# Tagged Questions

**7**

votes

**3**answers

682 views

### A conjecture about the entropy of matrix vector products

Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ ...

**7**

votes

**1**answer

138 views

### Distribution of entries of a doubly-sorted random matrix

Take an $n \times n$ random matrix whose entries are i.i.d. with uniform distribution in $[0,1]$. Look at the sums of the elements of each row and then permute the rows so that these sums form an ...

**3**

votes

**0**answers

133 views

### Kasteleyn, Gessel-Viennot and eigenvalues

The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by ...

**13**

votes

**2**answers

572 views

### Laws of Iterated Logarithm for Random Matrices and Random Permutation

The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then
$$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, ...

**10**

votes

**1**answer

358 views

### A simple proof for a theorem of Szekeres and Turán

Szekeres and Turán found in 1937 a formula for the sum of the squares and the sum of the fourth powers of determinants of all $n$ by $n$ matrices with $\pm 1$ entries. (The sum of squares case follows ...

**15**

votes

**0**answers

739 views

### The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...

**15**

votes

**4**answers

478 views

### The latice spanned by $m$ random 0-1 vectors of length $n$

Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...

**2**

votes

**4**answers

439 views

### Statistical computation in matrix. Rows before columns? riddle..

First I'll phrase the question as a riddle, and than as a general math problem.
We have 12 lettered vases $(A,B,...,L)$, in each vase there are 30 numbered balls (1-30). In each ball there is some ...

**9**

votes

**1**answer

559 views

### Average over Random Permutations

Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the ...

**11**

votes

**0**answers

315 views

### Lower Bound on the Volume of Certain Polytopes

Given a partition $\rho\in\mathcal{P}(n)$ with $k$ blocks
$$
\rho=\{B_1,B_2,\ldots,B_{k}\}
$$
we can define the set of equations
$$
E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in ...

**8**

votes

**0**answers

344 views

### Has the technique of “sprinkling” been used in studying random matrices?

In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of ...

**11**

votes

**2**answers

917 views

### Evaluation of a combinatorial sum (that comes from random matrices)

I'm looking for an elementary combinatorial/generating function/etc proof of the following result:
For nonnegative integers $r$,
$$\frac{1}{r!} = \sum_{p_0+p_1+\cdots = r} ...