Questions tagged [percolation]

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6 votes
1 answer
171 views

Origin of the term "connective constant"

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 views

Percolation: at what length scale do we see it?

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 votes
0 answers
34 views

Can I explore the infinite cluster of Bernoulli percolation in $\mathbb{Z}^2$?

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 vote
0 answers
20 views

How slowly can one be absorbed in the absorbing phase of the directed percolation universality class?

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 views

Infinite clusters for loopless percolation

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 ...
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0 votes
0 answers
62 views

Likelihood ratio of non-trivial cycles in an inhomogeneous random square lattice graph embedded on a toroidal surface

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 vote
2 answers
109 views

Which infinite random graphs with percolation threshold $p_c=0$ are transient?

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 vote
0 answers
82 views

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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9 votes
2 answers
394 views

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'...
9 votes
2 answers
984 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 ...
1 vote
0 answers
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 ...
1 vote
1 answer
199 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, ...
2 votes
0 answers
149 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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1 vote
1 answer
210 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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3 votes
0 answers
66 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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0 votes
0 answers
94 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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3 votes
1 answer
170 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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5 votes
1 answer
241 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 ...
3 votes
1 answer
142 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 $ ...
6 votes
0 answers
114 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 ...
3 votes
1 answer
1k 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 ...
8 votes
0 answers
138 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 ...
1 vote
1 answer
143 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 ...
1 vote
1 answer
221 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 ...
10 votes
0 answers
340 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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1 vote
1 answer
82 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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6 votes
0 answers
225 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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2 votes
0 answers
99 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 $(...
1 vote
0 answers
44 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 vote
0 answers
76 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 ...
0 votes
0 answers
101 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 ...
user avatar
2 votes
1 answer
82 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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3 votes
1 answer
163 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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6 votes
0 answers
103 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 vote
1 answer
108 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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1 vote
1 answer
144 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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2 votes
1 answer
314 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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2 votes
1 answer
86 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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0 votes
0 answers
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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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2 votes
0 answers
97 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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4 votes
0 answers
114 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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1 vote
1 answer
156 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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0 votes
0 answers
77 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
367 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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3 votes
2 answers
175 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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0 votes
0 answers
2k 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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6 votes
0 answers
106 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
258 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
512 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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11 votes
2 answers
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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