Direction-direction correlations of oriented polymers

Abstract

We calculate the direction-direction correlations between the tangent vectors of an oriented self-avoiding walk (SAW). LetJ ^μ(x) andJ ^v(0) be components of unit-length tangent vectors of an oriented SAW, at the spatial pointsx and 0, respectively. Then for distances |x| much less than the average distance between the endpoints of the walk, the correlation function ofJ ^μ(x) withJ ^v(0) has, ind dimensions, the form$\left\langle {J^\mu (x){\text{ }}J^\nu (0)} \right\rangle = k(d)(x^\mu x^\nu - \tfrac{1}{2}x^2 \delta ^{\mu \nu } )/\left| x \right|^{2d}$. The dimensionless amplitudek(d) is universal, and can be calculated exactly in two dimensions by using Coulomb gas techniques, where it is found to bek(2)=12/25π ². In three dimensions, the ε-expansion to second order in ε together with the exact value ofk(2)in two dimensions allows the estimatek(3)=0.0178±0.0005. In dimensionsd⩾4, the universal amplitudek(d) of the direction-direction correlation functions of an oriented SAW is the same as the universal amplitude of the direction-direction correlation functions of an oriented random walk, and is given byk(d)=Γ ²(d/2)/(d−2)π ^d.