1
$\begingroup$

I read the Wikipedia article on Percolation critical exponents. It says:

In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.

Let's consider the site percolation model on a square lattice. I'm specifically interested in whether the value of the critical exponent $\nu$ depends on local properties such as whether we consider NN connections or NN+NNN connections.

I know for a fact the percolation threshold itself depends on the neighborhood connectivity i.e. for NN the $p_c$ is 0.592 whereas for NNN+NN the $p_c$ is 0.407. C.f. Square-lattice site percolation at increasing ranges of neighbor bonds (Malarz & Galam, 2005). However, I suspect that value of $\nu$ will be same for both cases (NN and NNN+NN) i.e. $\nu=\frac{4}{3}$, since the dimension $d$ is $2$ for both. I think this should follow from the universality principle but I didn't find any good reference for this.

I did read through Levenshteĭn, M. E.; B. I. Shklovskiĭ; M. S. Shur; A. L. Éfros (1975). "The relation between the critical exponents of percolation theory" (PDF) but it doesn't really clarify this point well (at least, it's not clear to me after a couple of readings).

So, basically, my question is: Is there any good reference which states that the value of the critical exponent $\nu$ does not depend on local details such that the order of neighborhood connectivity. If there exist stronger results than this, I'd be interested in them too.

$\endgroup$

1 Answer 1

2
$\begingroup$

For a reference on the independence of critical exponents on the type of two-dimensional lattice (square, triangular, honeycomb, bond/site percolation, nearest-neighbor+next nearest neighbor, ...) you could cite Staffer and Aharony's Introduction to Percolation Theory (page 53). The reason for this "universality" is that the critical exponent describes the system in the limit that the correlation length diverges, so local details are lost.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.