# Tagged Questions

**3**

votes

**1**answer

122 views

### Random non-intersecting circles in the plane

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...

**7**

votes

**3**answers

312 views

### The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small?
More ...

**5**

votes

**0**answers

159 views

### Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?

Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...

**9**

votes

**1**answer

219 views

### A random variation on Polya's orchard problem

Polya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, ...

**1**

vote

**1**answer

121 views

### Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where ...

**1**

vote

**0**answers

63 views

### Quadrilaterals from a Unit Stick

This question could be seen as a coordinate-free variant of Sylvester's Four Point Problem (cf e.g. http://mathworld.wolfram.com/SylvestersFour-PointProblem.html):
Suppose one are given an ...

**15**

votes

**0**answers

427 views

### The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with ...

**27**

votes

**2**answers

753 views

### Shortest path through $\sqrt{n}$ points out of $n$

Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...

**4**

votes

**2**answers

123 views

### Connectivity of points sampled in a grid

Suppose that I partition an $n\times n$ square into $n^2$ squares $S_1 ,\dots, S_{n^2}$ each of area $1$, and then I sample a point $X_i$ uniformly at random in each $S_{i}$. Now fix a radius $r$ and ...

**14**

votes

**3**answers

644 views

### “Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ ...

**2**

votes

**2**answers

114 views

### Volume of specials sets on sphere $S^N$

Suppose I'm given $m$ points $\{q_i\}$ on the sphere $S^N$. I want to get a lower/upper bound for the volume of the following sets with respect to uniform probability measure $\mathbb{P}$ on the ...

**3**

votes

**1**answer

151 views

### Diameter of $n$-unit-vector closed scribble

Suppose one creates a random, closed, likely self-crossing polygon
from $n$ unit-length vectors arranged head-to-tail,
randomly oriented except for the requirement
that their sum is zero (so the ...

**6**

votes

**1**answer

266 views

### Limit of distance between two random points in a unit-radius $n$-sphere

This is a companion contrast to the earlier analogous question for unit $n$-cubes,
where the answer (provided by several respondents) is $\infty$ .
What is the limit, as $n \to \infty$, of the ...

**6**

votes

**2**answers

268 views

### Limit of distance between two random points in a unit $n$-cube

What is the limit, as $n \to \infty$, of the expected distance between two
points chosen uniformly at random within a unit edge-length hypercube
in $\mathbb{R}^n$?
For $n=1$, the average ...

**7**

votes

**0**answers

94 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 ...

**14**

votes

**2**answers

470 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}, ...

**4**

votes

**1**answer

427 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 ...

**2**

votes

**1**answer

153 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 ...

**0**

votes

**1**answer

119 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 ...

**16**

votes

**2**answers

405 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 ...

**1**

vote

**0**answers

135 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

62 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 $N$ ...

**2**

votes

**2**answers

230 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

101 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 ...

**4**

votes

**1**answer

177 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 ...

**6**

votes

**1**answer

162 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 ...

**33**

votes

**1**answer

2k views

### Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown.
I have already asked this question on ...

**19**

votes

**2**answers

882 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 $r>1$ ...

**0**

votes

**1**answer

69 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 ...

**6**

votes

**1**answer

158 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 ...

**1**

vote

**1**answer

79 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 ...

**6**

votes

**1**answer

287 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 ...

**6**

votes

**1**answer

231 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-$*$ ...

**3**

votes

**0**answers

119 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 ...

**0**

votes

**0**answers

101 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

93 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 ...

**17**

votes

**2**answers

879 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 ...

**14**

votes

**2**answers

245 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 ...

**9**

votes

**2**answers

485 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 ...

**6**

votes

**1**answer

205 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 ...

**1**

vote

**0**answers

43 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 ...

**8**

votes

**3**answers

438 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 ...

**9**

votes

**5**answers

389 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 ...

**8**

votes

**0**answers

384 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 ...

**14**

votes

**2**answers

589 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 ...

**7**

votes

**3**answers

251 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 ...

**1**

vote

**2**answers

282 views

### Terrain Generation: Infinite 2D space filled with Diffusion-limited aggregation clusters?

Disclaimer: I don't have a deep understanding of fractals or any higher math, I'm just personally interested in it, so please excuse me if I'm using wrong terms or if I'm being inaccurate. Making ...

**4**

votes

**1**answer

275 views

### Wasserstein geometry of measures on manifolds related to the generalized Legendre transform and $d^2/2$-convexity

Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class ...

**12**

votes

**1**answer

657 views

### Random polycube shapes

I am wondering if it is hopeless to obtain any firm results
on the following model of a "random polycube shape."
First, a polycube in $\mathbb{R}^3$
is a connected face-to-face gluing of unit cubes.
...

**36**

votes

**1**answer

2k views

### Rolling a random walk on a sphere

A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...