# Tagged Questions

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

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### What does it mean to know exact value of a number?

If we go to this page, https://en.wikipedia.org/wiki/Percolation_threshold, we find the statement "a tremendous amount of work has gone into finding exact and approximate values of the percolation ...
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### Placing spheres uniformly at random over $\mathbb{R}^3$

Put spheres uniformly at random all over $\mathbb{R}^3$, with density 1 sphere / unit cube. All spheres have the same radius $r$. What is the probability function $p(r)$, that that there is an ...
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Assume a unidirectional, unweighted network generated according to a degree distribution. Each node is given a value between 0 and 1 called threshold $\phi$. We topple some nodes, the neighbours will ...
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### Probability $u, v$ are connected in a random graph model

There is a random graph problem. Given an undirected graph $G(V,E)$ associated with $p_{uv}$ to denote the probability there is an edge between $u$ and $v$ ($p_{uv}$ are independent, maybe different ...
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### “Chemists triple point” in percolation theory

This is a vague question asking about the existence of a mathematical object, instead of properties of a well defined one. I am sorry if this is not the correct forum. I know if you have a random ...
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### Site percolation model that cannot be obtained from a bond percolation model

It is easy to obtain a site percolation model from a bond percolation model on a graph $G$ using the covering graph $G_c$ of $G$. I wondered if one can obtain any site percolation model from any site ...
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### Percolation events

Consider bond percolation on $\mathbb{Z}^d$. How can we prove that the set of configurations $A = \{ \text{there exist an infinite open cluster} \}$ is an event, i.e. that it belongs to the cyllinder ...
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### Is it possible to construct graphs with any critical bond percolation probability?

Given some probability $p\in[0,1]$ is it possible to construct a graph $(G,V)$ with critical bond percolation probability $p_c = p$? I know for example that I can get $\frac{1}{m}$ for any natural ...
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### Exponential decay

During my study of percolation I came across exponential decay and there are some parts I do not understand about this. The definition of exponential decay is as follows: $f(t)$ decays ...
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### Probability on Graphs. Percolation.

I am studying the book Probability on Graphs by Grimett. Grimett tells us that $\mu_1 \leq_{st} \mu_2$ if and only if $\mu_1(f)\leq\mu_2(f)$ for all increasing functions $f:\Omega\to \mathbb{R}$. I ...
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### Infrared bound and mean field theory in percolation theory

I have seen various references to the phrases "infrared bound" and "mean field theory", together or separately in the context of various lattice models. (Percolation, Ising Model, Interacting Particle ...
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### Stochastic domination preserved by dilution?

Consider an at most countable set $S$ and the corresponding bit space $\{0, 1\}^S$ that is often considered in percolation, interacting particle systems, and other lattice models. Suppose that $\le$ ...
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### How do you use renormalization techniques to get the critical exponent for percolation strength?

I've read this and this. The percolation strength, or $P(p) \sim |p-p_c|^{\beta}$, where $p_c$ is the critical probability, is the probability that an arbitrary site in a lattice is part of an ...
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### Bond percolation probability on a Bethe lattice

I derived the bond-percolation probability for a Bethe lattice and my result disagrees with a seemingly reputable reference by Albert & Barabasi (see below). I would be happy if somebody could ...
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### Percolation over the integers [closed]

Imagine a simple undirected graph with a countably infinite number of vertices, each labeled with an unique integer $v \in \mathbb{Z}$. Connect the vertices by edges randomly such that Each vertex ...
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### amenable groups versus amenable graphs

In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability In ...
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### Generalized percolation problem

Consider a simple site percolation problem on, for example, a 2D square lattice. Each vertex is randomly either there or not with some probability. If two neighbouring vertices are present, then the ...
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### Number of circuits that surround the square.

Consider a grid $G$ in the $\mathbb{R}^2$ plane formed by the points $(x,y)$ with integer coordinates i.e. $G=\{(x,y)\in\mathbb{R}^2: x\in\mathbb{Z},\;y\in\mathbb{Z} \}$. For $n>0$ let $B_n$ square ...
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### Books on random permutations

I'm looking for books or introductory/expository articles about random permutations, in particular with regards to their cycle structure. EDIT: I forgot to mention that I'm especially interested in ...
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### How to calculate the critical density estimation for “continuum” percolation model in “3D space” when we have “spatial correlation”?

I want to approximately estimate the critical density (lower bound for density) of balls in a cube to make sure that the upper and lower surfaces of the cube will be connected to each other through ...
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### First-hitting probability for the 2D critical site percolation on triangular lattice

Consider critical site percolation on the triangular lattice with mesh-size $\delta$. Let $D$ be the domain $H\backslash[0;i]$ where $H$ is the complex upper half plane. Compute the probability  \...
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### Reliability polynomial of Cartesian produt of graphs

The all-terminal reliability $R(p)$ of a graph is the probability that the graph remains connected after edges fail independently with probability $p$. Similar the two-terminal reliability $T_{u,v}(p)$...
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### Percolation Theory Basics: Open cluster size decay (Square Lattice)

I am trying to learn some stuff about percolation. On wiki (http://en.wikipedia.org/wiki/Percolation_theory) it says: "when $p<p_{c}$, the probability that a specific point (for example, the ...
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### The Gaussian moat problem and its extension to other rings in $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$

One of my favourite open problems in number theory, an area in which I enjoy only as a hobbyist, is the Gaussian moat problem, namely "Is it possible to walk to infinity in $\mathbb{C}$, taking ...
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### Percolation in a finite, 2D rectangular grid

Images: I'd like some hints to the ??? pointed out as well as how to approach the case in which the X,Y are opposite corners. P.S. This is a problem which I came up with after a conversation with ...
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### Total number of nodes in critical Galton-Watson process

Consider the following critical Galton-Watson process: initially there is a population of $Z(0) = z_0$. The distribution of children for each node follows a binomial law, with maximum value $d$; i.e. ...
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### Size of connected regions on a randomly-colored infinite chessboard

Consider an infinite chessboard where each square is colored white with probability $p$ and black with probability $1-p$. Suppose without loss of generality that the square at $(0,0)$ is white. We ...
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### Percolation and number of phases in the 2D Ising model.

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive After a long time I came back to try to understand an article on the Ising model. ...
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### A spaceship travelling to infinity while avoiding star collisions

Consider placing countably infinitely many points labeled $S_i$ randomly over $\mathbb{R}^2$, with asymptotic density points/area $Âµ$. Then, what is the largest $r$ such that we can find a a ...
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### How is percolation defined and measured in social networks?

From the wikipedia article on percolation it appears that the theory is applicable to graphs in general, and this presentation describes the theory nicely. This review article on complex networks ...
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### Edge percolation on $\mathbb{Z}^2$: probability that two neighbouring vertices are connected?

I'm considering edge percolation on $\mathbb{Z}^2$ with parameter $p$, so that edges are present with probability $p$. Is it known how to express the probability $P(p)$ that $(0,0)$ is in the same ...
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### Does Kolmogorov 0-1 law apply to every translation invariant event?

Let $\Lambda$ be a lattice in $\mathbb{R}^d$, $d \ge 2$, thought of as an infinite graph. In percolation theory we consider properties random subgraphs of $\Lambda$. In site percolation $\Lambda^s_p$ ...
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### percolation - number of cycles around the origin

I try to study Percolation Theory by "A mini course on percolation theory" of Jeffrey E. Steif. I am very curios about Exercise 2.4. Show that the number of cycles around the origin of length n is ...
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### Percolation theory BK inequality

Let $X_1,\cdots X_n$ i.i.d bernoulli(p) r'v's Let $A = \sum_{i=1}^nX_i \geq l$, and $B= \sum_{i=1}^n X_i \geq k$ Then $A \circ B = \sum_{i=1}^nX_i \geq l+k$ (disjoint occurance of A, B) Im ...
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### Branching process question

(Cross-posted from mathoverflow Q 93609) I am trying to analyze the following branching process. We start with a root (level 0) node. Each surviving node has two children, each of which may or may ...
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### Translation Invariant Random Variable (Percolation)

I'm currently studying the uniqueness of the infinite cluster in supercritical percolation. In the proof the rv $N$ counts the number of infinite clusters on the infinite 2-D square lattice. The claim ...
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### Percolation and hopping, has it been described?

This ultimately relates to a physics question, but It wasn't getting any answers, on physics.stackexchange. As it crosses the boundaries of each subject I was advised to post it here: Has a system ...
Let $\Omega\subset \mathbb C$ be a simply connected domain, $\tau = \exp(2\pi\mathrm i/3)$ and $a(\alpha),a(\tau\alpha),a(\tau^2\alpha)$ are some accessible points of $\Omega$. In this paper by S. ...