Questions tagged [percolation]

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In percolation on a lattice, how is "infected" status correlated for points in a region around the origin?

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....
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Are there more paths exiting a box in $\mathbb{Z}^2$ to the right if I remove some edges to the left

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'...
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943 views

An elementary question in bond percolation

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 ...
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55 views

Existence of a bigeodesic in last passage percolation is $0$-$1$ event

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 ...
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1answer
141 views

Understanding the wrapping criterion in percolation theory

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, ...
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142 views

Ask for some reference about isoperimetric constant on Voronoi diagrams?

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'...
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1answer
160 views

Continuum percolation in 1d

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}...
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63 views

super-critical percolation on $\mathbb{Z}^2$, number of corners in a directed open path

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 ...
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84 views

Ask for some percolation reference textbook

I try to learn Bernoulli percolation recently. Could anyone provide some lecture notes or textbooks to enter this field? Thanks.
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1answer
158 views

Bernoulli percolation, infinite path from (0,0) in a "cone"

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 ...
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1answer
220 views

Random walk on the hypercube with deleted edges

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 ...
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1answer
128 views

Does the union of two percolation measures satisfying the (FKG) inequality still satisfy (FKG)?

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 $ ...
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112 views

What can be said about percolation clusters after deleting a positive fraction of edges in general?

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 ...
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1answer
816 views

Understanding Finite Size Scaling in Percolation Theory

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 ...
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128 views

Pursuit-evasion with many slow pursuers

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 ...
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1answer
128 views

Figuring out a consistent definition for the percolation backbone

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 ...
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1answer
158 views

Percolation critical exponent $\nu$ does not depend on neighborhood connectivity. Does this follow from the universality principle?

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 ...
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330 views

Riemann–Hilbert-type problem

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 ...
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1answer
72 views

What is the expected distance between the sides of a random subgraph of the grid?

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 ...
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207 views

Gaussian square-free moat

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, ...
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97 views

Percolation-type question involving phase transition for graded acyclic directed graph

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 $(...
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43 views

Vertical and horizontal percolation on heterogeneous honeycomb lattice

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 "...
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74 views

Percolation and diameter of graph

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 ...
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90 views

How to mathematically justify the "sampling" over only $100$ random matrices to estimate percolation thresholds?

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 ...
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1answer
77 views

Generalization: (The "number" of) smaller sized clusters in large random binary matrices follow a descending order. Why?

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 ...
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1answer
150 views

Why is number of single cell clusters always greatest in a random matrix?

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 "...
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99 views

Length of optimal play in Hex as a function of size

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 ...
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1answer
106 views

Probability for a group of stones to live on an infinite Go board

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 ...
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1answer
124 views

Does there exist any analogous result for site percolation?

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 ...
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1answer
279 views

Proof and interpretation of the following percolation theory result for $n\times n$ square grid

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 ...
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1answer
81 views

References on the structure of bond percolation on the (finite) 2D-grid in the sub-critical regime (e.g p=1/10)

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.
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79 views

How to calculate the exact probability $p$ at which maxima occurs in the curves in an infinite system?

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 ...
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93 views

Is this correct: Inflection points of Euler number graph in Island-Mainland transition correspond to spanning cluster site percolation threshold?

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 ...
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102 views

Percolation in torus under threshold rule

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 ...
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1answer
127 views

KPZ relation $\chi = 2 \xi -1$ in a random geometric graph

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 ...
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73 views

Not exactly directed percolation

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 ...
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1answer
347 views

Critical Exponents for Island Mainland Transition (Percolation Theory)

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 ...
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2answers
156 views

Percolation on finite irregular trees

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 ...
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1k views

What is a self-consistent equation in percolation theory

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 ...
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105 views

Probability of a maximal chain in a random subposet of a finite poset

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 $...
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1answer
242 views

Equation of state for hard rods

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 ...
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455 views

What is the strongest known RSW result in planar percolation?

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 ...
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2k views

Why do we use hexagons in percolation?

In some cases, hexagons are used in percolation. Why do we use hexagons in percolation?
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55 views

k-dependent oriented percolation system with small closure

I am studying the next result Let $\Gamma=\{(m,n)\in\mathbb{Z}^{+}\times\mathbb{Z}\text{ such that } > m+n \text{ even }\}$, $\Omega=\{0,1\}^{\Gamma}$, and $\mathcal{F}$ the $\sigma$-algebra ...
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68 views

A random variable standing for the size of connected component including a given node in a tree

Suppose we have a tree $T = (V,E)$, in which each nodes $v_i \in V$ has a probability $p_i$ to vanish. Let $v_0\in V$, we define random variable $\boldsymbol{X} = \boldsymbol{X}(T, v_0)$ stands for ...
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252 views

Among regular graphs, do cliques have the highest infection rate?

Consider a graph $G$ with a particular node $i$ labeled as “infected”. Other nodes start uninfected, and will become infected over time according to the following process: To each edge of the graph, ...
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67 views

The invariant of a shape which determines percolation

Suppose we have a shape bounded by a simple closed curve $\gamma \subset \mathbb{C}$, with points $A,B,C,D$ in cyclic order on the curve. If we randomly color the interior of that shape in half red ...
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1answer
117 views

Berg-Kesten-Reimer inequality on infinite spaces?

See this link for a description of the van den Berg-Kesten-Reimer inequality. How important is the assumption that $\Omega_i$ are finite spaces? When Berg-Kesten state the inequality in their 1985 ...
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1answer
222 views

Not understanding a part of Hugo and Vincent's proof on Percolation

Kindly refer to this paper: https://arxiv.org/abs/1502.03050 In this paper, Hugo Duminil-Copin and Vincent Tassion have given an alternative proof of the well known results. I was reading this paper ...
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63 views

How many loops separate $(0,0)$ from $(n,0)$ in the site percolation on $\mathbb{Z}^2$?

I ran into this problem on the Bernoulli site percolation on $\mathbb{Z}^2$ coming from another area. I know there's a lot of theory on this and I'm hoping that mathoverflow might help point me in ...