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 $\lambda = {1 \over 18} \left[\psi^{ \prime}( 1/6)+\psi^{ \prime}( 1/3) \right]-{4\pi^ 2 \over 27}.$