# Questions tagged [percolation]

The percolation tag has no usage guidance.

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### 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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29 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 ...

**5**

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144 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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89 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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31 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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65 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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78 views

### How does the graph of percolation probability $\Pi$ vs. $p$ vary for different finite values of $L$?

This is a sequel to my previous question. @Carlo's response here (to my comment) prompted me to ask this question:
As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by ...

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57 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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**1**answer

71 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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votes

**1**answer

134 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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82 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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vote

**1**answer

96 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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vote

**1**answer

108 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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votes

**1**answer

181 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 ...

**2**

votes

**1**answer

65 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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77 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 link to the PDF ...

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78 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 ...

**4**

votes

**0**answers

80 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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votes

**1**answer

88 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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63 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 ...

**4**

votes

**1**answer

321 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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votes

**2**answers

135 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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315 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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89 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 $...

**4**

votes

**1**answer

164 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 ...

**7**

votes

**2**answers

305 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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**2**answers

1k 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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41 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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65 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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217 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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60 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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votes

**1**answer

87 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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votes

**1**answer

173 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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62 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 ...

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votes

**1**answer

90 views

### Dynamic site percolation of independent random walkers on 2-dimensional square lattice

I am stuck in a part of my research which I am not expert in.
I have a 2-dimensional square lattice with periodic boundary conditions(torus). I am placing one walker at each node at the beginning. It ...

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112 views

### Remaining models conjectured to converge to SLE(6) or CLE(6)

I am wondering which models are conjectured (eg. numerically) to converge to SLE(6) (Schramm-Loewner evolution with $\kappa=6$) or CLE(6) (conformal loop ensemble). I am searching for a research topic ...

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**1**answer

123 views

### Percolation on the hyperbolic plane and convergence to SLE(6) on hyperbolic plane

In "Percolation in the hyperbolic plane" the authors study the properties of percolation in the hyperbolic plane. Smirnov and others proved convergence of isotropic percolation to SLE(6).
Do these ...

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96 views

### Is there any known construction of IIC as a limit from supercritical phase?

Consider a nearest-neighbor percolation model on $\Bbb{Z}^d$, where each bond is occupied with probability $p$ independent of each other. Let $\Bbb{P}_p$ denote the corresponding law on the ...

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**2**answers

968 views

### How should a mathematician approach the physics literature concerning percolation?

I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...

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146 views

### Random Cluster Model only for bond percolation?

Can someone please tell me which of the following statements I make are true of the current state of the art:
The Random Cluster Model is a generalization of bond percolation (with possibly different ...

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116 views

### Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$.
We know that $C_n$ has a giant component whose size is of ...

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429 views

### First passage percolation on a random geometric graph in the large connectivity limit

Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...

**50**

votes

**5**answers

4k views

### Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...

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165 views

### Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice.
Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...

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votes

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155 views

### Semi-directed first-passage percolation on $\mathbb{Z}^2$ with deterministic vertical weights

Consider the following first-passage percolation problem on the plane grid $\mathbb{Z}^2$: all the horizontal edges are directed (pointing east) and carry an independent random weight, say standard ...

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votes

**1**answer

657 views

### percolation probability in a hexagonal region

Suppose one takes a large hexagonal region in the tiling of the plane by unit hexagons, with $n+1$ hexagons on each side, as seen in the figure below (taken from the COMAP website) for the case $n=5$. ...

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89 views

### Independent bond percolation on upper density zero subgraphs of the square lattice can have a non-trivial critical point ?

Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where edge set $\mathbb{E}^2$ is given by the pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$.
Let $V\subset\...

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votes

**1**answer

155 views

### Continuum limit of first-passage percolation paths

A few years ago, when I was working on first-passage percolation problems, I thought about the following problem. Recently it came back to my mind.
Consider, for some $\delta=n^{-1}>0$, the grid $\...

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votes

**1**answer

826 views

### Probability of two vertices to be connected in G(n,p)

A question I asked at math.SE without elliciting an answer.
Let $G(n,p)$ be an Erdős–Rényi graph on $n$ vertices. Is there an explicit expression for the probability $P_{n,p}(u,v)$ that two fixed (...

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217 views

### Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence,
a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit ...