Renormalised four - point coupling constant in the three - dimensional O ( N ) model with N → 0

We simulate self-avoiding walks on a cubic lattice and determine the second virial coefficient for walks of different lengths. This allows us to determine the critical value of the renormalized four-point coupling constant in the three-dimensional N -vector universality class for N = 0. We obtain ḡ = 1.4005(5), where ḡ is normalized so that the three-dimensional field-theoretical β-function behaves as β(ḡ) = −ḡ + ḡ2 for small ḡ. As a byproduct, we also obtain precise estimates of the interpenetration ratio Ψ, Ψ = 0.24685(11), and of the exponent ν, ν = 0.5876(2).