Influences and the random cluster model
1. Different models
Let denote the set of edges of the square planar grid. Study randomly chosen configurations . Denote by
the configurations obtained from by closing or opening one particular edge .
The influence of edge on event is
For every increasing event , is increasing.
Kesten, (shorter proof by Bollobas and Riordan) showed that vanishes for , is positive otherwise.
1.2. Voronoi percolation
Start with a Poisson process on , construct Voronoi cells and study site percolation on the resulting dual graph. Again, almost surely, .
1.3. Site percolation
Back to the square plane lattice. Draw diagonals, get -lattice. Then
1.4. Ising model
Let be a finite subset. Spin configurations are maps . The energy of a configuration is
Use Gibbs measure
where is a normalizing factor, is the inverse temperature.
small implies no long range order, correlation vanishes when tends to infinity.
Onsager 1944 : . is called Curie temperature.
Ising model is related to percolation as follows. Let . There exists such that if , spins percolate. There exists such that if , spins -percolate, and
1.5. Potts model
Ising model is a special case of Potts model. There, spins are chosen in , .
with probability distribution
The order parameter is
It vanishes for small , conjectured to be (known if and for .
1.6. Random cluster model
Let be a finite subset. Configurations are maps .
where is the number of clusters (connected components of union of open edges). It looks like bond percolation, without independence.
The probability distribution is
When , this is related to Potts model:
There is a correspondance. Percolation in the random cluster model exactly corresponds to long range order in the Potts model.
It is conjectured that relates to the random cluster model with
2. Analysis of the random cluster model
Use two coins, with respective and bias. Pick a coin uniformly at random and toss it times. The distribution of the sum of results has two peaks at and .
Let Majority. Then influences decrease exponentially with .
Let , , be positive measures on . If
then for every event ,
then for any events and ,
This shows that if and , .
For events , set
For a monotonic measure , define conditional influences
We have both KKL and Talagrand type results.
Theorem 1 (Graham, Grimmet) For a monotonic measure ,
In a standard manner, this implies a sharp threshold result
if is symmetric, , .
Proof: One compares conditional influences to traditional influences and applies KKL.
Let be a planar graph and its dual graph. The dual of the square lattice is again a square lattice. A -rectangle is dual to a -rectangle . In the random cluster model, a configuration on a lattice induces a configuration on the dual lattice. Therefore, if (resp. ) denote the events of a horizontal (resp. vertical) crossing in (resp. ),
This implies that
The probability that a given bond be pivotal tends to , implying a sharp threshold, and
This is due to Vincent Beffara and Hugo Duminil-Copin.