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 $p_N \left\langle R^2 \right\rangle^{ d/2}_Nμ^{ -N},$ (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 $μ$ 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 $λ$ appearing to second order in $\varepsilon$ in all critical phenomena amplitude ratios is equal to $λ= {1 \over 18} \left[ψ^{ \prime}( 1/6)+ψ^{ \prime}( 1/3) \right]-{4π^ 2 \over 27}.$

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