All Questions
11 questions
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Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
- 156
2
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2
answers
228
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Minimal conditions on random vector $X \in R^n$ to ensure that $\lim_{t\to 0^+}\sup_{\|w\|_p = 1}\sup_{u \in \mathbb R}\mathbb P(|X'w-u| \le t)=0$
Let $X$ be a random variable on $\mathbb R^n$ and let $S_p^n := \{w \in \mathbb R^n \mid \|w\|_p = 1\}$ be the unit-sphere w.r.t to the $\ell_p$-norm in $\mathbb R^n$. We will be particularly ...
- 5,474
3
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1
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155
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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 ...
- 1,765
2
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0
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458
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Random variable matrix exponential
I am trying to find out the distribution of a matrix exponential which is a function of a random variable. My mathematics background is very limited and I hope I can receive some help from here.
What ...
2
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1
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1k
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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 ...
- 7,514
5
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1
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A general formula for Gaussian integrals over matrix elements
The question I have is quite specific. So in the hope that this post might help others in the future, my problem boils down to solving the following integral:
$$I_\tau=\int \prod_{i, j=1}^{N} d J_{i ...
- 188k
9
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1
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652
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Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 128k
13
votes
2
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656
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Random matrix with given singular values
Let $\sigma_1\geq\sigma_2\geq...\geq\sigma_n\geq0$ be any deterministic sequence of positive real numbers such that $\sum_{i=1}^n\sigma_i^2=1$. Let
$$D=diag\{\sigma_1,...,\sigma_n\}\in\mathbb{R}^{n\...
- 4,361
0
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1
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622
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Is there any way to compare between diagonals of a resolvent and a Cauchy transform?
Say $A$ is a symmetric matrix of $n$ dimensions. Then let the ``resolvent" of $A$ be the matrix valued function $R_A(z) = \frac{1}{z-A}$ and its Cauchy transform be the real valued function $C_A(z) = ...
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- 1
5
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Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
- 6,206
15
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2
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3k
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What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
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