**Influences and the random cluster model**

**1. Different models **

** 1.1. Percolation **

Let denote the set of edges of the square planar grid. Study randomly chosen configurations . Denote by

and

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

Compatible means…

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 .

** 2.1. Estimates **

**Holley’s inequality**.

Let , , be positive measures on . If

then for every event ,

**FKG inequality**.

If

then for any events and ,

This shows that if and , .

For events , set

Then

** 2.2. Results **

For a monotonic measure , define conditional influences

Then

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.

** 2.3. Application **

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.