All Questions
Tagged with lattices pr.probability
15 questions
27
votes
7
answers
9k
views
Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 387
16
votes
4
answers
597
views
The lattice 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 ...
- 24.7k
11
votes
2
answers
466
views
Defining measures over frames in place of $\sigma$-algebras
Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...
- 5,502
10
votes
3
answers
644
views
Models with SLE scaling limit
What discrete processes/models have been proven to have scaling limits to $\text{SLE}(\kappa)$, for various $\kappa$?
I know about loop-erased random walk and uniform spanning trees.
What about ...
- 85.6k
9
votes
1
answer
735
views
Where has this structure been observed?
$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...
- 6,741
6
votes
1
answer
180
views
Expected value of the length of the shortest non-zero vector in a lattice?
$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(...
- 609
6
votes
3
answers
425
views
Probability a random matrix contains a short integer vector in its kernel
Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
- 3,377
4
votes
1
answer
829
views
Probability that a "closable" self-avoiding random walk forms a polygon
Consider a self-avoiding random walk on an infinite graph (for concreteness, the grid of 2-dimensional lattice points $\mathbb{Z}^2$), in which on each step, the next position is chosen uniformly at ...
- 141
3
votes
0
answers
185
views
Lattice points in a rotated product-of-balls
Fix $U$ unitary over $\mathbb{R}^{K},$ take $U_n=I_{n\times n}\otimes U$ and denote the unit ball at 0 in $\mathbb{R}^n$ as $B^n$. For $d_1,\dots,d_K>0$, fix $S_n:=U_n\left(\prod_{k=1}^K d_k B^n\...
2
votes
1
answer
267
views
Expected number of identical vertex pairs with the same Euclidean distance on a randomly colored rectangular lattice
Imagine I have an $N$ by $M$ rectangular lattice where I randomly assign one of $k$ colors to every vertex in the lattice. I then write down a list of the ${N*M}\choose{2}$ possible unordered pairs ...
- 23
1
vote
1
answer
555
views
The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself
What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
- 55
1
vote
1
answer
242
views
Two types of random walkers on square lattice
Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index $1$...
user82424
1
vote
0
answers
66
views
Random lattice always has trivial automorphism group?
In example 2.5 of a paper [LS17] written by Lenstra and Silverberg, it is written that “Random” lattices have $Aut(L) = \{ \pm 1 \}$, I guess the 'Random' here refers to the distribution in Siegel's ...
- 253
1
vote
0
answers
78
views
Canonical representation of the a probability distribution for Hammersley Clifford Theorem
I'm reading the following paper
http://www2.stat.duke.edu/~scs/Courses/Stat376/Papers/GibbsFieldEst/BesagJRSSB1974.pdf
On page 7 they give the result that
$$Q(\textbf{x}) = \sum_{1 \leq i \leq n} ...
- 903
0
votes
1
answer
289
views
Dither in Leech lattice quantization!
Can you please help me how to generate a dither signal $\mathbf{U}$, where $\mathbf{U}$ is a random vector of length 24 that is uniformly distributed over the Voronoi region of the Leech lattice.
Best,...
- 197