Percolation theory: What is the critical amplitude for the “backbone” of a 2-D network?

Question

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. The probability that a bond is "open" is given by $p$ and the corresponding probability that a bond is "closed" is given by $1-p$. For an infinite lattice, there exists the critical threshold $p_c$ where a single cluster of open bonds will form an infinite cluster. Additionally, there is some probability $P(p)$ that any given bond will belong to this infinite cluster.

When $p < p_c$, then $P(p) = 0$ since no infinite cluster exists. However, when $p\geq p_c$, then

$$P(p)\approx A(p-p_c)^\beta .$$

Most of the above info comes from pg 196 of Clerc et al., 1990, The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models.

The issue is, Clerc et al. never define $A$ or $\beta$!

Looking on Wikipedia, I believe the critical exponent I'm looking for is called the "backbone" critical exponent and has a value of $\beta = 1.64$ (denoted $d_B$ in the Wikipedia article).

However, I am not sure what the value of $A$ is. I have looked in many places and many authors simply say that $P(p)$ is proportional to $(p-p_c)^\beta$ without ever defining what the scaling factor is.

My question: What is the value of $A$ for a 2-D square network of bonds and sites as defined above?

Any help is appreciated.

Answer 1

The coefficient $\beta$ of the percolation strength is defined as $P_\infty \propto (p-p_c)^\beta$ as $p \rightarrow p_c^+$. It is necessary to note here that the function $P_\infty$ is asymptotically equal to $(p-p_c)^\beta$ in the appropriate limit, not proportional in the usual sense. Therefore in the limit you can take $A=1$.

The strict definition of asymptotic equality between functions $f(x)$ and $g(x)$ at $x_0$ is that:

$$f(x) \propto g(x)$$ as $x \rightarrow x_0$ only if: $$\lim_{x\rightarrow x_0} \frac{f(x)}{g(x)} = 1.$$

Assuming what you said really held for all $p\geq p_c$ for the particular lattice (please double check that), you could easily find $A$ by noting that at $p=1$ every bond must belong to the infinite cluster, i.e. $P=1$. Hence $$A = (1-p_c)^{-\beta}.$$

Answer 2

In the notation of wikipedia's percolation critical exponents article, the critical exponent for the probability of a bond being part of the backbone is the conductivity exponent $t$, not $d_B$.¹ The "backbone" essentially consists of those bonds along which there is actual flow, i.e. those bonds that are not part of a dead-end branch. $d_B$ is the fractal dimension of the backbone, so the number of backbone bonds within a radius $r$ of an arbitrary point scales as $r^{d_B}$.

The proportionality constant $A$ is the critical amplitude, and unlike the critical exponents, critical amplitudes are not in general universal, i.e. their value can depend on the details of the lattice. Ratios of different critical amplitudes corresponding to different critical exponents may, however, be universal, e.g. see J. Viti's 2012 Ph.D. thesis "Universal properties of two-dimensional percolation". Since the critical exponents may only describe behaviour near the critical point, one can't determine the critical amplitude by simple normalization arguments.

$\beta$ is the critical exponent for the probability of bonds belonging to an infinite cluster, and the critical amplitude associated with $\beta$ is usually called $B$, not $A$, i.e. $$P(p)=B(p-p_c)^\beta$$ e.g. See Eq. 16 of "Recent advances in percolation theory and its applications" by A.A. Saberi, Physics Reports 578 (2015) 1-32 or Eqs. 4.39-4.42 of Viti.

The values of critical amplitudes are not trivial to calculate and depend on details such as the lattice type (e.g. square, triangular, honeycomb, …), and whether bond percolation or site percolation is being studied. Some reported values of $B$ for bond percolation on a two-dimensional square lattice are $1.55$ ^2,3, $1.4138\pm0.0015$ ⁴, and $1.39\pm0.01$ ⁵.

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References

(1) "Conductivity exponent and backbone dimension in 2-d percolation", P. Grassberger, Physica A 262 (1999) 251-263.

(2) Eq. 2.2, "Percolation processes in two dimensions IV. Percolation probability", by M. Sykes, D. Gaunt, M. Glen, J. Phys. A: Math. Gen. 9 (1976)725-730.

(3) Table 4, "Scaling theory of percolation clusters", by D. Stauffer, Phys. Rep. 54 (1979) 1-74.

(4) Table 11, "On two-dimensional percolation", A.R. Conway and A.J. Guttman, J. Phys. A: Math. Gen 28 (1995) 891-904.

(5) Table 7, "Universality of amplitude combinations on two-dimensional percolation", D. Daboul, A. Aharony and D. Stauffer, J. Phys. A: Math. Gen 33 (2000) 1113-1137.