proof using coupling that the site percolation probability function is monotone (thing)

This writeup talks about percolation, so it might help to skip through that node first. And it proves something that critical percolation alludes to, so you might want to read that node after.

Consider the setting of percolation on a connected infinite transitive graph G=(V,E). The real function θ(p):[0,1]→[0,1] is the probability for any given vertex of the graph to be a member of an infinite connected open cluster, when edges are open with probability p. (The same argument works in the dual case, for edge percolation, when vertices are open with probability p.)

The claim we wish to prove is that if p≤q, then θ(p)≤θ(q) -- increasing the probability that an edge is open will not decrease the probability of finding an infinite cluster. This is intuitively obvious: it feels like we only open more edges at q than we do at p, so naturally clusters can only grow.

Unfortunately the proof is a bit more involved. The problem is that the probability space Ω_p for percolation on G with edges open with probability p is a different probability space than Ω_q, the one for percolation on G with edges open with probability q. It is not clear how to transfer events from one probability space to another.

Coupling is a technique that can help. We will find a probability space Ω with two probability-preserving transformations

Φ_p: Ω → Ω_p
Φ_q: Ω → Ω_q

(i.e., preimages under the transformation have the same probability: if P is probability on Ω, then for any event A⊆Ω_p, P(Φ_p^-1(A)) = P_p(A)). This will connect Ω_p with Ω_q.

We will show that the event "v∈V has an infinite open connected cluster" in Ω_p:

C_p={ω∈Ω_p: the connected open component of v in ω is infinite}

has smaller probability that the corresponding event C_q in Ω_q, by realising that

Φ_p^-1(C_p) ⊆ Φ_q^-1(C_q).

The larger coupling space Ω will be the space of IID random variables {x_e}_e∈E that are uniformly distributed on [0,1]. A mapping Ω→Ω_p must give a 0/1 value to each edge; we define

Φ_p(e) = 0 [e is closed], if x_e > p;
Φ_p(e) = 1 [e is open], if x_e ≤ p;

In other words: We assign a random number 0≤x_e≤1 to each edge e. We transform this assignment into percolation with probability p by considering an edge open in Ω_p if the random value assigned to in Ω is between 0 and p.

Obviously this describes percolation on Ω_p. Equally obvious, this is a measure-preserving transformation. By the coupling argument presented above, the intuitively obvious -- that the percolation probability increases with the edge probability -- is proved.