(Probability theory and statistical mechanics:)

See percolation for brief initiation into the subject, as well as more details on some of the notation. Consider percolation on some infinite (countable) connected graph G=(V,E) (since we'll be looking for an infinite connected cluster, it makes sense to start from an infinite connected graph). Additionally, we require that the graph G be a (vertex-) transitive graph (although many of the results can be obtained with somewhat weaker conditions on G).

In this writeup, we'll consider all events to be based on the parameter 0≤p≤1, and accordingly add it in a subscript to our probability measure Pp(⋅). We shall also keep the notation C = {there exists an infinite connected open cluster} of percolation and Cv = {there exists an infinite connected open cluster containing v} (for any v∈V). This is of course the simplest possible scenario, and we study the simplest possible parameter (pc, below). However, that shall already be enough to take us to some of the edges of what is known today. The full study of percolation, and even of critical percolation, is of course considerably richer that what I present here.

Let G be, in addition, a transitive graph. Define the site percolation probability to be the probability for a site v to be contained in an infinite connected open cluster:

θ(p)=Pp(Cv).
What does the function θ:[0,1]→[0,1] look like?

  1. θ(p)>0 iff Pp(C)=1.

    Note that by Kolmogorov's 0-1 law Pp(C) is either 0 or 1, so also Pp(C)=0 iff θ(p)=0.

    There are 2 cases to prove:

    1. Suppose θ(p)=0. Clearly
      C = v∈V Cv
      (there exists some infinite connected open cluster iff there is some vertex v∈V which is in an infinite connected open cluster), so by summing probabilities we see that
      0 ≤ Pp(C) = Pp(v∈V Cv) ≤ v∈V Pp(Cv) = v∈V θ(p) = 0.
      and Pp(C)=0 as required.
    2. Otherwise, θ(p)>0. Take some vertex u∈V. Then
      Pp(C) ≥ Pp(Cu) = θ(p) > 0.
      In particular, Pp(C) ≠ 0. But by Kolmogorov's 0-1 law, Pp(C) is either 0 or 1; since it cannot be 0, we must have Pp(C) = 1.
  2. θ(p) is (weakly) monotone increasing.

    This means that if p<q then θ(p)≤θ(q): as the probability of an edge remaining open increases, the probability of v being included in some infinite connected open cluster does not decrease. This is "obviously" true, yet requires some trick to prove: The problem here is that Pp describes one model of probability (each edge of G is kept open with probability p), while Pq describes another model. It's not clear how to compare the two. Here's a proof using coupling that the site percolation probability function is monotone.

  3. A phase transition.

    Note that, if G is infinite and connected, θ(0)=0 (when all edges are closed, there is no infinite open cluster) and θ(1)=1 (when all edges are open, G is the infinite open cluster containing any vertex v). Define the critical probability for percolation on G

    pc = inf { p: θ(p)>0 } = sup { p: θ(p)=0 }.
    (Both formulae give the same number, due to "B" above).

    By definition, if p < pc then θ(p)=0 and (by "A" above) Pp(C)=0, and if p > pc then θ(p)>0 and Pp(C)=1. pc is the phase transition between almost always not having an infinite connected open cluster and almost surely having one.

  4. The critical point.

    Thus far I've been silent about what happens at p=pc. If θ(pc)=0, then Ppc(C)=0 -- there is no percolation at the critical probability. Otherwise θ(pc)>0, and Ppc(C)=1 -- percolation occurs at the critical point. Which case occurs might be a property of G. However, note that if Pc=1 then we must have that θ(pc)=1. In particular, if G=Z is the "line" graph, then pc=1 (because for all p<1 there are (almost always) no infinite connected open clusters, and indeed the expected size of the connected open cluster of a vertex is 2/(1-1/p)-1<∞, so the probability of an infinite connected open cluster has to be 0...) -- so critical percolation occurs on G=Z, but is wholly uninteresting. The same occurs for any similarly "one-dimensional" graph, such as the "ladder graph" G=Z×{0,1}.

    In fact, all known cases of (transitive, connected) G with pc<1 have θ(pc)=0. This is conjectured to be true:

    (*) Conjecture: For any transitive, connected infinite graph G with pc<1, there is no critical percolation: θ(pc)=0.

    For G=Z2, Harris and Kesten showed that pc=1/2 (each showed a different half, separated by 20 years), and Kesten showed that indeed no percolation occurs at this value. The proof is heavily based on uniquely two-dimensional properties, such as dual graphs, and probably cannot be generalised. For G=Zd, where d≥19, Hara and Slade showed that no percolation occurs at pc (whose value is distinctly lower). Their technique is said to work for lower dimensions d≥6 when many edges are added to Zd, connecting nodes distant in the original lattice. Unfortunately, it is also known that this technique won't work for d<6. This leaves the cases d=3,4,5 open -- and these include precisely the cases of greatest physical interest!

    What about other graphs? Benjamini, Lyons, Peres and Schramm's paper Critical Percolation on Any Nonamenable Group Has No Infinite Clusters (1999) showed (despite the title) that the conjecture (*) above is true if G is a nonamenable graph which is also unimodular.

    On my M.Sc. thesis, I showed that (*) is true for two examples of nonamenable graphs G which are not unimodular: (*) holds for the grandparented tree (and, more generally, for any decorated tree, unimodular or not) and for Diestel and Leader's kinship graph whenever it is nonunimodular. It is not clear whether the proofs are powerful enough to be generalised to all nonunimodular graphs.

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