# Proof and interpretation of the following percolation theory result for $n\times n$ square grid

While I was discussing this question with @JamesMartin, he mentioned a result here that:

In a $n\times n$ finite square grid, if $p\geq p_c+\epsilon$, such that $\epsilon>0$ and $p_c$ is the critical point for percolation, then with a high probability there is a single large component of linear size i.e. $O(n)$ and all other components are small i.e. $O(\log(n))$.

This result looks extremely interesting but I couldn't manage to find it's statement or proof in any of the standard percolation books (it is possible that I missed it, but I did try looking for it as much as possible), like Bollobas and Riordan or Grimmett.

By the way, one thing I'm confused about is that James mentioned $p_c$ is that critical point where $\theta(p)$ becomes positive for the first time. He gave the definition of $\theta(p)$ as:

Fix the grid $\Bbb{Z}^{2}$. The origin is $(0,0)$. $\theta(p)$ is the probability that $v$ (any other point on the square grid) is contained in an infinite open cluster. This is the same for every point $v$ if the model is translation invariant.

So, my questions are:

1. I'm not sure what James meant by $p_c$ is the point where $\theta(p)$ becomes positive for the first time. Isn't $\theta(p)$ are probability? How can a probability be negative in the first place?

Moreover, the definition of $\theta(p)$ is not clear to me.

The picture above shows a part of an infinite square grid with some open edges and some closed edges. Say the blue dot in the centre is our origin $(0,0)$ and the pink dot in the centre is the point $v$. Then is $\theta(p)$ the probability that that pink dot is part of the infinite cluster of open edges? Or is the definition something else?

1. How to prove the theorem/result which James stated for the $n\times n$ finite grid? (If anyone could point me to an article or exact location of any textbook where the proof and statement are given, that would be very beneficial for me.)

2. What does the theorem mean by "large component"? Does it refer to a large cluster of open edges? Or does it refer to a large cluster of sites (represented by black dots in the image above)? I'm yet not sure if the theorem is for site percolation or bond percolation, which leads to my question 4, below.

3. This stated theorem seems to be about bond percolation. Is there any equivalent theorem for site percolation? Or can this theorem be restated for site percolation?

• In answer to your first question, $p_c = \operatorname{inf}\{p : \theta(p) > 0\}$. So, if $p<p_c$, then $\theta(p)=0$. In other words, $p_c$ is the point where $\theta(p)$ becomes strictly greater than $0$. – Andrew Uzzell May 2 '18 at 16:04
• @AndrewUzzell Thanks. :) Could you also explain the exact definition of $\theta(p)$ to be in that case? It's a part of question (1) – user123818 May 2 '18 at 16:06
• @JosephO'Rourke "We present a method of general applicability for finding exact or accurate approximations to bond percolation thresholds" Uhh, but is that paper about finding approximate bond percolation thresholds? Because I'm not concerned about that, but rather the theorem James stated – user123818 May 2 '18 at 17:32
• It seems that this paper, Section 2, also contains the statement along with some references. – Sangchul Lee May 3 '18 at 13:48
• @j.c. I would be happy if I can find some references with details. Anyway, in the first paper I linked, Lemma 2.1 and Remark 2.2 at page 5 tells that w.h.p. the largest cluster has size of order $n^d$ and all the other clusters have size $\mathcal{O}(\log n)$. ($A_{\kappa}^n$ in the statement is defined as the event that any occupied path of diameter $>\kappa\log n$ is contained in the largest cluster.) I tried to check the references therein to find some comprehensive proof of this, though they were hard to skim over. – Sangchul Lee May 3 '18 at 18:02

Here is a reference for part of the theorem referred to by James Martin.

Borgs, C.; Chayes, J. T.; Kesten, H.; Spencer, J., The birth of the infinite cluster: Finite-size scaling in percolation, Commun. Math. Phys. 224, No. 1, 153-204 (2001). ZBL1038.82035.

The DOI link seems to be broken right now so you can also try this or this.

Theorems 3.2 states that with probability going to 1 as $n\rightarrow\infty$, the largest open cluster in an $n\times n$ box is of size $\theta(p)n^2$, which is the first part of what James Martin states (except that $O(n)$ was a typo for $O(n^2)$). (More precisely, this is proved under a set of postulates which are then proven to hold for $\mathbb{Z}^2$ in theorem 3.6.)

Thus in this reference, the answer to question 3 is that "large component" simply means "largest component".

The notation is a bit dense but note that $W^{(i)}_{\Lambda_n}$ is the number of vertices in the $i$th largest open cluster for percolation in the box $\Lambda_n=\{v\in\mathbb{Z}^d|-n\leq v_i<n,i=1,\dots,d\}$, $P_\infty(p)$ is the same as what you call $\theta(p)$, and $s(n)$ is defined in equation 2.17, and $L_0(p)$ is defined in equation 2.20.

I couldn't immediately find a reference for the statement that the next largest components are of logarithmic size, but Theorem 3.1(iii) in the paper cited above at least proves that the expectation value of the second largest cluster is sublinear in the total number of sites.

• Thank you! Could you please also address part (4) of my question? – user123818 May 2 '18 at 19:49
• Also, how do you define $\theta(p)$ for a finite box? – user123818 May 2 '18 at 19:59
• @Blue $\theta(p)$ is not defined for a finite box. It is defined as usual, as the probability that the origin in $\mathbb{Z}^d$ is part of an infinite cluster. It just turns out that the asymptotic size of the largest cluster in finite boxes can be expressed simply in terms of $\theta(p)$ (and this should be at least heuristically clear at least if you think of $\theta(p)$ as the density of the infinite open cluster in $\mathbb{Z}^d$ -- the size of the largest open cluster in a large box should be that density multiplied by the size of the box). – j.c. May 2 '18 at 20:08
• @Blue Regarding part (4), certainly there should be some analogue of these results for site percolation but I don't know of any place where this is explicitly stated or proved. I don't think that the main ideas of the paper I cite depend much on the distinction between bond vs site percolation (though I could be wrong as I haven't gone through the proofs in this paper). – j.c. May 2 '18 at 20:17
• Another question. Theorem 3.2 states that $$\lim_{n\to\infty}\frac{W^{(I)}_{\Lambda_n}}{|\Lambda_n|P_\infty(p_n)} = 1$$. What does $|\Lambda_n|$ stand for? What does the $|.|$ symbol mean here? – user123818 May 3 '18 at 9:54