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Questions tagged [percolation]

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10
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248 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 ...
1
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0answers
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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0answers
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, ...
2
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0answers
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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0answers
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 "...
1
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0answers
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 ...
1
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0answers
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 ...
0
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0answers
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 ...
2
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1answer
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 ...
3
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1answer
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 "...
6
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0answers
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 ...
1
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1answer
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 ...
1
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1answer
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 ...
2
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1answer
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
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1answer
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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0answers
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 ...
2
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0answers
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
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0answers
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 ...
0
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1answer
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 ...
0
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0answers
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
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1answer
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 ...
3
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2answers
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 ...
0
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0answers
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 ...
5
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0answers
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
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1answer
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
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2answers
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 ...
9
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2answers
1k views

Why do we use hexagons in percolation?

In some cases, hexagons are used in percolation. Why do we use hexagons in percolation?
2
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0answers
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 ...
0
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0answers
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 ...
9
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0answers
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, ...
1
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0answers
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 ...
0
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1answer
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 ...
2
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1answer
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 ...
5
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0answers
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 ...
4
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1answer
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 ...
3
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0answers
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 ...
2
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1answer
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 ...
5
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0answers
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 ...
10
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2answers
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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0answers
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 ...
2
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0answers
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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0answers
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 ...
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5answers
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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0answers
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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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0answers
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 ...
6
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1answer
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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1answer
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\...
2
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1answer
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 $\...
6
votes
1answer
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 (...
6
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0answers
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 ...