unitary group $U(n)$,...
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Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function

Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
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Defining an involution on the free product of unital algebras

Let $A$ and $B$ be unital complex algebras (so the ground field is $\mathbb{C}$) and consider their free unital product $A*B$ together with the canonical inclusions $i_A: A \hookrightarrow A * B$ ad , ...
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72 views

Limiting value of $\dfrac{1_n^\top B^{-1} A B^{-1} 1_n}{d}$, where $A=WW^\top + a I_n$, $B = WW^\top + b I_n$, and $W \sim N(0,\Sigma_d)$

Let $n$ and $d$ be positive integers with $$ n,d \to \infty, \quad n/d \to \rho \in (0,\infty). $$ Let $\Sigma_d$ be a psd matrix such that $\mbox{trace}(\Sigma_d) = 1$. $\|\Sigma_d\|_{op} = \mathcal ...
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Asymptotics of $\mathbb E_W 1_n^TX_1^{-1}X_2X_1^{-1}1_n$, for $X_i = \mu_i\mu_i^T + b_i WW^T + c_i I_n$, and $W=(w_1,\ldots,w_n) \sim N(0,C)$

Let $n$ and $d$ be positive integers such that $$ n,d \to \infty,\quad d/n \to \gamma \in (0,\infty). \tag{1} $$ Let $W$ be a random $n \times d$ matrix with entries from $N(0,\Sigma_d)$, where $\...
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20 votes
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Is there a noncommutative Gaussian?

In classical probability theory, the (multivariate) Gaussian is in some sense the "nicest quadratic" random variable, i.e. with second moment a specified positive-definite matrix. I do not ...
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Find $a,b,c \in \mathbb R$ s.T $\|f(ZZ^\top)-(a I_n+bXX^\top + c 1_n1_n^\top)\|_{op} \to 0$, where $x_1,\ldots,x_n \sim N(0,C)$ and $z_i=x_i/|x_i|$

Let $f:\mathbb R \to \mathbb R$ be a "sufficiently smooth" function. Let $n$ times $d$ be comparably large positive integers. For example, assume $$ n,d \to \infty,\quad d/n \to \gamma \in (...
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Bounds on trace of ratio of rank-1 perturbed Wishart matrices

Let $a,b,d,e,f \ge 0$ and $c>0$. Let $X$ be a random $n \times m$ matrix with iid entries from $N(0,1/m)$ and set $$ \begin{split} W &:=XX^\top,\\ D &:= W \circ I_n = \mbox{diag}(w_{1,1},\...
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  • 5,580
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2 answers
159 views

Free multiplicative convolution of two random matrices

Given two freely independent random hermitian matrices $A$ and $B$ following laws $\mu, \nu$, one can compute the empirical spectral distribution of $AB$ by their free multiplicative convolution $\mu\...
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Q-Gaussian processes and probability space

It is well known that for isonormal Gaussian processes, one can decompose the space $L^2(\Omega)$ into Wiener chaos so that the space $L^2(\Omega)$ is isomorphic to the direct sum of the Wiener chaos, ...
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Probabilistic lower and upper-bounds for a certain random quartic form involving gaussian random matrices

Let $d,m \to \infty$ (integers) with $m/d \to \rho \in (0, \infty)$. Let $C$ be a $d \times d$ psd matrix with $trace(C)=\mathcal O(1)$, and let $w_1,\ldots,w_m$ be iid uniformly distributed on the ...
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2 votes
0 answers
55 views

Asymptotic lower and upper bounds for the eigenvalues of hadamard product $W \circ W$, where $W$ is a large Wishart matrix

Let $n$ and $d$ be large positive integers with $n,d \to \infty$ such that $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ random matrix with iid copies of log-concave isotropic ...
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3 votes
1 answer
118 views

Combinatorial formula to compute the moments of the product of two free random variables

I found in the PhD thesis Moments method for random matrices with applications to wireless communication the following combinatorial formula to compute the free moments of the product of two random ...
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1 vote
1 answer
160 views

How to compute the first moment of the distribution of the convolution of Marcenko-Pastur law with a not iid matrix?

Let $\mathbf{F}$ denote an M × N matrix whose entries are independent zero-mean complex random variables, the limiting eigenvalue distribution is given by the Marchenko Pastur law $MP_{\beta}$, where $...
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1 vote
1 answer
108 views

Limiting eigenvalue distribution of $YY^\top$ where $Y_{ij} = X_{ij} + a$ and $X$ has iid rows from an isotropic log-concave distribution

Let $a \in \mathbb R$ be a determinstic scalar and let $X$ be and $n \times d$ such that the $n \times n$ psd random matrix $S=XX^T$ has limiting eigenvalue distribution $\mu$, when $n,d \to \infty$ ...
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2 votes
1 answer
124 views

For fixed $\lambda \ge 0$, Integrate the function $f_\lambda(x):=x/(x + \lambda)^2$ w.r.t. Marchenko-Pastur density

In trying to solve another the problem posed in the question https://www.mathoverflow.net/q/385777/78539, I'm led to consider the following problem. Let $\mu_\gamma$ be the Marchenko-Pastur ...
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1 vote
1 answer
237 views

High-probability lower bound for norm of least squares solution when both design matrix $X$ and response vector $y$ are random (and independent)

Let $n,d \to \infty$ with $n/d \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ matrix independent rows uniformly distributed on the the unit-sphere in $\mathbb R^d$ and let $y$ be a ...
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3 votes
0 answers
182 views

Using linearization trick (free probability) to compute limiting singular-value density of $R=XY+Z+A$ (or equivalently, of $RR^\top$)

Disclaimer. I only started learning the subject of free probability $1$ day ago, and I'm still trying to absorb the fundamentals, while applying them to my own specific problems arizing in the ...
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3 votes
3 answers
360 views

Quick derivation of classical probability theory from von Neumann algebraic framework

Watching (the begining of) a lecture on free probability theory by Dimitri Shlyakhtenko https://www.youtube.com/watch?v=F8Urtr39jM0, I'm led to consider the following question Question. How can one ...
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  • 5,580
1 vote
1 answer
121 views

Bounds for the extreme singular-values of random matrix with thresholded entries

Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random ...
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8 votes
1 answer
362 views

Is $L(\mathbb{Z}*\mathbb{Z}_{2})$ a free group factor?

This is a reference request for something that is likely to be well-known to operator algebraists. I will not, therefore, include the technical definition of free product of finite von Neumann ...
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2 votes
1 answer
374 views

Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data

Let $Z=[z_1, \dots z_n]$ be a $d \times n$ matrix, where the $z_i$'s are iid random vactors with mean $\mu \in \mathbb{R}^d$ and $d \times d$ (population) covariance matrix $\Sigma$, but the entries $...
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2 votes
1 answer
106 views

Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the "features" are correlated: references

Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
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0 votes
0 answers
59 views

Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?

Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
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0 votes
1 answer
79 views

Marcenko Pastur law when the dimensionality/sample size ratio $p/n \to 0, \infty$? Lack of resources?

Let $X: \Omega \to \mathbb{R}^{p \times n}$ be a random matrix so that each entry $X_{ij}$ is a random variable with $\mathbb{E}X_{ij}=0, \mathbb{E}X_{ij}^2=\sigma^2$ I was wondering what would ...
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1 vote
1 answer
277 views

The problems of global asymptotic freeness

Let $X_{N}\in\mathcal{M}_{N}\big(L^{\infty-}(\Omega,\mathbb{P})\big)$ be a $N\times N$ random complex matrix such its entries $(x_{ij}, 1\leq i, j\leq N)$ be $i.i.d.$, centred with variance $1$. $X_{...
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2 votes
0 answers
112 views

How high to build a dam in Amsterdam in order that the probability of a flooding within the next 100 years be less than 1%? [closed]

In the preface of the monography, Pr. D. Voiculescu Wrote: "Free probability and operator algebras The well-known question about how high to build a dam in Amsterdam in order that the probability ...
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  • 139
0 votes
1 answer
159 views

Are they the $N\times N$ random matrices with real or complex entries asymptotically free?

The $N\times N$ random matrices with real or complex entries are generalizations of non-hermitian gaussian ensembles, also known as Girko ensemble: the entries are independent and identically ...
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  • 139
3 votes
2 answers
273 views

Does free probability have anything to say about the eigenvalue correlations of random matrices?

Free probability provides a compact route to compute the average eigenvalue density for various families of random matrices in the large $N$ limit. Does it provide any route to eigenvalue correlations,...
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0 votes
1 answer
237 views

Berry-Esseen type theorem for Monotonic independence

The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases ...
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1 vote
0 answers
160 views

Convergence in probability in the setting of free probability

Let $A_n$ and $B_n$ be sequences of positive-definite random matrices whose empirical spectral distributions converge to (possibly different) limiting spectral distributions $\mathcal A$ and $\mathcal ...
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2 votes
1 answer
216 views

The fundamental group of the von Neumann algebra of a free group of infinite rank

It is well-known that that the fundamental group (in the sense of Murray and von Neumann) of the factor $L(F_{\mathbb{N}})$ is $\mathbb{R} \smallsetminus \{0\}$. I think that by the cutting and ...
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5 votes
1 answer
181 views

The definition of amalgamated free product for general von Neumann algebras

When discussing the amalgamated free product von Neumann algebras, people often assume that the algebras are $\sigma$-finite. I am wandering if there is any literature on the amalgamated free product ...
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5 votes
3 answers
690 views

Why a random variable is better described by its cumulants than by its characteristic function?

It is a classical and well known problem that a random variable $X$ is not uniquely determined by its moments $\mathbb{E}(X_n)$. The moment problem is the problem of determining the probability ...
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  • 139
1 vote
1 answer
158 views

Brown measure of left shift operator

Let $L$ be the left shift operator on $\ell^2(\mathbb{Z})$ with trace $\tau(T) := \langle T \delta_0, \delta_0 \rangle$. How can I show that the Brown measure of $L$ is the uniform measure on the ...
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  • 113
25 votes
3 answers
3k views

Why is free probability a generalization of probability theory?

Note: This question was already asked on Math.SE nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope ...
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  • 2,478
1 vote
4 answers
500 views

Independence of two noncommutative observables

If two observables are free, you can find the joint distribution of these two observables. But, by Heisenberg's Uncertainty Principle it is impossible unless $X$ and $Y$ are such that $XY=YX$. Is ...
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  • 139
4 votes
1 answer
227 views

Can we solve the FGF problem by finding an appropriate action?

If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
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3 votes
0 answers
198 views

Uniqueness of the reduced free product of unital completely positive maps

For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = *...
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3 votes
1 answer
134 views

Existence of free operators, independent and with given distributions

Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...
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  • 411
4 votes
1 answer
219 views

Spacing of the largest singular values of Wishart matrix

Let $X \in \mathbb{R}^{n \times p}$ consist of iid $\mathcal{N}(0,1)$. Assume that $n/p$ converges to a positive constant. Denote by $\sigma_1 \ge \sigma_2 \ge \cdots \ge \sigma_{\min(n,p)} \ge 0$ the ...
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1 vote
1 answer
240 views

Relating the R-transform in free probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
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  • 1,259
10 votes
1 answer
322 views

Non-abelian freeness of SU_2

The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution. The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution. ...
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8 votes
0 answers
320 views

C* algebras of free semicircular systems

It was shown by Pimsner and Voiculescu in 1982 that the reduced group $C^{*}$-algebras $C^{*}_{r}(\mathbb{F}_{n})$ and $C^{*}_{r}(\mathbb{F}_{m})$ are isomorphic if and only if $n = m$ (here, $\mathbb{...
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4 votes
1 answer
614 views

Distribution of sum of freely independent Marchenko-Pastur measures

Given freely independent random variables $X_i$ with Marchenko-Pastur measures $\mu_i$, $i\in\{1,\dots,n\}$ how can we find the distribution of the scaled sum of these random variables $\sum_ia_iX_i$....
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24 votes
3 answers
2k views

Why did Voiculescu develop free probability?

I was recently asked why Voiculescu developed free probability theory. I am not very expert in this and the only answer I was able to provide is the classical one: he was challenging the isomorphism ...
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10 votes
2 answers
2k views

Intuition behind the spectral density of random matrices

Hi, I have read that the spectral density of an NxN random matrix consisting of iid random variables with zero mean and unit variance converges as N goes to infinity to the uniform distribution on ...
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3 votes
0 answers
206 views

Recursive formula for joint moments in free probability

Suppose $\mathfrak{A}$ is an algebra (over $\mathbb{C}$, let's say), $\phi$ a linear functional on $\mathfrak{A}$, and $A_1, A_2 \subset \mathfrak{A}$ subalgebras which are $\phi$-freely independent. ...
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