Correlation function of the fractional spread-out Ising models at their upper critical dimension and critical temperature

Abstract

We give a renormalisation-group argument for the critical free two-point function of the spread-out Ising model with kinetic operator (−Δ_R)^d/4 in dimensions 1 ≤ d ≤ 4. For sufficiently large fixed spread radius R, the critical correlation is asymptotic to a positive constant times |x|^−d/2.

Main theorem

For each d ∈ {1, 2, 3, 4} and all sufficiently large fixed spread radii R, consider the Ising model with kinetic operator (−Δ_R)^d/4, where Δ_R is the spread-out Laplacian. At the critical inverse temperature τ_c(d, R), the infinite-volume free two-point function satisfies

⟨σ₀σ_x⟩^f_τc ∼ C_d,R|x|^−d/2, with C_d,R > 0, as |x| → ∞.

In d = 4, the kinetic operator is the spread-out Laplacian and the decay exponent is 2.