Questions tagged [percolation]

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Connectivity of random geometric graph with open boundary conditions

I have a question regarding the existence of a closed-form solution of the connectivity in terms of the radius of vertices (disks) in a two-dimensional ($d=2$) random geometric graph (RGG) with open ...
Johannes Nauta's user avatar
2 votes
0 answers
45 views

How to show random cluster models with non-integer $q$ have no local description?

It is known that the random cluster model with $q = 1$ corresponds to bond percolation, and $q = 2, 3, ... $ corresponds to the $q$-state Potts model. Both of these have a local description. What ...
tclin's user avatar
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Boundary/contour tracing of a binary image in Python

I'm studying percolation on a square lattice. I identify clusters on the binary lattice using the SciPy API ndimage.label. My goal is to study the external perimeter (hull) of each of these clusters. ...
2 votes
1 answer
88 views

When $h>0$ and $0<T<\infty$, do up domains percolate in the Ising model?

I'm considering the Ising model in a field on a square lattice in $d$ dimensions: $$H = -\sum_{\langle i j \rangle} \sigma_i \sigma_j - h \sum_{i} \sigma_i$$ As usual, $\langle i j\rangle$ refers to ...
user196574's user avatar
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Union of Percolations

I also asked this in Math Stackexchange, but since the motivation is from statistical models such as Potts or loop $O(n)$ models, I would also like to ask this here. Consider a finite graph $G=(V,E)$. ...
Andrew Yuan's user avatar
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2 votes
1 answer
138 views

Correlation length for percolation theory [closed]

I am currently running a numerical simulation for site percolation. Using periodic boundary conditions I am attempting to determine the correlation length following the method set out in this paper ...
jore1's user avatar
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Local statistical model for percolation

I am interested in classical statistical models with degrees of freedom with a finite configuration set on regular lattices, and Boltzmann weights depending on the configurations in a constant-size ...
Andi Bauer's user avatar
1 vote
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62 views

Is there any book on percolation theory that analyse bond percolation mathematically?

Introduction to percolation theory, by Stauffer and Aharony, deals with site percolation only. Is there any book which describes mathematically the cluster size distribution, mean cluster size etc for ...
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If a truncated power law distribution still has no characteristic length scale?

I do know that a power law distribution can extend from 0 to $+\infty$, so due to the shape of the distribution, there is no way to define an average value (this might be a characteristic length scale ...
Tingchang Yin's user avatar
4 votes
0 answers
71 views

Question about 1D Percolation Theory [closed]

I am currently reading "Introduction to Percolation Theory" by Stauffer and Aharony and am doing the problems. Question 2.2 wants me to calculate a closed-form expression for the $k$-th ...
Brasswyrm's user avatar
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48 views

Coffee quality and temperature

The question is about adjusting the size of particles of coffee powder to temperature. It is said that properly grinding the coffee grains is cricual for making good coffee (I have an espresso ...
Roger V.'s user avatar
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2D Ising model and FK-percolation

Consider the 2D Ising model on the finite lattice $\Lambda$ with $+$ boundary conditions, i.e., all spins outside of $\Lambda$ are $=+1$. Let $\mathscr{E}_\Lambda^b$ denote the edges in $\Lambda$ and ...
Andrew Yuan's user avatar
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6 votes
0 answers
298 views

Percolation universality class

There's very good table of different universality classes: Ising model lies in the same universality class with $\phi^4$ theory. Ising in $d≥4$ have critical exponents for free scalar field. But I ...
Nikita's user avatar
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1 vote
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A question on subcritical percolation

Consider some 2D lattice of infinite extent where each site can either be $0$ or $1$ with probability $p$. It is known there exists some critical probability $p_c$ below which clusters of $1$ values ...
kevinkayaks's user avatar
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Is it possible to implement the Invaded Cluster Algorithm on a network of Ising spins rather than a lattice?

My main concern is how to know if a cluster has percolated the network. For a periodic square lattice it is easy to determine if the cluster percolates by looking at the size of the cluster (as given ...
Pratyush Kollepara's user avatar
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1 answer
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Percolation of electrical conductivity

When a metal is made of different conductivity parts, current passes the least resistive percolation path. My question is “does this path depend on the amount of current?” I think the path doesn’t ...
Long's user avatar
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Percolation in Bethe Lattice with Alternating Neighboor Numbers

Suppose we have a Bethe Lattice with alternating numbers of neighbours,starting from 3 and then 4 and so on. So the zeroth point has 3 neighbours, each of those 3 first neighbours has 4 neighbours (3 ...
Arbiter's user avatar
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2 answers
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Percolation theory: What is the critical amplitude for the "backbone" of a 2-D network?

Disclaimer: I am just learning about percolation theory for the first time, so I am not too familiar with some of the terminology. Suppose you have a 2-D square lattice with bonds connecting sites. ...
Darcy's user avatar
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6 votes
1 answer
364 views

Percolation theory: Is there a clear relationship between "probability of connection" and "effective porosity"?

Disclaimer: I am a geophysicist, not a physicist. Sometimes we speak a slightly different language and use different terms. Suppose you have an $N$ x $N$, 2-D rectangular lattice with bonds which ...
Darcy's user avatar
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1 vote
1 answer
217 views

Physical interpretation of power law cluster size distribution in percolation problem

In the site percolation problem, when the occupation probability $p \rightarrow p_c$, where $p_c$ is the critical probability. The characteristic length diverges, and assuming the usual scaling ansatz ...
Akerai's user avatar
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2 votes
2 answers
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Is there a link between Percolation Theory and phase transitions?

Is there a direct link between phase transitions in physics and Percolation Theory (and the critical points in both cases) in the sense that the mechanism in one of them could be a special or ...
ali's user avatar
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0 answers
56 views

What's that theorem? All RG operators flow to directed percolation?

I can't for the life of me get this theorem straight. I can remember neither the name nor it's statement and would be grateful for anyone who wants to toss out some wisdom. It pertains to the ...
Jojker's user avatar
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4 votes
0 answers
180 views

Percolation in Ultimate Frisbee (and Rugby, American Football, Basketball etc)

Question Has anyone ever investigated weather game dynamics in certain sports ever experiences a percolation-type transition with catching probability as the driver? Details I recently started ...
Giorgio Torrieri's user avatar
1 vote
0 answers
33 views

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 ...
Klara.D's user avatar
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1 vote
0 answers
71 views

When does a finite size random graph percolate?

Assume we are simulating percolation on a 2d lattice. While the system is of finite size, we say that the critical state appears when a cluster connects two opposing ends of the lattice. The bigger ...
Dionysios Georgiadis's user avatar
0 votes
1 answer
193 views

Time for Two Reservoirs that are Connected with a Porous Medium to Equilibrate

I'm trying to calculate the time for the liquid in two connected reservoirs (open to the air) to equilibrate. The tricky part is the connecting region, which consists of a porous medium. I understand ...
SHL's user avatar
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0 answers
242 views

Speed of sound at phase transition -- does percolation theory describe it?

Tl;dr: How does the speed of sound vary across a phrase transition? Although a phase transition might be "instantaneous" in terms of temperature, if we instead control Pressure and Volume then we ...
Alex Meiburg's user avatar
1 vote
2 answers
91 views

Is there a formulation for (self-)accelerating fluid flow through permeable medium?

I have a permeable system where is an accelerating fluid flow. Imagine a sponge that is squeezed. The fluid starts at rest, accelerates and flows out from the sponge. How to calculate the fluid speed? ...
Juha's user avatar
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3 votes
2 answers
445 views

Phase transition without the Peierls' counter argument

Is there any proof of the existence of phase transition in models of statistical mechanics of the Ising type models without using the Peierls' argument and its variations? By models of the Ising type ...
MathOverview's user avatar
2 votes
3 answers
2k views

Why does correlation length diverge at the percolation threshold?

I'm reading a paper about electronic percolation. $p$ is the fraction of occupied bonds (or sites, depending on the model you're using, but I'll just use bonds), $p_c$ is the critical fraction of ...
F dot Floss's user avatar
1 vote
0 answers
71 views

Why do these papers show the wrong concavity to the conductivity near the percolation threshold?

I'm looking at two papers in particular: A. L. Efros and B. I. Shklovskii, Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non-Metal Transition Threshold, Phys. Status Solidi ...
YungHummmma's user avatar
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2 votes
1 answer
1k views

Numerical Ising Model: Swendsen–Wang algorithm, Percolation theory?

When you look at the original paper of Swendsen and Wang in 1987: "Nonuniversal critical dynamics in Monte Carlo simulations" it is somewhat mentioned that the proposed algorithm uses percolation ...
varantir's user avatar
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1 vote
0 answers
208 views

Physical, intuitive reason for divergence of dielectric constant at electronic percolation transition?

Several papers such as this (warning, PDF) and this (PDF again) talk about how, near the electronic percolation transition for a metallic 2D film, the real part of the dielectric constant diverges (...
YungHummmma's user avatar
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5 votes
1 answer
421 views

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. ...
MathOverview's user avatar
3 votes
0 answers
163 views

Random orientation percolation (Grimmett model) from the viewpoint of statistical mechanics

This is a rather soft question, but I would like to know how physicists would approach a problem which seems to be hard from the mathematical prospective. The Grimmett percolation model is defined ...
DmitryZ's user avatar
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5 votes
1 answer
482 views

The strong Markov property of Gibbs measures in 2D Ising Model

My background is that of a mathematician. I have a question about the two Dimensional Ising Model. I think the terminology I use is similar to the physical. I'm trying to understand the following ...
MathOverview's user avatar
10 votes
1 answer
2k views

Percolation in a 2D Ising model

For a 2D Ising model with zero applied field, it seems logical to me that the phases above and below T_c will have different percolation behaviour. I would expect that percolation occurs (and hence ...
Matthew Matic's user avatar
2 votes
1 answer
190 views

Have characteristics of a 'doubly percolated' lattice been described

Has a system where conducting sites can percolate by hopping over/tunnelling through a non-conducting site been described? If so what are the characteristics, and where can I find more details (such ...
AncientSwordRage's user avatar
3 votes
1 answer
291 views

Definition regarding percolation

in a homework sheet studying bond-percolation on the Bethe lattice, a function $g(r)$ is introduced as "the probability of finding two nodes separated by a distance $r$ on the same cluster". Now ...
Lagerbaer's user avatar
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15 votes
2 answers
166 views

Sampling typical clusters between distant points in subcritical percolation

I have on several occasions wondered how one might proceed in order to sample large subcritical Bernoulli bond-percolation clusters, say on the square lattice. More precisely, let's consider the ...
Yvan Velenik's user avatar
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