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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
1
vote
Eigenvectors of random unitary matrices
The crucial point is that the eigenvectors are not unique, as David Handelman commented. There is a gauge freedom since you can change their phase at will. If you want the map $U\to(D,V)$ to be biject …
0
votes
Sum of Square of the Eigenvalues of Wishart Matrix
The joint probability distribution of the eigenvalues of $S$ is proportional to
$$ \rho(S)=e^{-{\rm Tr}(S)}\det(S)^{a/2}|\Delta(S)|,$$
where $a=m-d-1$ and $\Delta(S)$ is the Vandermonde. The average v …
6
votes
Fourier transform of eigenvalue distribution of GUE matrices
Complementing the answer by Carlo, if you take all $k$'s equal you have
$$f_{\rm GUE(d)}(k,...,k)\propto \int dX e^{ik{\rm Tr}(X)}e^{-\frac{d}{2}{\rm Tr}(X^2)}.$$
Taking $x$ to be any real diagonal el …
1
vote
Spectral gap of $AA^{T}$ for Bernoulli random matrix A
The spectrum of random matrices has some universality properties, in the sense that they do not depend on details of the distribution of the matrix elements. For instance, the spectral gap, i.e. the s …
10
votes
Accepted
Probability of complex eigenvalues
The probability that a $n\times n$ real matrix (with elements that are independent random variables with standard normal distributions) has only real eigenvalues is given by
$$ 2^{-n(n-1)/4}$$
Refere …
11
votes
What is the Katz-Sarnak philosophy?
I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgo …
2
votes
Non combinatorial random matrix theory
I think this text by Eynard, Kimura and Ribault may interest you. There are some of those diagrams in chapter 2, but there are also nice connections to algebraic geometry, loop equations and integrabl …
2
votes
Moments of Matrix Gamma distribution
A good source on this topic is chapter 7 of Roob Muirhead's book, "Aspects of Multivariate Statistical Theory", or chapter 3 of Peter Forrester's "Log-Gases and Random Matrices". In particular, there …
7
votes
Accepted
Expected value of the largest singular value of a random matrix with entries in $N (0,1)$
If $A$ is a Gaussian random matrix as you describe, then the ensemble of matrices given by $A^TA$ is known as the Wishart ensemble, or the Laguerre ensemble. It has been extensively studied, and you c …
0
votes
Probability of positive definiteness of a random matrix
I think the following papers are closely related to what you are looking for:
How many eigenvalues of a Gaussian random matrix are positive? (2011)
Index Distribution of Gaussian Random Matrices (20 …
20
votes
Accepted
A determinantal formula
The case $k=n$ is a consequence of the identity
$$\int \det(f_j(s_k))\det(g_j(s_k))\prod_{j=1}^N d\mu(s_j) = N!\ \det\left(\int d\mu(t) f_j(t)g_k(t)\right)$$
which I have seen under the names "And …
8
votes
Advanced reference and roadmap about random matrices theory
The Oxford handbook of random matrix theory (Oxford University Press, 2011), edited by G. Akemann, J. Baik, P. Di Francesco, is an excellent reference, which covers a wide variety of properties and ap …
4
votes
What are fun elementary subjects in probability?
The book "The Drunkard's Walk: How Randomness Rules Our Lives" by Leonard Mlodinow is probably close to what you are looking for.
1
vote
Accepted
Alternative formula of a Green's function for average density of eigenvalues of random matrix
I'll post an answer to spell out all the details.
You have $$G(ω)=\frac{1}{N}E\left[{\rm Tr}\frac{1}{Iω−J}\right]=\frac{1}{N}E\left[\sum_\lambda\frac{1}{ω−\lambda}\right]$$
This can be written as $$ …
2
votes
Calculate correlation values of an ensemble of $N\times N$ real asymmetric random matrix fro...
Just expanding a bit on the comment by Beenakker. If you write the Gaussian measure in terms of the matrix elements,
$$ \prod_{ij}\exp[−\frac{N}{2(1−τ^2)}(J_{ij}^2-\tau J_{ij}J_{ji})]$$
You can see …