Percolation theory describes the behavior of connected clusters in a random graph.
Percolation theory describes the behavior of connected clusters in a random graph.
Questions tagged [percolation]
151 questions
45
votes
3 answers
A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?
A disc contains $n$ independent uniformly random points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random points and $7$ clusters, with an average cluster size…
Dan
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28
votes
2 answers
How to create mazes on the hyperbolic plane?
I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the lattice that's still connected (every cell can…
Steven Stadnicki
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22
votes
2 answers
The Gaussian moat problem and its extension to other rings in $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$
One of my favourite open problems in number theory, an area in which I enjoy only as a hobbyist, is the Gaussian moat problem, namely
"Is it possible to walk to infinity in $\mathbb{C}$, taking steps of bounded length, using the Gaussian primes as…
Bennett Gardiner
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20
votes
1 answer
The problem of the most visited point.
Represent the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ points as in the figures below for example. How to calculate the number of circuits that visit a chosen point in this rectangle? What is the…
Elias Costa
- 15,282
18
votes
1 answer
Size of connected regions on a randomly-colored infinite chessboard
Consider an infinite chessboard where each square is colored white with probability $p$ and black with probability $1-p$. Suppose without loss of generality that the square at $(0,0)$ is white. We can consider the entire connected region $W$ of…
MJD
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Take an $m \times n$ grid, and in each box pick two opposite corners at random to connect. What can be said about the resulting pattern?
Inspired by the upcoming book 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior.
Here's an example:
(Note that I'm considering the grey "negative space" here, not…
Daniel McLaury
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16
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2 answers
What is the average size of each color region in an infinite by infinite randomly colored grid with 3 colors?
You start out with an infinite by infinite grid of squares. Color each one randomly red, green or blue. Each one has a $1$ in $3$ chance of getting colored red, a $1$ in $3$ chance of getting colored green, and a $1$ in $3$ chance of getting colored…
Nico Zaczkowski
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13
votes
2 answers
Edge percolation on $\mathbb{Z}^2$: probability that two neighbouring vertices are connected?
I'm considering edge percolation on $\mathbb{Z}^2$ with parameter $p$, so that edges are present with probability $p$.
Is it known how to express the probability $P(p)$ that $(0,0)$ is in the same connected component as $(1,0)$ as an explicit…
Eckhard
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8
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Percolation and number of phases in the 2D Ising model.
Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive
After a long time I came back to try to understand an article on the Ising model.
The review article is Percolation and number of…
Elias Costa
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7
votes
1 answer
Site percolation model that cannot be obtained from a bond percolation model
It is easy to obtain a site percolation model from a bond percolation model on a graph $G$ using the covering graph $G_c$ of $G$. I wondered if one can obtain any site percolation model from any site bond and I read in the Geoffrey Grimmett's book…
José Milán
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0 answers
Number of circuits that surround the square.
Consider a grid $G$ in the $\mathbb{R}^2$ plane formed by the points $(x,y)$ with integer coordinates i.e. $G=\{(x,y)\in\mathbb{R}^2: x\in\mathbb{Z},\;y\in\mathbb{Z} \}$. For $n>0$ let $B_n$ square centered at $(0,0)$ whose sides have length…
Elias Costa
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6
votes
1 answer
Colored path in a randomly colored grid
A friend of mine asked this question a while ago which I couldn't find any appropriate answer for it. I'd appreciate any comment or help.
If one colors each unit square with black/white of an $m \times n$ grid according to the outcome of a coin…
Ehsan M. Kermani
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6
votes
1 answer
What is the average size of an island?
If you have a square grid, and each square* has probability $n$ of being ground. If the other squares are water, what is the average area of an island? If $n$ is small then the average island would have an area of about $1$. With large values of…
Infinite Planes
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6
votes
2 answers
amenable groups versus amenable graphs
In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability
In percolation theory and "random geometry" one is…
Jeff
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5
votes
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2d Brownian motion hitting a point
Let $\Omega\subset \mathbb C$ be a simply connected domain, $\tau = \exp(2\pi\mathrm i/3)$ and $a(\alpha),a(\tau\alpha),a(\tau^2\alpha)$ are some accessible points of $\Omega$.
In this paper by S. Smirnov he writes (Section 2, p.4, after equation…
SBF
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