## Notes of Ben Graham’s talk

Influences and the random cluster model

1. Different models

1.1. Percolation

Let ${E}$ denote the set of edges of the square planar grid. Study randomly chosen configurations ${\omega:E\rightarrow\{ 0,1 \}}$. Denote by

$\displaystyle \begin{array}{rcl} \omega^e (f)=1 \textrm{ if }e=f, \quad =\omega(f) \textrm{ otherwise}, \end{array}$

and

$\displaystyle \begin{array}{rcl} \omega_e (f)=0 \textrm{ if }e=f, \quad =\omega(f) \textrm{ otherwise}, \end{array}$

the configurations obtained from ${\omega}$ by closing or opening one particular edge ${e}$.

The influence of edge ${e}$ on event ${A}$ is

$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}(\omega^e \in A \textrm{ and }\omega_e \notin A). \end{array}$

For every increasing event ${A}$, ${p\mapsto \mathop{\mathbb P}_p (A)}$ is increasing.

Kesten, (shorter proof by Bollobas and Riordan) showed that ${\mathop{\mathbb P}_p (0 \textrm{ is connected to infinity})}$ vanishes for ${p\leq p_c =\frac{1}{2}}$, is positive otherwise.

1.2. Voronoi percolation

Start with a Poisson process on ${{\mathbb R}^2}$, construct Voronoi cells and study site percolation on the resulting dual graph. Again, almost surely, ${p_c =\frac{1}{2}}$.

1.3. Site percolation

Back to the square plane lattice. Draw diagonals, get ${*}$-lattice. Then

$\displaystyle \begin{array}{rcl} p_c^{\mathrm{site}}+p_{c}^{*,\mathrm{site}}=1. \end{array}$

1.4. Ising model

Let ${V\subset{\mathbb Z}^2}$ be a finite subset. Spin configurations are maps ${\sigma:V\rightarrow\{\pm 1\}}$. The energy of a configuration is

$\displaystyle \begin{array}{rcl} H(\sigma)=-\frac{1}{2}\sum_{xy\in E}\sigma(x)\sigma(y) -\frac{h}{2}\sum_{x\in V}\sigma(x) \end{array}$

Use Gibbs measure

$\displaystyle \begin{array}{rcl} \phi_{\beta}(\sigma)=\frac{e^{-\beta H(\sigma)}}{Z} \end{array}$

where ${Z}$ is a normalizing factor, ${\beta}$ is the inverse temperature.

${\beta}$ small implies no long range order, correlation ${\phi_{\beta}(\sigma(0)\sigma(x))}$ vanishes when ${x}$ tends to infinity.

Onsager 1944 : ${\beta_{c}=\log(1+\sqrt{2})}$. ${\frac{1}{\beta_c}}$ is called Curie temperature.

Ising model is related to percolation as follows. Let ${\beta<\beta_c}$. There exists ${h_c}$ such that if ${h>h_c}$, ${+}$ spins percolate. There exists ${h^*_c}$ such that if ${h>h^*_c}$, ${+}$ spins ${*}$-percolate, and

$\displaystyle \begin{array}{rcl} h_c +h^*_c =0. \end{array}$

1.5. Potts model

Ising model is a special case of Potts model. There, spins are chosen in ${[q]}$, ${q=2,3,\ldots}$.

$\displaystyle \begin{array}{rcl} H(\sigma)&=&-\sum_{xy\in E}1_{\{\sigma(x)=\sigma(y)\}} +const.\\ &=&\sum_{xy\in E}1_{\{\sigma(x)\not=\sigma(y)\}} \end{array}$

with probability distribution

$\displaystyle \begin{array}{rcl} \phi_{\beta,q}=\frac{1}{Z_{\beta,q}}e^{-\beta\sum_{xy\in E}1_{\{\sigma(x)\not=\sigma(y)\}}}. \end{array}$

The order parameter is

$\displaystyle \begin{array}{rcl} \lim_{x\rightarrow \infty}\phi_{\beta}(\sigma(0)=\sigma(x))-\frac{1}{q}. \end{array}$

It vanishes for small ${\beta}$, ${\beta<\beta_c}$ conjectured to be ${\log(1+\sqrt{q})}$ (known if ${q=2}$ and for ${q\geq 26}$.

1.6. Random cluster model

Let ${V\subset{\mathbb Z}^2}$ be a finite subset. Configurations are maps ${\sigma:E\rightarrow\{ 0,1 \}}$.

$\displaystyle \begin{array}{rcl} \mu_{p,q}(\{\omega\})=\frac{q^{k(\omega)}}{Z_{\beta,q}}\prod_{e\in E}p^{\omega(e)}(1-p)^{-\omega(e)}, \end{array}$

where ${k(\omega)}$ is the number of clusters (connected components of union of open edges). It looks like bond percolation, without independence.

The probability distribution is

$\displaystyle \begin{array}{rcl} \nu(\{\sigma,\omega\}=\frac{1}{Z_{\beta,q}}1_{\sigma,\,\omega \,\mathrm{ compatible}}\prod_{e\in E}p^{\omega(e)}(1-p)^{-\omega(e)}. \end{array}$

Compatible means…

When ${1-p=e^{-\beta}}$, this is related to Potts model:

$\displaystyle \begin{array}{rcl} \sum_{\omega}\nu(\sigma,\omega)=\phi_{\beta,q}(\sigma). \end{array}$

There is a ${1-1}$ correspondance. Percolation in the random cluster model exactly corresponds to long range order in the Potts model.

It is conjectured that ${SLE(\kappa)}$ relates to the random cluster model with

$\displaystyle \begin{array}{rcl} \cos(\frac{4\pi}{\kappa})=-\frac{1}{2}\sqrt{q}, \quad \kappa\in(4,8),\quad q\in(0,4). \end{array}$

2. Analysis of the random cluster model

Use two coins, with respective ${\frac{1}{3}}$ and ${\frac{2}{3}}$ bias. Pick a coin uniformly at random and toss it ${n}$ times. The distribution of the sum of results has two peaks at ${\frac{n}{3}}$ and ${\frac{2n}{3}}$.

Let ${A=}$Majority. Then influences decrease exponentially with ${n}$.

2.1. Estimates

Holley’s inequality.

Let ${\mu}$, ${\mu_1}$, ${\mu_2}$ be positive measures on ${\{ 0,1 \}^n}$. If

$\displaystyle \begin{array}{rcl} \mu_1 (\omega_1 \wedge \omega_2) \mu_1 (\omega_1 \vee \omega_2) \geq \mu_1 (\omega_1) \mu_2 (\omega_2) , \end{array}$

then for every event ${A}$,

$\displaystyle \begin{array}{rcl} \mu_1 (A)\leq\mu_2 (A). \end{array}$

FKG inequality.

If

$\displaystyle \begin{array}{rcl} \mu(\omega_1 \wedge \omega_2) \mu(\omega_1 \vee \omega_2) \geq \mu (\omega_1) \mu(\omega_2) , \end{array}$

then for any events ${A}$ and ${B}$,

$\displaystyle \begin{array}{rcl} \mu(A\cap B)\geq \mu(A)\mu(B). \end{array}$

This shows that if ${p_1 \leq p_2}$ and ${q\geq 1}$, ${\mu_{p_1,q}\leq\mu_{p_2,q}}$.

For events ${F\subset E}$, set

$\displaystyle \begin{array}{rcl} \mu_F^{\xi}=\mu_{p,q}(\cdot|\omega(e)=\xi(e) \textrm{ for all }e\in E\setminus F). \end{array}$

Then

$\displaystyle \begin{array}{rcl} \xi\leq\zeta\quad \Rightarrow\quad\mu_F^{\xi}\leq \mu_F^{\zeta}. \end{array}$

2.2. Results

For a monotonic measure ${\mu}$, define conditional influences

$\displaystyle \begin{array}{rcl} J_A (e)=\mu(A|\omega(e)=1)-\mu(A|\omega(e)=0). \end{array}$

Then

$\displaystyle \begin{array}{rcl} \frac{d}{dp}\mu_p (A)=\sum_{e}J_A (e). \end{array}$

We have both KKL and Talagrand type results.

Theorem 1 (Graham, Grimmet) For a monotonic measure ${\mu}$,

$\displaystyle \begin{array}{rcl} \max_{e}J_A (e)\geq C\,\mu_p (A)(1-\mu_p (A))\frac{\log N}{N} \end{array}$

$\displaystyle \begin{array}{rcl} \sum_{e}J_A (e)\geq C\,\mu_p (A)(1-\mu_p (A))\log(\frac{1}{2\max_{e}J_A (e)}). \end{array}$

In a standard manner, this implies a sharp threshold result

$\displaystyle \begin{array}{rcl} p_2 -p_1 \leq C\frac{\log(1/\epsilon)}{\log N} \end{array}$

if ${A}$ is symmetric, ${\mu_{p_1}(A)=\epsilon}$, ${\mu_{p_2}(A)=1-\epsilon}$.

Proof: One compares conditional influences to traditional influences and applies KKL. $\Box$

2.3. Application

Let ${R}$ be a planar graph and ${R'}$ its dual graph. The dual of the square lattice is again a square lattice. A ${(n+1)\times n}$-rectangle ${R}$ is dual to a ${n\times(n+1)}$-rectangle ${R'}$. In the random cluster model, a configuration on a lattice induces a configuration on the dual lattice. Therefore, if ${H}$ (resp. ${V}$) denote the events of a horizontal (resp. vertical) crossing in ${R}$ (resp. ${R'}$),

$\displaystyle \begin{array}{rcl} \mu_{p,q}(H(R))+\mu_{p_d ,q}(V(R'))=1. \end{array}$

This implies that

$\displaystyle \begin{array}{rcl} \frac{p}{1-p}\frac{p_d}{1-p_d}=q. \end{array}$

The probability that a given bond be pivotal tends to ${0}$, implying a sharp threshold, and

$\displaystyle \begin{array}{rcl} p_c (q)=\frac{\sqrt{q}}{1+\sqrt{q}}. \end{array}$

This is due to Vincent Beffara and Hugo Duminil-Copin.