31 votes
Accepted
What is the Katz-Sarnak philosophy?
The "Katz-Sarnak philosophy" is just the idea that statistics of various kinds for $L$-functions should, in the large scale limit, match statistics for large random matrices from some particular ...
- 52.4k
22 votes
Expected value of determinant of simple infinite random matrix
Very nice problem! Let me recall you that the determinant of $n \times n$ matrices with entries in $\{0,1\}$ is related to the one of $n+1 \times n+1$ matrices with entries in $\{-1,+1\}$: replace ...
- 1,825
19 votes
What is the Katz-Sarnak philosophy?
I'm going to give an answer that discusses some things that the other answers don't go into as much detail on. In particular let me try to explain why the results you mention on classical groups ...
- 162k
17 votes
Accepted
Counting eigenvalues without diagonalizing a matrix
Here is an efficient method. First of all, I must quote that diagonalizing $M$ is not a method, because there is no explicit way to carry this out. It amounts to calculating the roots of a polynomial !...
- 53.1k
16 votes
Accepted
Relative Entropy and p-norm
The argument below is not very elegant,but it is, indeed, a standard exercise. Let $g=\max(f-1,0)$. We shall prove that $$ f\log f\le 2g+\frac 2{p-1}g^p\,. $$ The integration and Holder then give the ...
- 63.9k
13 votes
Jensen Polynomials for the Riemann Zeta Function
The GUE random matrix model predicts that the zeroes should satisfy the local statistics of random matrices. It doesn't predict that the zeroes should satisfy the global statistics of random matrices, ...
- 162k
12 votes
Computing Haar measure of matrices sampled from SO(n)
Indeed, the distribution function of the eigenphases of a random matrix in $\operatorname{SO}(n)$ has a peak at 0 and at $\pm\pi$. It only becomes uniform for large $n$. The joint distribution ...
- 201k
12 votes
What is the implicit pseudorandomness conjecture behind the use of e.g. numpy.random() for probabilistic applications?
The standard definition of pseudorandomness in computational complexity is that the PRNG output cannot be distinguished from truly random output by any polyonmially bounded algorithm (typically, ...
- 88.1k
11 votes
Accepted
Gaussian integrals over the space of symmetric matrices
A recursion formula for the moments of the Gaussian orthogonal ensemble, M. Ledoux (2009). The desired recursion formula for the moment $b_p^N\equiv E\,[\,{\rm tr}\,(S_N^{2p})]$ is I notice a ...
- 201k
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 ...
- 2,572
11 votes
Accepted
Average of the maximum matrix element over the Haar measure
The answer to the question as stated (maximum of row elements) has been solved in Extreme statistics of complex random and quantum chaotic states, see also this MO posting: $$\int dU \max_j |U_{1,j}|...
- 201k
11 votes
Number of permutations with longest increasing subsequences of length at most $n$
There is an explicit determinental formula for these numbers due to Gessel in Symmetric functions and P-recursiveness (JCTA, 1990). Asymptotics were known much earlier and appear in a paper by Amitai ...
- 1,305
11 votes
Laplacian on manifolds and random matrix theory
There are quite a few connections. I will mention a result of mine where the connection is explicit and essential. Fix the metric $g$. Set $m=\dim M$ and assume that ${\rm vol}_g(M)=1$. Denote by $\...
- 35.8k
11 votes
Accepted
Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$
We have $$ C(W) = 2 A \circ A + v v^\top$$ where $v$ is the vector with entries $\|w_i\|^2$, $A$ is the Wishart matrix with entries $w_i^\top w_j$, and $\circ$ is the Hadamard product. From the Schur ...
- 120k
11 votes
Accepted
Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
Call $S$ the set of matrices with repeated eigenvalues and fix a hermitian matrix $A\not\in S$. In the vector space of hermitian matrices, any line through $A$ intersects $S$ in at most finitely many ...
- 3,236
11 votes
What are applications of asymptotic freeness of random matrices?
Here are some applications of free probability of random matrices: Neural networks: The asymptotic freeness assumption plays a fundamental role in the study of the propagation of spectral ...
- 201k
10 votes
Expected value of determinant of simple infinite random matrix
I agree with user39115! I will give a heuristic from random matrix theory because we know the global behaviour of the eigenvalue. First $$A=p 1 +\sqrt{N(p-p^2)}\frac{B}{\sqrt{N}} $$ where $1$ is the ...
- 4,517
10 votes
Reviews of Probability in High Dimension not by Van Handel
High-Dimensional Probability, An Introduction with Applications in Data Science, by Roman Vershynin (draft version freely available) The two texts by Van Handel and Vershynin are compared here: ...
- 201k
10 votes
Accepted
Scaling in Mehta's integral
Yes, this follows by the de la Vallée-Poussin necessary and sufficient condition for the uniform integrability. Indeed, suppose that \begin{equation} \gamma n^2\to a \end{equation} (as $n\to\infty$) ...
- 143k
9 votes
The expected square of the determinant of a random row stochastic matrix
Hidden in the comment by Victor Kleptsyn is a really nice argument. Since nobody upvoted that comment yet (I just did) it's probably worth expanding it. From a probabilistic approach it's more ...
- 4,748
9 votes
Accepted
A question about the paper "The Condition Number of a Randomly Perturbed Matrix"
One does not need to have $n^{-B-3/2}/2$ to be equal to $0.1$, it is enough for it to be less than or equal to $0.1$, which is certainly the case for $n$ large enough. Thanks for pointing out this ...
- 120k
9 votes
Jensen Polynomials for the Riemann Zeta Function
Unfortunately, I cannot comment. https://www.youtube.com/watch?v=HAx_pKUUqug 56:28
- 89
9 votes
Accepted
Eigenvalues and eigenvectors of Gaussian random matrices
This is the Ginibre ensemble, see Eigenvalue statistics of the real Ginibre ensemble for the eigenvalue distribution. For an $N\times N$ matrix with $N\gg 1$ there are on average $\sqrt{2N/\pi}$ ...
- 201k
9 votes
Laplacian on manifolds and random matrix theory
Much of the literature on random metrics constructs these as follows. Start from a reference metric $g_0$ on the compact Riemannian manifold $M$. The corresponding Laplacian $\Delta_0$ has ...
- 201k
9 votes
Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
Here is a (I think) mathematically correct, but clearly morally wrong, answer via extreme overkill. Upon multiplying by $i$, we may work instead with skew-Hermitian matrices, i.e., with $\mathfrak u(n)...
- 14k
9 votes
Accepted
Joint distribution of minor of Wigner Hermitian matrices
There is certainly no asymptotic independence between $\det M_{11}, \det M_{22}$. From the base times height formula for parallelepipeds we see that \begin{align*} \frac{|\det M_{12}|}{|\det M_{22}|} ...
- 120k
9 votes
Accepted
Taylor expansion of Stieltjes Transform
You certainly can't expand around $z=0$ if "$z$ has to be sufficiently large". Expand around $1/z =0$: $$ \frac{1}{N} \sum_{n=1}^{N}\frac{1}{z-\lambda_{n} } = \frac{1}{N} \frac{1}{z} \sum_{n=...
- 7,069
9 votes
Accepted
Spectral density of symmetrized Haar matrix
Since $O$ is orthogonal, $O^\top=O^{-1}$ commutes with $O$, hence the eigenvalues $\mu_n$ of $O+O^\top$ are related to the eigenvalues $e^{i\phi_n}$ of $O$ by $\mu_n=2\cos\phi_n$. The spectral density ...
- 201k
9 votes
Accepted
Wishart matrices: are eigenvalues and eigenvectors independent?
A proof is on page 80-81 and 90 of Forrester, the probability distribution function of $W=X^\top X$ is $\propto e^{-\tfrac{1}{2}\operatorname{tr}W}(\operatorname{det}W)^{(n-m-1)/2}$, for an $n\times m$...
- 201k
8 votes
Accepted
What is the Essential Difference Between Random Matrices and Random Graphs?
I suppose the main point is that the typically studied random graph models are not directed or weighted and they generally don't have self loops. Under your correspondence, this means they are ...
- 30.1k
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