Questions tagged [percolation]
The percolation tag has no usage guidance.
11
questions
57
votes
6
answers
5k
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 ...
2
votes
1
answer
386
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 ...
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 ...
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?
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 ...
8
votes
1
answer
2k
views
Van Den Berg-Kesten-Reimer inequality
Van Den Berg-Kesten-Reimer inequality
Let $n$ be a positive integer. For $i\in[n]$, let $\Omega_i$ be a finite set and $\mu_i$ a probability measure on it. Set $\Omega=\Omega_1\!\times\!\ldots\!\...
4
votes
0
answers
603
views
Expected number of components with multiple cycles in a subgraph of a square lattice
Short version
Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the edge-...
3
votes
1
answer
176
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 "...
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 ...
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 ...
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 ...