The critical two-point function of lattice φ⁴₂

Abstract

In this article, we show that the critical two-point correlation function of the lattice φ⁴₂ model in the strong coupling regime follows that of the Ising universality class.

Main theorem

Consider real spins on the square lattice with nearest-neighbor ferromagnetic interaction and single-site weight proportional to exp(−g(φ²−1)²). Write G_φ,β(x) for the infinite-volume free two-point function, β_c(g) for the critical inverse temperature, and G_I(x) for the critical nearest-neighbor Ising two-point function on the square lattice.

There exist g₀ < ∞ and ω > 0 such that for every fixed g ≥ g₀ there is Z_φ(g) > 0 with Z_φ(g) = 1 + O(g⁻¹) and, uniformly as |x| → ∞,

G_φ,βc(g)(x) = Z_φ(g)² G_I(x) + O_g(|x|^−1/4−ω).

In particular, the critical free two-point function is asymptotic to C_g|x|^−1/4 for some C_g > 0.