Easy Exercise - Robustness of percolation clusters

We are going to present some nice results which are very easy to show.

Consider Bernoulli bond percolation on ${\mathbb{Z}}^d$, $d\ge 2$. The phase transition is non-trivial with some critical probability $p_c\in (0,1)$. Take a percolation process $({\omega }_e{)}_{e\in E({\mathbb{Z}}^d)}$, where $P[e\text{ is open}]=P[{\omega }_e=1]=p$. If $p>p_c$, there exists a unique infinite connected component (cluster), which we denote by $C_{\infty }$. We ask: how robust is this $C_{\infty }$. One way to think about this is, if we randomly remove some edges, does it still contain an infinite connected subgraph?

One natural way to interpret this is the following. we resample each edge with probability $q$, meaning that for each $e$ with ${\omega }_e=1$, we associate another process ${\sigma }_e$ with $P[{\sigma }_e=1]=q$ and don’t do anything to closed edges (one can assume ${\sigma }_e=0$ for every $e$ with ${\omega }_e=0$). We ask, does $\left\{e\in C_{\infty }:{\omega }_e={\sigma }_e=1\right\}$ percolate? It is not hard to notice that this is equivalent to percolation with the parameter $pq$. So the infinite cluster survives if $q>\frac{p_c}{p}$.

Now, we look at a slightly different problem. Let $({\omega }_e^1{)}_{e\in E({\mathbb{Z}}^d)}$ and $({\omega }_e^2{)}_{e\in E({\mathbb{Z}}^d)}$ denote another two independent percolation processes with parameter $p,q>p_c$, respectively. We denote the respective infinite clusters by $C_{\infty }^1$ and $C_{\infty }^2$. It is easy to see (by Borel-Cantelli, for example) that they intersect at infinitely many points. But does their intersection $C_{\infty }^1\cap C_{\infty }^2$ still contain an infinite cluster? We were considering $C_{\infty }^1\cap \left\{{\omega }_e^2=1\right\}$ and now we are considering a smaller set. But globally, they should be quite similar. We conjecture that for $pq>p_c$, the intersection still percolates almost surely, while the opposite is true for $pq<p_c$.

Suppose $pq>p_c$. Then both $p$ and $q$ are greater than $p_c$ and the two infinite clusters exist. Now, we consider a third percolation configuration, $\hat{\omega }$, where ${\hat{\omega }}_e=1$ if and only if ${\omega }_e^1={\omega }_e^2=1$. Same as before, we see that this is in fact equivalent to percolation with parameter $pq$. Since $\{{\hat{\omega }}_e=1\}\subseteq \left\{{\omega }_e^1=1\right\}$, we know that $\widehat{C_{\infty }}\subseteq C_{\infty }^1$, and the same is true for ${\omega }^2$. So $\widehat{C_{\infty }}\subseteq C_{\infty }^1\cap C_{\infty }^2$.

Now suppose $pq<p_c$ (note that we still have $pq>p_c^2$). There are no infinite clusters in $\left\{\hat{\omega }=1\right\}$, which contains $C_{\infty }^1\cap C_{\infty }^2$, so this contains no infinite connected component.

It is straightforward to generalise the above to resampling multiple times and intersections of multiple infinite clusters.

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Let $({\phi }_x{)}_{x\in {\mathbb{Z}}^d}$ denote the discrete Gaussian free field on ${\mathbb{Z}}^d$ for $d\ge 3$. The excursion set $\left\{\phi \ge h\right\}$ contains an infinite cluster for every $h<h_{\ast }$ for some $h_{\ast }\in (-\infty ,\infty )$. We continue to denote the infinite cluster by $C_{\infty }$. We know that random walk on $C_{\infty }$ is transient for almost every $\phi$, hence we can yet still define a GFF on $C_{\infty }$, which we denote by $\psi$. What can we say about $\left\{\psi \ge h\right\}$? Is there ‘physical significance’ for this?