# Questions tagged [percolation]

Percolation theory describes the behavior of connected clusters in a random graph.

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### Triangle Hexagon Duality

https://arxiv.org/pdf/1004.1435.pdf In this paper below equation 6, a dual relationship is presented between the triangular lattice and the hexagonal lattice. I would like to understand how the ...
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### On the probability of the existence of path from one corner to its opposite

Consider a $n\times n$ grid, whose nodes are randomly colored black and white (with probability $p$ and $1-p$ respectively). Let $A$ be the event that there exists a path of all black nodes connecting ...
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### Book recommendation for introductory percolation theory

I want some recommendation on introductory level books on the mentioned topics. if someone recommend good lecture notes/tutorials on the mentioned topic that also appreciable. If someone share some ...
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### Limit of a sequence involving floor function

I am reading an article, and in a certain point I need to estimate the following limit $$\xi_p=\lim \limits_{k \to +\infty} \frac{k}{\lfloor\frac{k}{n}\rfloor+1},$$ where $n \ge 1$ is fixed. The ...
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### How does one interpret last passage percolation?

To the best of my knowledge, the question that first passage percolation tries to address is whether/when something (say a fluid) will reach a certain destination from some given source. A paper I was ...
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### Find $B_n$ such that $\mathbb{P}_p(A \mathbin{\Delta} B_n)=0$

I have a question about percolation. Show that any measurable event can be approximated by events depending on finitely many edges, in the sense that for any $A$ in the product $\sigma$-algebra, ...
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### Help with visualizing the matching lattice of a triangular lattice

I'm reading Creswick, Farach, and Poole's book called Introduction to Renormalization Group Methods in Physics (unfortunately, it's out of print). Despite the book being about physics, I was hoping ...
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### Why is the Erdos-Renyi model called a mean field theory of percolation?

ER can be viewed as edge percolation on a complete graph. But some sources say that physicists like to refer to it as a 'mean field theory' of percolation. Why is that so? Does it correspond to some ...
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### Expected size of colored block on chessboard?

Randomly color the squares of an $m\times n$ chessboard red or black (each square has a fifty-fifty chance of being red or black). A monochromatic region is a set of squares that are connected along ...
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### What is the average size of an island? [closed]

If you have a square grid, and each square* has probability $n$ of being ground. If the other squares are water, what is the average area of an island? If $n$ is small then the average island would ...
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### Finding lower bound for standard deviation

I have a random variable $R_n$ and a constant $w_n$ (which are related to a oriented percolation problem from https://arxiv.org/abs/1610.10018 on section 4.1(ii)) with the following properties: (...
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### Strict stochastic domination of “thinned out” random cluster model

Fix some $q\geq 1$ and denote by $X_p$ a random variable sampled from the law of the random cluster model with parameters $p,q$ on some graph $G$ and with, say, free boundary conditions. Define the "...
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### Percolation related counting problem

I was trying to look into the following problem, which I intend to use for a lemma for a bigger problem. The question is: For the 2-dimensional integer lattice, what are some good lower and upper ...
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### Percolation Textbook Recommendation

I was wondering if someone could recommend a Percolation textbook for undergraduates. I have looked at Percolation by Grimmett and it seems quite dense. I was looking for a book that I could self-...
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### $a_{n+m+2} \leq a_m+a_n+g(n)$ with $g(n) = o(n)$. Show that $a_n \geq (n+2)\lambda-g(n)$ where $\lambda = \lim \frac{a_n}{n}$

I was working on the proof of the exponential decay on supercritical percolation as shown in Grimmett's Percolation (1999, 2 ed. pg 206 - 210) and he uses as a lemma a form of the subadditive theorem (...
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