for percolation in the 2D high temperature Ising Model

(1)

Scaling relations

for percolation in the 2D high temperature Ising Model

Yasunari Higuchi (Kobe University)

Masato Takei

(Osaka Electro-Communication University)

This talk is based on joint work with Yu Zhang (University of Colorado).

We consider the percolation problem for Ising model on the two-dimensional square lattice Z². For T > T_c and h ∈ R, there exists a unique Gibbs measure µ_T,h. The (+)-cluster containing the origin is denoted by C⁺₀. For each T > 0, the critical external ﬁeld is deﬁned by

h_c(T) := inf{h:µ_T,h(#C⁺₀ =∞)>0}.

It is known that h_c(T) > 0 whenever T > T_c. Hereafter we ﬁx a T > T_c, and abbreviate µ_T,h toµ_h and h_c(T) toh_c, respectively. The expectation under µ_h is denoted by E_h.

The following power laws are widely believed to hold:

I Percolation probability θ(h) :=µ_h(#C⁺₀ =∞)≈(h−h_c)^β ash&h_c. I Mean cluster size:

χ(h) :=E_h[#C⁺₀ : #C⁺₀ <∞]≈ |h−h_c|⁻^γ as h→h_c. I Correlation length:

ξ(h) :=

[ 1 χ(h)

∑

v∈Z²

|v|²µh

(O↔⁺ v, #C⁺₀ <∞)]1/2

≈ |h−hc|⁻^ν as h→hc.

＊ForS(n) = [−n, n]², we deﬁne

L(h, ε₀) :=

{min{

n :µ_h(

LS ofS(n)↔⁺ RS of S(n))

≥1−ε₀}

(h > h_c), min{

n :µ_h(

LS ofS(n)↔⁺ RS of S(n))

≤ε₀}

(h < h_c).

Then ξ(h)³L(h, ε₀).

I One-arm probability: π_h_c(n) :=µ_h_c(

O↔⁺ ∂S(n))

≈n⁻^1/δ^r. I Connectivity function: τ_h_c(n) := µ_h_c{O↔⁺ (n,0)} ≈n⁻^η.

We derive some scaling relations, provided the exponents exist; for 2D Bernoulli percolation, these relations are proved by Kesten.