All Questions
13 questions
36
votes
0
answers
2k
views
Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 188k
20
votes
0
answers
3k
views
+200
What does a product of many Gaussian matrices converge to?
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 ...
- 2,252
6
votes
1
answer
274
views
Spectrum asymptotics for a product of $k$ random matrices?
How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
- 2,252
6
votes
1
answer
353
views
Quaternion Wishart matrices of half-integer dimension?
For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
$$P_\beta(\...
- 188k
4
votes
2
answers
2k
views
Advanced reference and roadmap about random matrices theory
There is few posts on MO that asked about reference on this topic, and I found some difficulty during the process of getting myself into the subject so here is the question.
I really want to hear ...
4
votes
2
answers
2k
views
Eigenvalues of random Hamiltonian matrices
A real $2n\times 2n$ Hamiltonian matrix has the general form
$$H=\begin{pmatrix}
A & B \cr
C & -A^T
\end{pmatrix}
$$
where $A$, $B$ and $C$ are $n\times n$ matrices, and $B$ and $C$ are ...
- 1,038
3
votes
1
answer
155
views
What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?
The Tracy–Widom distributions admit many interpretations.
One of them is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ...
- 9,501
2
votes
1
answer
1k
views
Marchenko-Pastur Law under general covariance structure
Let $x_1,...,x_n\in\mathbb{R}^p$ be i.i.d. random vectors with mean 0 and covariance $\Sigma_p$. Let $S_{n,p}=\sum_{i=1}^nx_ix_i^T/n$ be the sample covariance. We consider the asymptotics of the ...
- 1,495
2
votes
1
answer
187
views
Compute the limit of trace of inverse of square of rank-1 perturbation of Wishart matrix
Let $a \ge 0$, $b,c>0$ be fixed constants, and let $X$ be an $m \times d$ random matrix with entries drawn iid from $N(0,1/d)$. Consider the random psd matrix $S := a 1_m 1_m^\top + b XX^\top + c ...
- 6,853
2
votes
0
answers
129
views
Large deviation principle for product of iid bounded symmetric random variables
Let $n$ and $k$ be positive integers. Let $X$ be the empirical mean of $n$ iid Rademacher random variables. Note that the distribution of $X$ is symmetric about 0, and also $|X| \le 1$ w.p 1. Let $X_1,...
- 6,853
1
vote
0
answers
129
views
Concentration of a combinatorial sum
Let $X=(x_1,\ldots,x_p)$ be an $p \times n$ random matrix with iid entries from $\{\pm 1\}$, distributed so that $\mathbb P(x_{ij} = 1) \equiv 1/2$, where $x_i=(x_{i1},\ldots,x_{in})$. Let $y$ be a ...
- 6,853
0
votes
1
answer
369
views
How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X
Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
0
votes
0
answers
112
views
Additivity of purity of random matrix products
Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as
$$\rho(M)=\frac{n \sum_i \sigma_i^4}{\...
- 2,252