Questions tagged [percolation]
The percolation tag has no usage guidance.
92
questions
6
votes
0
answers
85
views
The uniform odd and even subgraph of $\mathbb{Z}^2$
Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
2
votes
0
answers
73
views
Multi-scale 3- and 5-arm exponents for critical planar percolation
Consider critical site percolation on the planar triangular lattice. Denote by $A_j(m,n)$ the event that there are $j$ arms (paths from the inner boundary to the outer boundary) of alternating colour ...
2
votes
0
answers
99
views
The fluctuations of a random path
Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...
13
votes
1
answer
946
views
A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?
A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points.
For example, here are $20$ random ...
0
votes
0
answers
26
views
Do finite components get smaller as supercritical random graphs with an arbitrary degree sequence get denser?
I asked this question a few weeks ago on MSE but did not receive any responses so I am going to ask a related but more specific question here.
First some notation.
Let $\mathbb{G} = \mathbb{G}(n,\...
1
vote
0
answers
45
views
Locality and restriction properties for self-avoiding and loop-erasing random walks
This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa
I ...
1
vote
0
answers
32
views
Crossing a slightly longer box in Bernoulli percolation
Consdier critical Bernoulli bond percolation on $\mathbb{Z}^2$. Given $a, b \in \mathbb{N}$ denote by $p(a,b)$ the probability that there is an open left-right crossing in the box $[0,a]\times [0,b]$....
3
votes
0
answers
161
views
Topology of level sets for meromorphic function
Let $F$ be a meromorphic function on $\mathbb{C}$.
I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ ...
3
votes
0
answers
66
views
The boundary between infinite clusters connected by closed and open bonds
In the following, I'll heuristically describe a boundary between two infinite clusters arising in percolation on the triangular lattice. I expect this concept has been well-studied before. My hope is ...
6
votes
1
answer
109
views
What's the minimum ratio of positive cells such that the player has a positive probability to reach the boundary of a large random map?
A map with $(2n-1)$ rows and $(2n-1)$ columns of square cells is generated randomly as follows: the value of each cell is chosen from {1, -1} randomly and independently with probability {$p$, $1-p$}. ...
6
votes
1
answer
206
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
228
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
52
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
26
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
2
answers
404
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 ...
0
votes
0
answers
69
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 ...
2
votes
2
answers
159
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
87
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....
9
votes
2
answers
446
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 vertices $(n,m)$ and $(n',m')$ if and only if
$\vert n-...
9
votes
2
answers
1k
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
59
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
305
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
159
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'...
1
vote
1
answer
243
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}...
3
votes
0
answers
76
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 ...
0
votes
0
answers
122
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.
3
votes
1
answer
178
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 ...
5
votes
1
answer
283
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
155
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
117
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
2k
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
151
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
175
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
288
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
349
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
vote
1
answer
92
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 ...
6
votes
0
answers
247
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
votes
0
answers
102
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
50
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
88
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
113
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
votes
1
answer
87
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
votes
1
answer
175
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
votes
0
answers
116
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
114
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
vote
1
answer
153
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
votes
1
answer
381
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
95
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.
0
votes
0
answers
80
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 ...
2
votes
0
answers
101
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 ...