All Questions
Tagged with pr.probability geometric-probability
164 questions
4
votes
2
answers
2k
views
Do Random Walks on the Hexagonal Lattice have a limit?
For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn
induces a tiling of $\mathbb{R}^...
- 3,245
4
votes
1
answer
2k
views
What is the nearest-neighbor distribution in this picture?
Consider the following process: sample $n$ points uniformly at random in the unit square, and for each point $i$, let $d_i$ be the distance from $i$ to its nearest neighbor. Finally, let $z_i = d_i\...
- 4,049
4
votes
1
answer
243
views
Does there exist a scale invariant random packing of circles in the plane?
I want to construct a scale invariant random packing of the plane with circles.
Here is a way to construct a rotationally invariant, but not scale invariant random packing of the plane with circles:
...
- 1,054
4
votes
1
answer
447
views
Area enclosed by Brownian motion (without winding number)
The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
- 29.8k
4
votes
2
answers
175
views
Almost independence of $x^\top a$ and $x^\top b$ for $x$ uniform on the sphere in $\mathbb R^d$ and $a,b \in \mathbb R^d$ with $a^\top b = 0$
Let $d$ be a large positive integer. Let $x$ be uniformly distributed on the unit-sphere in $\mathbb R^d$ and let $a$ and $b$ be perpendicular vectors in $\mathbb R^d$, i.e such that $a^\top b=0$. Let ...
- 6,853
4
votes
1
answer
218
views
Mass distributions for high dimensional simplex and cross polytope
In this question, it is shown that the radial mass distribution of an $n$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $n$-cube to the cube's ...
- 956
4
votes
1
answer
116
views
Distribution of the $pn$ shortest edges out of $n$ uniform points, $p\to 0$
Suppose I sample $n$ points independently and uniformly at random in the unit square, and then I select the $pn$ shortest edges between all pairs of points, for fixed $0<p<1$. For large $n$ and ...
- 4,049
4
votes
2
answers
1k
views
Total progeny of a Galton-Watson branching process - standard textbook question
While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.
...
- 396
4
votes
1
answer
539
views
Buffon's needles revisited
Hi,
I recently came across the famous Buffon's needle problem (http://en.wikipedia.org/wiki/Buffon%27s_needle), and there is no doubt that the problem as well as its answer are elegant.
However, the ...
- 169
4
votes
1
answer
1k
views
Defining a probability distribution on each tangent space of a manifold?
I've been reading about probability on manifolds. What bothers me is that there's no clear way to generalize the Gaussian distribution to manifolds. If we instead assign a probability distribution to ...
- 518
4
votes
1
answer
209
views
Is $\int_{-c}^c |A \cap (x + A)|\, dx$ maximized when the measurable subset $A \subseteq \mathbb R$ is an interval centered at the origin?
Let $A$ be a nonempty measurable subset of $\mathbb R$, with Lebesgue measure $|A|=1$, and let $c>0$. Define the scalar $I(A)$ by
$$
I(A) := \int_{-c}^c |A \cap (x + A)|\, dx,
$$
where $x+A := \{x +...
- 6,853
4
votes
1
answer
133
views
Union of random half spaces cover a ray
Let $x, y \in \mathbb{R}^{n}$ be two fixed unit vectors with angle $\alpha \in (\frac{\pi}{2}, \frac{3\pi}{4})$. Define the positive half space associated with a vector $z$ to be $\mathcal{H}(z) = \{h ...
- 83
4
votes
1
answer
910
views
Uniformly Random Independent Unit Vectors Inner Product Limit
Suppose $V$ is a $N\times n$ matrix the columns of which are independently distributed uniformly on $\mathbf S^{N-1}$ the surface of the unit sphere in $\mathbf R^N$. I conjecture that $V^TV$ ...
- 2,239
4
votes
1
answer
286
views
Upper tail concentration of sample covariance matrices
I'm interested in concentration of the following random matrix sum in spectral norm
$\frac{1}{m}\sum_{k=1}^m b_k^2\mathbf{a}_k\mathbf{a}_k^*$
Here $\mathbf{a}_k\in\mathbb{R}^n$ are i.i.d. standard ...
- 363
4
votes
2
answers
343
views
Concentration of $k$-th pairwise distance of random points in a unit square
For $1\leq i \leq n$, let $X_i\sim \text{Uniform}(0,1)$, $Y_i \sim \text{Uniform}(0,1)$ be $n$ points chosen uniformly in the unit square. Denote the $k$-th smallest pairwise distances across the $n$ ...
- 662
4
votes
1
answer
356
views
Random spherical caps cover a spherical cap
Let $S^{n-1}$ be the unit sphere in $n$ dimensional Euclidean space. Define the spherical cap at $x \in S^{n-1}$ with angle $\theta$ to be $C(x,\theta) = \{z \in S^{n-1} \mid z^\top x \geq \cos(\theta)...
- 83
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
4
votes
0
answers
52
views
Isomorphism of Wasserstein space implies isomorphism of base spaces?
Assume $(X_i,d_i)$ are polish spaces (or compact metric spaces) for $i=1,2$.
Further assume that the 1- Wasserstein spaces $(P_1(X_1),W_1)$ and $(P_1(X_2),W_1)$ are isometrically isomorphic. Does that ...
4
votes
0
answers
259
views
Malliavin calculus and geometric interpretation of $\nabla \cdot ({\nabla F(x)}{\|\nabla F(x)\|^{-2}})$, with regards to the surface $S = \{F = 0\}$
Let $F:\mathbb R^n \to \mathbb R$ be a "sufficiently regular" function. For any $k \ge 1$ and $x \in \mathbb R^n$, define
$$
\alpha_k(x) := \nabla \cdot \left(\dfrac{\nabla F(x)}{\|\nabla F(...
- 6,853
4
votes
0
answers
180
views
Volume difference in random approximation of polytope
The following easily stated problem has arisen in my research. However, it's outside my field, and I'm unfamiliar with the literature. I would greatly appreciate any references.
Let $K\subset \...
- 211
4
votes
0
answers
481
views
The probability distribution for the number of pairwise distances $\leq$ some threshold for points uniformly placed in a sphere
If I place place $N$ particles in a sphere of radius $R$, selecting positions across the sphere's volume with uniform probability, what is the exact probability distribution for the number of pairwise ...
- 41
3
votes
2
answers
590
views
The Largest Piece of Circumference
We add $n$ random lines to the unit disc. We do so by adding two points on the disc, transcribing a line between them and extending that line to the boundary of the disc, namely the unit circle. Each ...
- 133
3
votes
1
answer
473
views
On 4 random points in a rectangle [closed]
Given a bounded rectangular area, I generate 4 random points. What is the probability that the fourth point lie within a triangle formed the first 3?
How would I attack this problem? The goal is to ...
- 133
3
votes
1
answer
403
views
Regularity of transport map
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...
- 5,405
3
votes
1
answer
206
views
Random planes separating points in $\mathbb{R}^3$
We are given a unit origin-centered sphere $S$ in $\mathbb{R}^3$, and three points $\mathbf{x},\mathbf{y},\mathbf{z}\in S$. Let $\mathbf{h}$ be a point selected uniformly at random from $S$ and let $H$...
- 1,677
3
votes
1
answer
114
views
Lower-bound for $\underset{p \le \gamma_d(A) \le q}{\inf} \gamma(A^\epsilon)$, where $\gamma_d$ is the standard gaussian distribution on $\mathbb R^d$
Let $\gamma_d = \gamma_1 \otimes \ldots \otimes \gamma_1$ be the standard Gaussian distribution on $\mathbb R^d$, where $d$ is a large positive integer. Given $\epsilon \ge 0$ and a measurable $A \...
- 6,853
3
votes
2
answers
442
views
What happens to the Gaussian volume of a Borel set when it is translated?
Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$, $A \subseteq \mathbb R^n$ be Borel and $c \in \mathbb R^n$. Define the translate $A_c := c + A := \{c+a \mid a \in A\} = \{x \in \...
- 6,853
3
votes
1
answer
182
views
How tight is the bound $P(\|X\|^2 \ge t |\langle a,X\rangle|) \ge 1 - t\sqrt{\frac{2}{m-1}}$, where $X \sim N(0, I_m)$ and $\|a\| = 1$?
Let $X$ be a random vector in $\mathbb R^m$ with iid $N(0,1)$ coordinates and let $a$ be a fixed unit vector in $\mathbb R^m$. In another post (SE link here https://math.stackexchange.com/a/3792730/...
- 6,853
3
votes
2
answers
207
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 ...
- 31
3
votes
1
answer
492
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? ...
- 903
3
votes
1
answer
321
views
Is disintegration continuous?
Let $X,Y$ be Polish spaces and suppose that $X$ is compact. Denote by $\mathcal{Mes}(X,\mathcal{P}(X\times Y))$ the set of (Borel) measurable functions from $X$ to the set of Borel probability ...
- 5,405
3
votes
1
answer
189
views
Isoperimetry on $[0, 1]^n$ w.r.t $\ell_p$ distance, with $p \in [1,\infty]$
Let $A$ be a measurable subset of the metric space $\mathcal X = ([0, 1]^n,\ell_p)$ with $1 \le p \le \infty$, and define its $\varepsilon$-blowup by $A^\varepsilon:=\{x \in \mathcal X \mid \|x-a\|_p \...
- 6,853
3
votes
0
answers
230
views
Expectation of angle between two vectors in the image of a gaussian random matrix
Let $m$ and $n$ be large positive integers (going to infinity), and let $W$ be a random matrix of size $n \times m$ with iid entries from $N(0,1/m)$. Let $x,y \in \mathbb R^m$ be deterministic vectors,...
- 6,853
3
votes
0
answers
132
views
Probability that a Voronoi cell contains exactly k random points
Consider two independent point processes in the unit square $[0,1]^2$. The two point processes are identically independent and typically binomial/Poisson. One, say $\Phi^*$, is used to generate a ...
- 137
3
votes
0
answers
253
views
Metric ($f$-divergence) on space of probability measures that satisfies pythagorean theorem
Let $E$ be a polish space, $\mathcal{P}(E)$ the Borel probability measures on $E$ with the topology of weak convergence and $\mathcal{Q} \subset \mathcal{P}(E)$ a convex and compact set.
First, the ...
- 1,095
3
votes
0
answers
234
views
Are random convex polygons on a sphere themselves sphere-like?
Say $\mathbb{R}^n$ is divided by $k>n$ randomly chosen hyperplanes. Each connected region away from the hyperplanes is the intersection of $k$ half-spaces, so it is a convex cone. It is known that ...
3
votes
0
answers
85
views
Expected Largest Area with Random Lines
We take the unit square and have it divided by $n$ lines which are chosen randomly. We choose the lines as follows, choose one of the four sides of the square at random and then choose a random point ...
- 133
3
votes
0
answers
133
views
Partitioning a sphere with a random tessellation
I have a $d$-dimensional sphere of radius $1$. I now tessellate the space with $d$-dimensional cubes of side $L$ and uniform distribution of the origin of the tessellation. Thus the size of the ...
- 162
3
votes
0
answers
134
views
Algorithm to calculate moments of uniform distribution on convex polyhedra
There is system of linear inequalities
$$
Ax \leq K,
$$
$$
x\geq a, x\leq b.
$$
$A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions ...
3
votes
1
answer
528
views
Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
- 161
2
votes
1
answer
396
views
Manhattan distance vs. absorption time on an unbounded integer lattice
Imagine I have unbounded $d$-dimensional integer lattice where I take two vertices, $v_a$ and $v_b$, separated by a fixed Manhattan distance $L$, and I release a random walker at $v_a$ and allow for ...
- 173
2
votes
1
answer
201
views
Dispersion of a "random" subset of $[-1,1]^2$
Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|...
- 261
2
votes
1
answer
149
views
Volume computation using probabilistic approach
Let $\mathbb{S}^{d-1}=\{v\in\mathbb{R}^d:\|v\|_2=1\}$, namely $d-$dimensional sphere. It is well-known that if a random vector $X$ is distributed uniformly on $\mathbb{S}^{d-1}$, then there exists i.i....
- 1,096
2
votes
1
answer
275
views
Does this KL divergence inequality hold?
Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that
$$
\frac{y\left( x \right)}{p\left( x \right)}=\frac{\...
- 49
2
votes
1
answer
81
views
Distribution of scaled Johnson-Lindenstrauss transforms
Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that
$$
\...
- 269
2
votes
2
answers
322
views
Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere
This question is related to: https://math.stackexchange.com/q/4270522/168758
Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
- 6,853
2
votes
1
answer
319
views
Minimum distance to a sampled point with given pdf
Let $f(x)>0$ be a probability density function defined on the unit square $[0,1]^2$ in $\mathbb{R}^2$. Suppose that we take $N$ independent samples, $X_1,\dots,X_N$, of $f$. Now, sample a point $...
- 645
2
votes
2
answers
571
views
The consequence of overlap sharing for the length-distribution of rods randomly placed on a line
Please imagine that one populates a finite line of unit length, or circle with unit length contour (to avoid edge-effects), with $N$ one-dimensional 'rods' such that their LHS-ends, at positions $(p_1,...
2
votes
2
answers
195
views
Random partition of an interval – Dirichlet distributed?
Let $X_1, \ldots, X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$.
What is the distribution ...
- 57
2
votes
1
answer
149
views
Tightly placed sphere caps
Problem. Fix a positive real $r>0$ and a positive integer $n>0$. Consider an independent, identical sample $X_1, \ldots X_n$ drawn from the uniform distribution over the unit $d$-sphere $\mathbb ...
- 903