All Questions
Tagged with random-matrices integration
18 questions
12
votes
1
answer
628
views
A function with unexpectedly simple Legendre transformation
Let $I(x) = \frac{1}{2\pi} \int_{-2}^2 \sqrt{4-y^2}\ln|x-y|dy$. Then $I(x)$ is a concave function and
\begin{equation}
I(x)=
\begin{cases}
\frac{1}{4}x^2-\frac{1}{2}, &\text{if } |x|\leq2 \\
\...
CommunityBot
- 1
1
vote
0
answers
199
views
Integration with respect to Haar measures normalised over a subspace
Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work
$\int_{\mathcal{U}(d)} \frac{\...
- 63.9k
3
votes
0
answers
346
views
Gaussian integral with Vandermonde determinant
I want to compute the following integral, which contains a Gaussian piece and a Vandermonde determinant:
$$
\int d^Nx \,e^{-\frac{1}{2} \sum_{k=1}^N a_k x_k^2 + \sum_{k=1}^N b_k x_k} \Delta(x),
$$
...
- 133
4
votes
1
answer
356
views
Haar integral of rational function of unitaries
I'm trying to compute the following Haar integral over the unitary group:
$$
\int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU.
$$ Is there anything known about the ...
- 63.9k
4
votes
0
answers
75
views
Marginalization of Wishart distribution
Consider the following Wishart distribution
$$
f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1}
$...
0
votes
0
answers
112
views
Solving integral equation with an unknown probability distribution
Considering this system of integral equations, where $\gamma \in \mathbb{R} $ and $\alpha\in \mathbb{C}$ are the unknown to solve :
$$ 1=\int_{-\infty}^{\infty} p(u) \frac{ -1}{\gamma-\left(u-z^{*}+\...
- 117
1
vote
0
answers
103
views
Spectrum of large random asymmetric matrices with correlation
Background:
In their paper, Sommers Crisanti Sompolinsky and Stein derive the spectral distribution of large random matrices $\mathbf{J}$ by studying the following integral:
\begin{equation}
I=\left[\...
- 117
1
vote
1
answer
571
views
How can we do a Gaussian integral over matrix elements?
I am integrating the following Gaussian over all possible matrix elements $J_{ij}$:
$$ I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm{...
- 188k
2
votes
0
answers
55
views
Solving the inverse of a matrix under a uniform distribution
I am looking to solve the following equation:
$$\left(\begin{array}{cc}{ g_{11}} & { g_{12}} \\ { g_{21}} & { g_{22}}\end{array}\right)=\int_{a}^{b} \frac{1}{b-a}\left(\begin{array}{cc}{- g_{...
- 117
1
vote
0
answers
38
views
The condition for random positive matrice integration
For a $k \times k$ positive matrix $V=(v_{ij}), $ write $V=\Gamma'D\Gamma,$ where $D=diag(d_1,d_2,\ldots,d_k)$ with $d_1>d_2>\cdots>d_k,$ and $\Gamma$ is orthogonal matrix. From the result of ...
CommunityBot
- 1
0
votes
1
answer
96
views
Probability of a quantity from Ginibre ensemble
I'm doing a project on random matrices and its applications. I have the joint probability density and want to calculate the probability of $s=\sum_{j=1}^N\lambda_j^2$. So we have
$$P(s)=C_{N,K}\int.....
- 188k
3
votes
0
answers
99
views
Evaluating a Fermi gas problem for a SO(2N+1) matrix integral
I have the following multiple integral derived from a random matrix calculation I wish to evaluate
$$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$
where the $\rho$ functions ...
- 181
1
vote
0
answers
126
views
Dixon-Anderson-Selberg integral variant
I am trying to evaluate or at least obtains bounds for the following integral for $0<\gamma^{2}<2$
$$ \int_{[0,1]^{2n}}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2-\...
- 5,474
2
votes
2
answers
435
views
Non-trivial examples of taking the exponential of an integral
This question is inspired by a recent course I did on random matrix theory and also from common mistakes high-schoolers make in algebra :).
In random matrix theory, one often encounters somewhat ...
- 188k
3
votes
1
answer
1k
views
A Gaussian integral over complex variables by a defined Green's function for a Gaussian ensemble of random matrix
We construct an $N\times N$ matrix $J$ whose elements are drawn from Gaussian distribution with zero mean and variance $\frac{1}{N}$. Since we want to have different variances for different columns, ...
CommunityBot
- 1
2
votes
1
answer
467
views
A slight generalization of Mehta's integral.
I am trying to find the value of following integral
$$\int_{-\infty}^{\infty}\dots\int_{-\infty}^{\infty}\prod_{i=1}^ne^{-\frac{t_i^2}{2}+\alpha_i t_i}\prod_{1\le i<j\le n}\left|t_i-t_j\right|^{2\...
- 121
6
votes
2
answers
3k
views
Weak convergence of random measures
Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
- 255
0
votes
1
answer
382
views
A particular kind of Cauchy Principal Value integral
I am sorry to bother the community with such a narrow question, it may perhaps be a little specific. As I study Random Matrix Theory, I often have to solve integrals of the form
$$\mathcal{P} \int_a^...
- 188k