# Questions tagged [random-matrices]

Statistics of spectral properties of matrix-valued random variables.

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### Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression

From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
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### Trace of product of two Wishart matrices

Let $A,B$ be two independent complex Wishart matrices, $A,B\sim CW_p(\mathbf{I},n)$, that is $A=\frac1n GG^\dagger$& $B=\frac1n QQ^\dagger$ where $G$ and $Q$ are independent $p\times n$ complex ...
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### Random walk on matrix until singularity

Consider a random walk on matrices, where one starts with the matrix $M=I_n$ and at each step randomly chooses an entry of $M$ to increase by $1$. I’m interested in two things about this walk: What’s ...
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### Eigenvalues of the 1D lattice with diagonal disorder?

Consider the adjacency matrix $\mathbf{A}$ of a one dimensional lattice of size $N$. That is, $A$ is a $N\times N$ matrix with $A_{ij}=1$ if vertex $i$ adjacent to vertex $j$ (there exists an edge ...
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### Uniform distribution on pairs of unitary matrices

This question has two parts. In Part 1, I would like to know if the following distribution on pairs of $d$-dimensional unitary matrices has popped up in the literature: "Uniform distribution on ...
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### Random matrices may be asymptotically free but never free themselves?

It is well known that independent $N\times N$ unitarily-invariant random matrices (or independent families of random matrices) may be asymptotically free as $N\to \infty$ with respect to the ...
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### 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 ...
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### Behavior of $\operatorname{Tr}[H(I-H)^s]$ for random positive definite $H$
Suppose $H$ is a random positive definite matrix normalized to have norm 1. Can we say anything about behavior of $f(s)$? $$f(s)=\operatorname{Tr}[H(I-H)^s]$$ Taking $H=A^T A$ with entries of $A$ ...
### How to bound $P(\frac{1}{N}\sum_{i=1}^N \sigma^i X_i^2\ge ax)$ for eigenvalues of a normalized $N\times N$ GOE matrix?
Let $A$ be a normalized $N\times N$ GOE matrix. Let $\sigma^1<\sigma^2<\dots <\sigma^N$. We know that the largest eigenvalue converges $\sigma^N$ to 2 almost surely. Assume that \$X_1,\dots, ...