# Questions tagged [percolation]

The percolation tag has no usage guidance.

The percolation tag has no usage guidance.

82
questions

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1
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Let $G$ be a vertex-transitive locally finite graph and $c_n$ the number of self-avoiding walks in $G$ starting from some fixed vertex $v_0$. One can easily see that $c_{m+n} \leq c_m c_n$ and hence ...

4
votes

1
answer

151
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Consider classical bond percolation on $\mathbb{Z}^d$. Each edge is included with probability $p$ and deleted with probability $1-p$. As is well known, there is a $p_c(d) \in (0,1)$ such that $p>...

0
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0
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34
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In Bernoulli percolation on the square lattice $\mathbb{Z}^2$ every edge is kept with a probability $q$ and erased with probability $1-q$. It is classical that whenever $q > \frac{1}{2}$ there is ...

1
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0
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20
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In absorbing state transitions on a lattice, one often considers "active" and "inactive" sites, with update rules describing how activity spreads or decays. When the lattice sites ...

6
votes

1
answer

242
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I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is ...

0
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0
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62
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Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...

1
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2
answers

109
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I am interested in long-range percolation models with heavy-tailed degree distributions such as DHH13, GLM21, Y6. The simplest example is scale-free percolation in which vertices are elements $\mathbb{...

1
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0
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82
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Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster....

9
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2
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394
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Suppose that I am given the graph $G = (V,E)$ where $V = \{ 1, 2, \dots 2N+1 \} \times \{ 1, 2, \dots 2N+1 \} $ and there is an edge between two vertice $(n,m)$ and $(n',m')$ if and only if $\vert n-n'...

9
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2
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984
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Consider a locally finite, connected graph and "bond (edge) percolation" on this graph. Each edge is open with probability $p.$ There is a parameter $\alpha$, $0<\alpha<1.$
The ...

1
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0
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55
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On the bottom of page two of This paper, the authors remark the following:
'...by translation invariance and ergodicity, we know that existence of a bigeodesic is a $0−1$ event and hence it follows ...

1
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1
answer

199
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Context:
When studying percolation in finite sized systems, there exist various definitions and criteria for determining when a given system is percolating, i.e., given a definition for connectivity, ...

2
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0
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149
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Given a Poisson point process $\mathcal{P}$ in $\mathbb{R}^2$, the $\textbf{Voronoi cells}$ of a point $p\in \mathcal{P}$ is defined by
$$V(x):=\{y\in \mathbb{R}^2: \|x-y\|=\min_{x'\in \mathcal{P}}\|x'...

1
vote

1
answer

210
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What is known about continuum percolation in 1d?
By this, I mean, for $d \in \mathbb{N}$, the Poisson-Boolean model of disks of radius $r_0 \in \mathbb{R}$ with centres arranged randomly in $[0,1]^{d}...

3
votes

0
answers

66
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Define the planar percolation where each unit edge is open with probability $p$ very close to $1$.
Looking at the event where there exists a directed open path between $(0,0)$ and $(n,n)$. This event ...

0
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0
answers

94
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I try to learn Bernoulli percolation recently. Could anyone provide some lecture notes or textbooks to enter this field? Thanks.

3
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1
answer

170
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Look at Bernoulli percolation on $\mathbb{Z}^2$ with $p> p_c$ ($p$ can be arbitrarily close to 1).
I am interested in the probability that there exists an infinite cluster starting at $(0,0)$ and ...

5
votes

1
answer

241
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Let $G$ be the $n$-dimensional boolean hypercube, i.e. the graph on $\{0,1\}^n$ where two vertices are adjacent iff they differ on exactly one coordinate. Consider a graph $G'$ obtained by deleting a ...

3
votes

1
answer

142
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Let a percolation measure be a measure on $\{0,1\}^n$. We have a natural partial order on $\{0,1\}^n$ given by comparing all coordinates. An event $A$ is called increasing if for all $ \omega \in A $ ...

6
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0
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114
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Start with a bond-percolation process just above criticality, say $p=1/2+\varepsilon$ on the graph $\mathbb Z^2$ with $\varepsilon>0$.
Sample $D\in\{0,1\}^E$ from an independent product measure ...

3
votes

1
answer

1k
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Fundamental results in percolation theory are all based on the assumption that the system sizes are infinite, as the spanning/percolating cluster is by definition an infinitely sized cluster that ...

8
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0
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138
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Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...

1
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1
answer

143
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In the context of percolation, e.g., bond/site percolation, random graph connectivity in 2-3 dimensions, etc., once the percolation threshold is reached, that is the system is spanned by an infinite ...

1
vote

1
answer

221
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I read the Wikipedia article on Percolation critical exponents. It says:
In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of ...

10
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0
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340
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Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides
of $P$ going in the counterclockwise order. We are ...

1
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1
answer

82
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Let $G$ be the $n \times n$ grid, in which each vertex is connected to the vertices above it, below it and on either side. Let $G_p$ be the random subgraph of $G$ obtained by keeping each edge with ...

6
votes

0
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225
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Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free
Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$
For the analogous problem with Gaussian primes instead, ...

2
votes

0
answers

99
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Let $G$ be an acyclic directed graph with $MN$ vertices arranged into $M$ generations of $N$ vertices each. We stipulate that edges may only go from generation $j$ to generation $j+1$, so there are $(...

1
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0
answers

44
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I have a regular honeycomb lattice where a bond in the unit cell aligns with $(1,0)$; call this the horizontal direction. Each horizontal bond in the lattice is open with probability $p$ and each "...

1
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0
answers

76
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Is the critical probability in percolation and diameter of graph related. I guess larger the diameter higher the probability. Is there any result like this?
By critical probability I mean the ...

0
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0
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101
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As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by Stauffer et al., the variation of spanning cluster percolation probability $\Pi$ in a finite $L < \infty$ square ...

2
votes

1
answer

82
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This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...

3
votes

1
answer

163
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Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...

6
votes

0
answers

103
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Consider Hex on an $n \times n$ board without a swap rule, so that the first player wins. Assume the first player tries to minimize the length of the game, and the second player tries to maximize the ...

1
vote

1
answer

108
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Suppose on an infinite two dimensional Go board the tengen is occupied by a black stone, and every other grid point is occupied by a black stone, or a white stone, or nothing, with probability 1/3 ...

1
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1
answer

144
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This is a sequel to the question: Proof and interpretation of the following percolation theory result for $n\times n$ square grid
In the paper: The Birth of the Infinite Cluster:Finite-Size Scaling ...

2
votes

1
answer

314
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While I was discussing this question with @JamesMartin, he mentioned a result here that:
In a $n\times n$ finite square grid, if $p\geq p_c+\epsilon$, such
that $\epsilon>0$ and $p_c$ is the ...

2
votes

1
answer

86
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Would appreciate references to the most up-to-date results for the structure of bond percolation on the (finite) 2D-grid in the sub-critical regime (e.g, $p=1/10$).
Thank you.

0
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0
answers

79
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I'm writing with respect to this paper: Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice
Here's a ...

2
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0
answers

97
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I'm writing with respect to the paper Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice.
Here's a link to a PDF ...

4
votes

0
answers

114
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As part of my graduate research I am currently studying the last section in the paper "Random Majority Percolation" by Balister, Bollobas et al. The paper itself is very complicated but the last two ...

1
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1
answer

156
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If I have $n$ points uniformly distributed on the surface of a torus, and form a graph by adding an edge between pairs whenever they are within a unit distance (induced by the Euclidean metric), I ...

0
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0
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77
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Is the following problem known/well-studies? I'm looking for references or a name that I can look up.
I start with $N$ cell, each one divides into two cells, each one of the new cells either dies ...

4
votes

1
answer

367
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I was looking at this paper “Islands in Sea” and “Lakes in Mainland” phases and related transitions simulated on a square lattice on Percolation theory. The concept of phase transition used here seems ...

3
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2
answers

175
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Consider a rooted tree of height $h$, such that all the leaves are at last layer. We perform the following random process: each edge is deleted with probability $0.5$, and otherwise it is retained. We ...

0
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0
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2k
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I was reading papers about percolation theory in which I was confused by the expression "self-consistent equation", for example in Temporal percolation in activity-driven networks. I read some ...

6
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0
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106
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Let $P$ be a finite poset, and let $0<p<1$. Choose a random subposet
$Q$ of $P$ by letting each $t\in P$ belong to $Q$ with probability
$p$. What is the best way to compute the probability that $...

4
votes

1
answer

258
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Some context:
For ideal gases, the thermodynamic equation of state is the well-known:
$$
pV = nRT \tag{1}
$$
where $n$ is the amount of substance, $R$ the universal gas constant and $P,V,T$ are ...

7
votes

2
answers

512
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One of the weakest estimates conjectured to hold for critical planar percolation models (and proved in many cases) is the so-called RSW estimate. RSW estimate is the statement that the probability of ...

11
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2
answers

2k
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In some cases, hexagons are used in percolation. Why do we use hexagons in percolation?