Self-avoiding-walk contacts and random-walk self-intersections in variable dimensionality

Abstract

The average number of nearest-neighbor (NN) contacts 〈m〉 of self-avoiding walks (SAW's) on a hypercubic lattice is calculated using direct enumeration and 1/d expansion methods, where d is the spatial dimension. These calculations are compared with exact analytic determinations for the asymptotic number of random-walk (RW) self-intersections in the limit of long chains n\ensuremath{\rightarrow}\ensuremath{\infty}. The number of RW (binary, ternary, etc.) self-intersections is a function of the probability ${\mathit{C}}_{\mathit{d}}^{\mathrm{*}}$ that a RW escapes from the origin to infinity and an accurate tabulation of ${\mathit{C}}_{\mathit{d}}^{\mathrm{*}}$ is given in the dimension range 2d10. We find that the number of SAW NN contacts 〈m${\mathrm{〉}}_{\mathrm{SAW}}$ has an asymptotic behavior (〈m${\mathrm{〉}}_{\mathrm{SAW}}$\ensuremath{\sim}${\mathit{a}}_{\mathrm{\ensuremath{\infty}}}$n) similar to that for the number of RW self-intersections (〈m${\mathrm{〉}}_{\mathrm{RW}}$\ensuremath{\sim}${\mathit{a}}_{\mathrm{RW}}$n), as first suggested by Domb, but the corrections to this leading scaling differ for the RW and SAW problems. The ``contact amplitude'' ${\mathit{a}}_{\mathrm{\ensuremath{\infty}}}$, determined from direct enumeration data and ratio extrapolation, exhibits a maximum near d=3 dimensions as does ${\mathit{a}}_{\mathrm{RW}}$. Comparison of the numerical estimates for ${\mathit{a}}_{\mathrm{\ensuremath{\infty}}}$ to the 1/d expansion calculation of ${\mathit{a}}_{\mathrm{\ensuremath{\infty}}}$ shows significant deviation for d5, reflecting the strong fluctuations in contacts that arise in lower dimensions. The correction to the scaling exponent ${\mathrm{\ensuremath{\Delta}}}_{\mathit{m}}$ for SAW NN contacts exhibits a maximum near d=2 dimensions, a behavior similar to previous observations for the SAW exponent \ensuremath{\gamma}. Estimates of the \ensuremath{\theta} point for interacting SAW's, the critical temperature of the O(m) model, and other lattice constants (e.g., bond and site percolation thresholds) are obtained in terms of SAW and RW lattice parameters.