Questions tagged [percolation]

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

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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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59 views

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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74 views

Phase transition threshold for 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 $(...
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120 views

Examples of graphs that are amenable and non-amenable

The amenable graph $G=(V, E)$ is a graph that satisfies the following $$ \inf\limits_{K \subset V,\, |K|< \infty} \frac{\partial K}{|K|}=0$$ I know for example that $\mathbb{Z}^2$ is amenable ...
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How to make the description of my probabilistic binary lattice model more precise and succinct?

I am supposed to write up a report on a percolation theory project which I worked on, over the summers. I'm trying to describe my mathematical model below: We report some results regarding certain ...
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54 views

How to rigorously prove this very simple fact about percolation?

I speak about the last sentence. It is clear by saying we can model the openning at the probability $p_2$ by an openning at the probability $p_1$ followed by another at the probability $p_2- p_1$. But ...
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37 views

Good reference on solving the Smoluchowski coagulation equation with generating functions

I am looking at a paper "Exact Solutions of a Model For Crowding and Information Transmission in Financial Markets" by R.D'Hulst and G. J. Rodgers. They use a model from bond percolation theory based ...
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1answer
138 views

On bernoulli percolation, increasing events and Russo's formula

I am very new to this particular branch of probability theory, I try to be as formal as possible. In this question I consider bernoulli percolation as it is usually introduced as a first model (see ...
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54 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 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 link to the PDF ...
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1answer
103 views

Estimating the critical probabilities $\mathrm{P_{c1}}$ and $\mathrm{P_{c2}}$ mathematically for the infinite system case

Suppose I have an $\mathrm{N\times N}$ square matrix consisting of only $0$'s and $1$'s. (I'm considering that $0$ represents "white" and $1$ represents "black".) The probability of a certain element ...
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48 views

Does a relation exist between the 2d random walk and edge percolation on a 2d lattice

I apologize for the title gore. I was recently studying percolation theory to work on improving the theoretical work behind some research I'm doing with a professor. While working on it, a thought ...
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Average time bootstrap percolation

Presentation of the model: we consider the regular lattice created from $\mathbb{Z}^d$. At $t=0$, each site is said "active" independently with a probability $p$, "inactive" otherwise. At $t$, if a ...
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117 views

How to show that the probability that $\mathbb{Z}^d$ is internally spanned is equal to 1? (bootstrap percolation)

Presentation of the model: we consider the regular lattice created from $\mathbb{Z}^d$. At $t=0$, each site is said "active" independently with a probability $p$, "inactive" otherwise. At $t$, if a ...
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Explain why $P(\{\# C^p(y)=\infty\})=P(\{\# C^p(0)=\infty\})$. (Percolation)

We're in $\mathbb{Z}^d$, and we have a collection $(X_e^p)_{e \in E}$ where $P(X_e^p=1)=p=1-P(X_e^p=0)$, i.e. the probability of an edge $e$ being open is $p$. Let $C^p(y)$ is a random open cluster, ...
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26 views

Area of Shaded Regions in Randomly Generated Plots

Suppose you took an $n \times n$ array of cells, and for each cell there is a 0.5 probablity of being shaded or not. Suppose we defined "connected regions" as regions for which all shaded cells shared ...
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Probability of Path on Randomly Edge-Coloured Graph

I begin with a graph $G=(V,E)$. For each edge $e \in E$, I colour the edge with probability $p_e$. I'm looking for the probability that two vertices $v,w \in V$ are the endpoints of a coloured path. ...
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37 views

Determining if a graph subset is percolated or not

Assume you are presented with a undirected graph without self-loops, with a given arbitrary degree distribution. The nodes are in one of two states, $A$ or $B$. How is it possible to determine ...
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Dynamic site percolation of independant random walkers on 2d square lattice

Really appreciate your help, I am stuck in a part of my research which I am not expert in. I have a 2-dimensional square lattice with periodic boundary conditions(torus). I am placing one walker at ...
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Distribution of Last Passage Percolation on $\mathbb{Z}^2$.

Suppose I have square lattice $\mathbb{Z}^2$ and I endow each site $x$ with a weight $W_x$, which is an independent exponential random variable of rate $\mu$. An oriented path between $(1, 1)$ and ...
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366 views

Is there a book or lecture notes on Percolation Theory containing exercises?

I have seen Grimmett's Percolation Theory and I have also seen a few online lecture notes. But they don't have exercises. I understand it is stupid to ask of exercises in such a recent and hot ...
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37 views

Is Site Percolation with Bernoulli variables i.i.d. independent and identically distributed?

I cannot understand the identically distributed part in the i.i.d assumption. Consider a site percolation where each event is a Bernoulli variable. Does this mean ...
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How is the threshold probability $p_c$ defined for oriented site percolation?

The threshold probability for unoriented site percolation is such that \begin{eqnarray*} \mathbb{P}_{p} & = & \underset{v\in\mathbb{\mathbb{L}}^{d}}{\prod}\mu_{v}\\ \theta(p) & = & \...
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1answer
39 views

Site Percolation and S-clusters with $n\times m$ grid where $n\not =m$?

Consider a site percolation but change the dimension of the lattice from $n\times n$ to $n\times m$ where $n\not = m.$ S-clusters are defined for $n\times n$ lattice. The occupation of each site are ...
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1answer
159 views

Probability for Connection of LHS-RHS in Path of Bond Percolation?

Problem: Probability that that the left side and the right side are connected by a path Graph Consider a square lattice in a bond percolation. There are two sides which are connected together by a ...
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Sites always closed in Bond percolation?

The page 2 of Percolation by Bollobas et all (2009) contains this picture where the left is for the site percolation and the right for the bond percolation. The filled circles on the left are open ...
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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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Condition for global cascade

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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103 views

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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71 views

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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151 views

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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71 views

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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117 views

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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42 views

$k$ points of contact for percolation

In Grimmett's book on percolation near the beginning of Ch. 7, he summarizes the plan of proof of the result on percolation of slabs. We are in $\mathbb{Z}^d, d\ge2$, with his usual notation that $...
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What may be the number of connected components of this randomly generated topological space?

I hope there exists some answer to this but I am not very good in probability so i do not know how to tackle this problem. It's contemporary-art related so it might not have a good answer. Furthermore,...
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2answers
336 views

Translation invariant event in percolation theory in $\mathbb{Z}^d$

At my probability theory proseminar I had a speech about the percolation theory and one of the topics I presented was the uniqueness of an infinite open cluster in Bernoulli bond percolation in $\...
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347 views

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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77 views

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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109 views

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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108 views

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 $T_{u,v}(p)$...