Hyperscaling for polymer rings

Abstract

The statistics of a long closed self-avoiding walk (SAW) or polymer ring on a $d$-dimensional lattice obeys hyperscaling. The combination $p_N{⟨R^2⟩}_N^{d/2}{\mu }^{-N},$ (where $p_N$ is the number of configurations of an oriented and rooted $N$-step ring, ${⟨R^2⟩}_N$ a typical average size squared, and $\mu$ the SAW effective connectivity constant of the lattice) is equal for $N⟶\infty$ 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 $\epsilon \equiv 4-d.$ 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 $\epsilon$ in all critical phenomena amplitude ratios is equal to $\lambda =\frac{1}{18}\left[{\psi }^{\prime }(1/6)+{\psi }^{\prime }(1/3)\right]-\frac{4{\pi }^2}{27}.$

References in corpus (4)

• Addendum-erratum to: ``Nonasymptotic critical behavior from field theory at d=3. II. The ordered-phase case. Phys. Rev. B35, 3585 (1987)
• Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails
• Static Scaling Behavior of High-Molecular-Weight Polymers in Dilute Solution: A Reexamination
• Renormalization Theory for Interacting Crumpled Manifolds

Cited by in corpus (4)

• Self-consistent equation for an interacting Bose gas
• Ring polymers in crowded environment: conformational properties
• Swelling of asymmetric pom-pom polymers in dilute solutions
• Some approximate analytical methods in the study of the self-avoiding loop model with variable bending rigidity and the critical behaviour of the strong coupling lattice Schwinger model with Wilson fermions