Mean Square End-to-End Distance of a Semiflexible Polymer on the Bethe Lattice

A combinatorial method to calculate the mean square end-to-end distance 〈R2〉 of a polymer on a Bethe lattice is used. The case of an anisotropic lattice and semiflexible polymers is considered. The distance on the Cayley tree is defined by embedding the tree on an N-dimensional Euclidean space considering that every bend of the polymer defines a direction orthogonal to all the previous ones. The semiflexible polymer is effectively equivalent to a flexible one if one considers an effective (noninteger) coordination number. Although an analytical calculation is performed, a closed expression for 〈R2〉 is possible only for the isotropic case. Numerical results are shown for the anisotropic case. Plots of 〈R2〉 against N for different values of the anisotropy parameter y are shown. The power dependence for N does not depend on the anisotropy as expected, but the linear coefficient increases on increasing the anisotropy. The anisotropy tends to stretch the polymer.