All Questions
Tagged with pr.probability mg.metric-geometry
223 questions
8
votes
1
answer
314
views
Longest induced cycles in random geometric graphs near criticality
We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge ...
- 7,857
15
votes
2
answers
571
views
Spearing rolling hula hoops
Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, d-\...
- 151k
5
votes
1
answer
1k
views
Probability distribution or the distance between two points in $n$-dimensional Euclidean space after a random perturbation of one point
Take two points, $p_0$ and $p_k$, in $n$-dimensional Euclidean space, where $d(p_0,p_k)$ is the distance between the points. Now, draw an $n$-sphere of radius $r$ centered on $p_0$ and uniformly ...
- 55
2
votes
2
answers
1k
views
Distance measure for noisy $SE(3)$ transforms
I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...
- 21
0
votes
1
answer
136
views
What is the area of the piece of an $n$-sphere within a given angle of a vector? [closed]
Let $x$ be the unit vector $(1,0,0,\ldots,0)$ in $\mathbb{R}^n$, and let $A(\theta)$ be the subset of $\mathcal{S}^{n-1}$ whose angle to $x$ is less than $\theta$, i.e.
$$ A(\theta) = \left\{ y \in \...
- 256
17
votes
2
answers
1k
views
What kind of probability distribution maximizes the average distance between two points?
If $f$ is a probability distribution on the unit disk in $\mathbb{R}^2$, and $X_1$ and $X_2$ are two independent samples from $f$, then what is the distribution $f^*$ that maximizes the average ...
- 173
0
votes
0
answers
404
views
When is the median closest nearest-neighbor distance larger than the mean closest nearest-neighbor distance?
Consider a random Poisson process in an $d$-dimensional cube of arbitrary size (alternatively, consider an arbitrarily large $(d-1)$-dimensional sphere in an $d$-dimensional space). If we have a ...
0
votes
0
answers
104
views
Minimum distance larger than a fraction $f$ of the closest nearest-neighbor distances for points placed by a random Poisson process?
Consider a random Poisson process on arbitrarily large volume in $R^d$ enclosed by an $R^{(d-1)}$ dimensional sphere. The process terminates when a density of points $\rho$ is achieved (letting $N$ ...
- 1
2
votes
2
answers
331
views
what's the best way to characterise the distribution of prime elements in simple perfect squared squares
DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
0
votes
1
answer
376
views
Probability of disc-disc overlap for discs placed with uniform probability on a surface until a density $\rho$ is achieved
Imagine I place discs of radius $r$ on a two-dimensional plane, selecting their positions with uniform probability across the surface of the plane, and stop when I reach a disc density $\rho$. As a ...
- 13
9
votes
1
answer
784
views
Width of a random convex polygon
Consider a planar (2D) random walk comprised of N steps.
Consider the minimum convex polygon enclosing the N points visited by the random walker.
Assume the definition of the width of a convex polygon ...
- 91
6
votes
1
answer
330
views
Best and worst centrally symmetric convex covering shapes
Suppose you have a centrally symmetric convex 2D shape $C$ of area $A$, and you randomly throw
down copies of $C$ on the plane so that each $C$-center lies within a given unit square $S$,
until $S$ is ...
- 151k
60
votes
1
answer
7k
views
Probability that a stick randomly broken in five places can form a tetrahedron
Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...
- 7,831
21
votes
2
answers
1k
views
Probability that a convex shape contains the unit ball
This probability problem seems interesting and I don't know if it has been solved before.
If you pick $n$ points uniformly at random from the surface of a $d$ dimensional sphere of radius $r>1$ ...
- 3,377
0
votes
1
answer
89
views
Is a $CD(K,\infty)$ space a length space?
Let $(X,d)$ be a complete and separable metric space endowed with a nonnegative Borel measure $\mu$ with support $X$ and satisfying
\begin{eqnarray}
\mu(B(x,r))<\infty,\quad\mbox{for every }x\in X\...
- 11
6
votes
1
answer
273
views
Proof of a statement from Steele's "Probability theory and combinatorial optimization"
I am reading "Probability theory and combinatorial optimization" by J.M. Steele and am hung up on a statement made in Section 2.2 of Chapter 2, "Easy size bounds", in which it is stated (paraphrasing ...
2
votes
1
answer
121
views
Expected length of a certain kind of nearest-neighbor graph
Suppose I have sets of points $Z_1,\dots,Z_N$, such that $|Z_i|=m$ for all $i$, and where all $m\times N$ points are independently distributed uniformly at random in the unit square. Can someone give ...
9
votes
1
answer
642
views
Twisted random walks
Suppose the points of two random walks in $\mathbb{R}^2$ are given the
step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$.
Here are several pairs of walks of $n=...
- 151k
6
votes
1
answer
396
views
Does a metric refine the weak-* topology on a dual space?
Let $X$ be a topological affine space over $\mathbb C$, with no additional assumptions. Let $X^*$ denote its dual space of continuous affine functionals $X \to \mathbb C$, equipped with the weak-$*$ ...
- 8,512
4
votes
0
answers
173
views
On understanding Discrete-Valued Stochastic Processes( time series, panel data )
It seems to me that a significant proportion of work in probability theory, statistics and machine learning are on understanding continuous-valued, relatively weakly dependent, or linear dependent ...
- 41
19
votes
2
answers
569
views
Repeated random two-steps in $\mathbb{R}^3$: unbounded?
I created a random isometry $T$ of $\mathbb{R}^3$ by generating
a random orthogonal matrix $M$,
uniformly distributed among all such,
and a random displacement $v$, whose coordinates
are drawn from a ...
- 151k
32
votes
5
answers
6k
views
What is a good method to find random points on the n-sphere when n is large?
As part of a more complex algorithm, I need a fast method to find random points of the n-sphere, $S^n$, starting with a RNG (random number generator). A simple way to do this (in low dimensions at ...
- 15.3k
0
votes
1
answer
197
views
Area on the unit sphere swept out by big circles corresponding to a curve
For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the ...
4
votes
0
answers
128
views
Metrized categories
Motivation: Let $\Gamma = (V,E)$ be a directed graph. To each edge $e \in E$, choose a value $\kappa^e \in \mathbb R$, representing the cost of transporting one unit of "stuff" through the edge. Let $\...
- 8,512
20
votes
3
answers
1k
views
How can I randomly draw an ensemble of unit vectors that sum to zero?
Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question:
Is there a way to sample ...
- 7,633
17
votes
2
answers
406
views
Random rings linked into one component?
Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly (...
- 151k
3
votes
1
answer
248
views
"Trapping" of discs after random sequential adsorption
Imagine I perform Random Sequential Adsorption (RSA) of discs of some radius $r$ on $[0, 1]^2$, eventually covering the surface to some density $Q \leq 0.543$ with $N$ total discs (where $\approx 0....
- 73
4
votes
2
answers
124
views
Simulating random sequential adsorption in reverse
Please consider two processes:
Process 1 - I simulate random sequential adsorption of discs on the unit square in the continuum limit, randomly selecting real number coordinates and rejecting the ...
- 73
4
votes
2
answers
882
views
The probability distribution for vertex degree in a unit disc graph generated from random points on a plane
Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx A*\...
4
votes
1
answer
275
views
Nontrivial lower bounds on Cheeger inequalities for Markov chains
For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the size of $K$ when ...
- 51
9
votes
2
answers
519
views
The fraction of the sphere a fixed distance from a subspace
The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a $k$-...
- 91
2
votes
1
answer
715
views
The largest circle that encloses no points on a plane with points placed at $N$ random coordinates
I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in [...
- 75
3
votes
1
answer
443
views
What is the expected value for this
If there are $8$ random points in the plane whose horizontal coordinate
and vertical coordinate are uniformly distributed on the open interval
$\left(0,1\right)$, what is the expected largest size of ...
- 31
6
votes
1
answer
333
views
Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
- 111
1
vote
1
answer
516
views
Properties of the minimum circumscribing circle for points sequentially placed within a circle of radius $r$ of previously placed points on a 2D plane
Imagine I perform the following procedure:
[1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$.
[2] At time point $t_2$, I center a ...
- 197
1
vote
0
answers
50
views
Volume estimates of rooted embedded tree containing certain subtrees.
Consider a rooted embedded tree of $n+1$ vertices. It is known that around the root for small $r$, volume of the ball of radius $r$ grows like $r^2$. Now suppose we are given that a certain subtree is ...
- 141
5
votes
2
answers
472
views
Product of random diagonals on the unit circle
Let $P_1, P_2, ..., P_n$ be points randomly placed on a unit circle from a uniform distribution. Consider the product $D$ of all pairwise distances:
$D=\displaystyle \prod_{1\leq i < j \leq n} \...
- 413
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
8
votes
3
answers
1k
views
Expected distance between two points in the plane
Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$...
- 83
10
votes
5
answers
509
views
Path length of ball on tilted, perforated plane
Imagine that an $\epsilon$-radius hole is punched in the plane centered
on every integer-coordinate point.
Now a point "ball" is dropped on the plane at a random spot $p$.
If $p$ has not already ...
- 151k
2
votes
1
answer
235
views
The distance between the centroid of $P$ points and the centroid of a subset of the points
Imagine I have an $(p_1, ..., p_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R_c$, with a spacing between the points ...
- 55
2
votes
1
answer
350
views
The equilibrium position for the body of a spider-like spring system after randomly perturbing the anchor positions of its legs
Take $N$ springs, $(s_1, ..., s_N) \in S$ of length $(l_1, ..., l_N)$, and for each spring, label one end "A" and one end "B". Connect the "A" ends of the $N$ springs to a point-like particle on a ...
- 55
6
votes
2
answers
568
views
Number of neigbour Voronoi cells for a random set of points on S^k or cube [-1, 1]^k?
Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ).
Consider Voronoi cell around each point.
How many neighbours would a cell have ...
- 24.9k
0
votes
5
answers
1k
views
Generate points of a (n-2)-sphere on a n-hyperplane [duplicate]
Possible Duplicate:
Efficiently sampling points uniformly from the surface of an n-sphere
I'm trying to generate random points of a (n-2)-sphere on a n-hyperplane so basically the intersection of ...
- 11
11
votes
0
answers
601
views
High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
17
votes
3
answers
923
views
Random permutations from Brownian motion
Let $B(t)$ be a Brownian motion. The ordering of $(0, B(1), ..., B(n-1)) $ is a random permutation in $S_n$. This is not uniform for $n>2$ since the probabilities of the identity permutation $[123.....
- 28k
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
3
votes
0
answers
146
views
The mean number of vertices in small connected components of random geometric graphs
I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, ...
7
votes
3
answers
377
views
Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$
Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...
- 151k