Percolation theory describes the behavior of connected clusters in a random graph.

Percolation theory describes the behavior of connected clusters in a random graph.

Percolation theory describes the behavior of connected clusters in a random graph.

Percolation theory describes the behavior of connected clusters in a random graph.

113 questions

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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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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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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

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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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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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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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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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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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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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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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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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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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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…

Ilya

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I try to study Percolation Theory by "A mini course on percolation theory" of Jeffrey E. Steif.
I am very curios about Exercise 2.4.
Show that the number of cycles around the origin of length n is at most $n4(3^{n−1})$.
I need to this on lattice…

com

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Put spheres uniformly at random all over $\mathbb{R}^3$, with density 1 sphere / unit cube. All spheres have the same radius $r$. What is the probability function $p(r)$, that that there is an infinite component, that is an infinite sequence of…

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