Questions tagged [percolation]
39 questions
15
votes
2 answers
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 lattice $\mathbb{Z}^2$, and open each edge…
Yvan Velenik
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10
votes
1 answer
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 there is a spanning cluster) below T_c and that it…
Matthew Matic
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6
votes
0 answers
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 don't know field theories, which belong the…
Nikita
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6
votes
1 answer
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 have a probability $p$ of being "open" and a…
Darcy
- 183
5
votes
1 answer
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.
The review article is Percolation and number of…
MathOverview
- 268
5
votes
1 answer
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 passage in the article on the arXiv about strong…
MathOverview
- 268
4
votes
0 answers
Question about 1D Percolation Theory
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 moment $M_k$ of the cluster number $n_k$ in one dimension.…
Brasswyrm
- 51
4
votes
0 answers
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 playing Ultimate Frisbee, at the beginner level, and I…
Giorgio Torrieri
- 49
- 1
3
votes
0 answers
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 as follows. Consider the $\mathbb{Z}^2$ lattice and…
DmitryZ
- 131
3
votes
2 answers
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 I want to mean Ising models of first and…
MathOverview
- 268
3
votes
1 answer
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 just to see if I get this definition right, for the…
Lagerbaer
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2
votes
0 answers
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 about a random cluster model with non-integer $q$?
I…
tclin
- 49
2
votes
1 answer
When $h>0$ and $0
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 nearest neighbor couplings.
When $h>0$ and $0< T <…
user196574
- 2,482
2
votes
1 answer
Correlation length for percolation theory
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 https://arxiv.org/pdf/1902.03708.pdf.
The Figure…
jore1
- 99
2
votes
0 answers
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 machine in mind), as it affects the efficiency of the…
Roger V.
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