Hyperscaling for polymer rings

The statistics of a long closed self-avoiding walk (SAW) or polymer ring on a $ d $-dimensional lattice obeys hyperscaling. The combination $ p_N \left\langle R^2 \right\rangle^{ d/2}_N\mu^{ -N}, $ (where $ p_N $ is the number of configurations of an oriented and rooted $ N $-step ring, $ \left\langle R^2 \right\rangle_ N $ a typical average size squared, and $ \mu $ the SAW effective connectivity constant of the lattice) is equal for $ N \longrightarrow \infty $ to a lattice-dependent constant times a universal amplitude $ A(d). $ The latter amplitude is calculated directly from the minimal continuous Edwards model to second order in $ \varepsilon \equiv 4-d. $ The case of rings at the upper critical dimension $ d=4 $ is also studied. The results are checked against field theoretical calculations, and former simulations. As a consequence, we show that the universal constant $ \lambda $ appearing to second order in $ \varepsilon $ in all critical phenomena amplitude ratios is equal to $ \lambda = {1 \over 18} \left[\psi^{ \prime}( 1/6)+\psi^{ \prime}( 1/3) \right]-{4\pi^ 2 \over 27}. $