Skip to main content
MathOverflow

Questions tagged [free-probability]

Ask Question

The tag has no summary.

75 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
0 votes
0 answers
52 views

I am trying to learn random matrix theory myself from the book https://zhenyu-liao.github.io/book/. For a random $p\times p$ symmetric matrix $M$, let $m_{\mu_M}(z)$ denote the Stieltjes ...
alex1998's user avatar
  • 71
4 votes
1 answer
356 views

I am new to Random Matrix Theory (RMT), and I am self reading the book Zhenyu Liao's book on RMT https://zhenyu-liao.github.io/book/. A fundamental object of study is the so-called Stieltjes transform ...
alex1998's user avatar
  • 71
0 votes
0 answers
70 views

Let $G=(V, E)$ be a graph on $L$ nodes. To each vertex $i\in V$, we associate a symbol/generator $g_i$. Two generators $g_{i}, g_{j}$ commute if $\{i, j\}\in E$. Let $\Sigma$ denote the set of all ...
Raghav's user avatar
  • 371
0 votes
1 answer
138 views

Let $\{\mathcal{A}_i\}_{i=1}^3$ be a collection of unital free sub algebra in a noncommuttaive probability space $(\mathcal A,\varphi)$ (see https://arxiv.org/pdf/0911.0087 for definitions). Is it ...
A beginner mathmatician's user avatar
  • 1,980
0 votes
0 answers
68 views

There are Isotropic/Anisotropic local laws for Wigner matrices. I have also seen many works such as The Isotropic Semicircle Law and Deformation of Wigner Matrices on analysis of eigenvalues of low ...
No one 's user avatar
  • 1
0 votes
0 answers
56 views

I am trying to understand the method of proof of Proposition 4 in Demni, N. Free Jacobi Process. J Theor Probab 21, 118–143 (2008). https://doi.org/10.1007/s10959-007-0110-1 . If I understand the ...
mortenmcfish's user avatar
  • 1
2 votes
0 answers
107 views

Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
user3826143's user avatar
  • 303
1 vote
1 answer
351 views

Let $X$ and $Y$ be two $n\times n$ random matrices that have zero measure over degenerate matrices. For a positive definite matrix with eigen-decomposition $A = U \Lambda U^\top$, let $\log(A) = U \...
kvphxga's user avatar
  • 187
1 vote
1 answer
209 views

Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
dohmatob's user avatar
  • 7,043
1 vote
1 answer
189 views

Let $\mu$ be a probability measure and $\mu_N$ be a sequence of probability measures. For simplicity we may assume them to have compact supports contained in $[-1,1]$. Define $$G_\mu(z):=\int\frac{\mu(...
JJJZZZZZ's user avatar
  • 400
0 votes
1 answer
147 views

Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
dohmatob's user avatar
  • 7,043
1 vote
2 answers
414 views

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
Yaroslav Bulatov's user avatar
  • 2,394
1 vote
0 answers
113 views

Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$ ...
Panda Jonas's user avatar
  • 31
10 votes
2 answers
760 views

In around 1990 Voiculescu showed asymptotic freeness of certain random matrices, i.e., free independence when the matrix size goes to infinity. Since then this link between free probability and random ...
Bart's user avatar
  • 201
10 votes
3 answers
1k views

This question is cross-posted from MSE.$\newcommand{\E}{\mathbb{E}}$ Near the end of "Finite Operator Calculus" (1976), G.C. Rota writes: Note that one can define cumulants relative to any ...
Daigaku no Baku's user avatar
  • 635
8 votes
3 answers
606 views

From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
Samuel Johnston's user avatar
  • 534
2 votes
0 answers
192 views

It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
user510187's user avatar
  • 21
23 votes
0 answers
3k views

Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$. Is ...
Yaroslav Bulatov's user avatar
  • 2,394
3 votes
2 answers
353 views

Consider a random i.i.d matrix $\mathbf{A}_{m\times n}$ with entries generated from a complex Gaussian distribution with zero mean and unit variance. I am interested in the large dimension analysis of ...
Math_Y's user avatar
  • 199
2 votes
0 answers
118 views

Let $F=F_{m,d}$ be a random $m \times d$ matrix with iid entries from $N(0,1)$. Let $A=A_d$ and $B=B_d$ be deterministic $d \times d$ positive-definite matrices. In case it helps, it may be assumed ...
dohmatob's user avatar
  • 7,043
1 vote
0 answers
89 views

Let $A=A(d)$, and $B=B(d)$ be (sequences of) deterministic positive-definite $d \times d$ matrices and let $X$ be an $n \times d$ random matrix with iid rows from $N(0,A)$. Let $R$ be the resolvent of ...
dohmatob's user avatar
  • 7,043
1 vote
0 answers
102 views

Let ${\mathbb F}_{q}$ be a given finite field. How many couples of $n\times n$ matrices $\left(A,B\right)$ over ${\mathbb F}_{q}$, such that $\gcd\left(\mu_{A}\left(\lambda\right),\mu_{B}\left(\lambda\...
Yossi Peretz's user avatar
  • 21
9 votes
1 answer
455 views

It is said by Halmos, P.R.; in "Lectures on ergodic theory" "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
Kadi Harouna Illia's user avatar
  • 101
5 votes
1 answer
213 views

Let $X$ be an $n \times d$ random matrix with iid entries from $N(0, 1/d)$. Let $S:=X^\top X/n$, a $d \times d$ Wishart matrix and let $T = e^{S} := \sum_{k=0}^\infty \dfrac{S^k}{k!}$ be its ...
dohmatob's user avatar
  • 7,043
3 votes
0 answers
111 views

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 ...
dohmatob's user avatar
  • 7,043
5 votes
0 answers
185 views

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 $U(n)$,...
user447643's user avatar
  • 163
0 votes
0 answers
115 views

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,\...
dohmatob's user avatar
  • 7,043
1 vote
1 answer
152 views

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 ...
dohmatob's user avatar
  • 7,043
20 votes
2 answers
2k views

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 ...
Pulcinella's user avatar
  • 6,191
0 votes
2 answers
626 views

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\...
Shadumu's user avatar
  • 95
0 votes
1 answer
208 views

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, ...
Chris's user avatar
  • 321
1 vote
1 answer
171 views

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 ...
dohmatob's user avatar
  • 7,043
2 votes
0 answers
207 views

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 ...
dohmatob's user avatar
  • 7,043
3 votes
1 answer
260 views

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 ...
Andrea Tani's user avatar
  • 87
2 votes
1 answer
371 views

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 $...
Andrea Tani's user avatar
  • 87
1 vote
1 answer
161 views

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 $n,d \to \infty$ ...
dohmatob's user avatar
  • 7,043
3 votes
1 answer
360 views

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 ...
dohmatob's user avatar
  • 7,043
2 votes
1 answer
504 views

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 ...
dohmatob's user avatar
  • 7,043
3 votes
0 answers
280 views

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 ...
dohmatob's user avatar
  • 7,043
4 votes
3 answers
1k views

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 ...
dohmatob's user avatar
  • 7,043
1 vote
1 answer
184 views

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 ...
dohmatob's user avatar
  • 7,043
8 votes
1 answer
483 views

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 ...
Jon Bannon's user avatar
  • 7,177
2 votes
1 answer
1k views

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 $...
Mathguest's user avatar
  • 1,691
2 votes
1 answer
292 views

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 ...
Mathguest's user avatar
  • 1,691
0 votes
0 answers
157 views

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 ...
Mathguest's user avatar
  • 1,691
0 votes
1 answer
513 views

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 ...
Mathguest's user avatar
  • 1,691
1 vote
1 answer
454 views

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_{...
Iliyo's user avatar
  • 147
2 votes
0 answers
132 views

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 ...
Iliyo's user avatar
  • 147
0 votes
1 answer
284 views

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 ...
Iliyo's user avatar
  • 147
4 votes
2 answers
452 views

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,...
Austen's user avatar
  • 1,038

1
2