Questions tagged [percolation]

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

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Cluster growth by random enumeration

Consider the $\mathbb{Z}^2$-lattice and the $l^p$-Ball subgraph $\Lambda_r^p = (V_r^p, E_r^p)$ with $ V_r^p = \{v\in\mathbb{Z}^2 \colon \|v\|_p\leq r \}$ and $E_r^p = \{e = \{v,w\}\colon v,w \in V_r^p ...
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Existence of a bigeodesic is a tail event!

In the Last passage percolation in $\mathbb{Z}^2$ (with iid exponential weight), an infinite up/right path $\gamma$ is called a bigeodesic if for every pair of vertex $x, y\in \gamma$ the restriction ...
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88 views

Probability of infinite cluster on this specific random geometry.

Context We consider the plane, $\mathbb{R}^2$, which we split into squares of size $1$. Each square has a probability $p$ of being black and $1-p$ of being white, independently for each square. Let $F$...
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Exact sampling of the $\mathbb{Z}^2$ - Ising measure on finite subsets

Consider the standard Ising model on $\mathbb{Z}^2$ for some inverse temperature $\beta\leq \beta_c$ (such that the corresponding inifinite volume measure $\mu$ is unique). For a finite subset $A\...
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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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1answer
34 views

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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49 views

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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1answer
114 views

How to prove that two probability measures are equal in the described scenario?

Let $(\mathbb{Z}^d, \mathbf{E}^d)$ be a graph with vertex set $\mathbb{Z}^d$ and edge set $\mathbf{E}^d$, such that $\mathbf{E}^d = \{(x, y) \in \mathbb{Z}^d \times \mathbb{Z}^d : \sum_{i = 1}^{d} |...
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In an $n\times n$ grid with diagonals, how to count number of paths along the diagonals?

Given an $n\times n$ grid, with two diagonals in each unit square. I am interested in the number of (directed) paths or walks from one side of the grid to the opposite side, walking along the ...
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75 views

Probability of crossing $n\times n$ grid with random diagonals; and bond percolation critical threshold $p_c$

You can always cross an $n\times n$ grid with random diagonals, from one side of the grid to the opposite side of the grid. So the probability of this crossing is $1$. Here random diagonals means you ...
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Exponential Growth and Decay Model for Human Genealogy (Common Ancestor)

First time poster, and, as my post will intimate, not a mathematician, just someone searching for answers. My question has two parts: 1) In a genealogical chart for a single individual (called an ...
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1answer
64 views

Hexagonal grid elementary simulation of percolation (Cardy's formula) using adjacency matrix. Looking for improvements

We consider here the probability to percolate (i.e. find a path) in a triangular shape made of hexagons as given in Fig. 1. Fig. 1 : Departure line in red (hexagons $1, 2, 4, 7, 11, 16, 22$). ...
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Why is the percolation threshold not well-defined in a finite lattice?

I am not a mathematician but I want to get some familiarity with percolation theory for an application to my job. I am reading this text, where it is implied on page 5 that the percolation threshold ...
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Percolation theory critical density simple proof!

Is there a simpler proof for the existence of infinite connected component in 2D lattice (percolation theory) if the probability of connection exceeds a critical threshold? Currently, I am reading the ...
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15 views

Optimizing parameter to obtain a lower bound to a probability

I am reading a probability paper and I am having trouble in a proof. My doubt is in the proof of Lemma 10 (page 15). Basically, the lema says, that, for any $\delta, \eta >0$, there exists $\...
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3answers
130 views

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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1answer
42 views

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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29 views

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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81 views

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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815 views

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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81 views

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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52 views

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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111 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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97 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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1answer
539 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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1answer
78 views

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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59 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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43 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
258 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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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
107 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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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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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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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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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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525 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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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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51 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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263 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 ...