Questions tagged [percolation]

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

72 questions
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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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