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_NR²_N^d/2μ^−N (where p_N is the number of configurations of an oriented and rooted N-step ring, R²_N a typical average size squared, and μ the SAW effective connectivity constant of the lattice) is equal for N å ∞ to a lattice-dependent constant times a universal amplitude A(d). The latter amplitude is calculated directly from the minimal continous Edwards model to second order in 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 λ appearing to second order in in all critical phenomena amplitude ratios is equal to .