Self-avoiding walks on two-dimensional square and triangular lattices are considered in order to test numerically theoretical conjectures for the leading correction-to-scaling exponent. \(N\)-step self-avoiding walk (SAW) is a sequence of distinct points \(\omega_0, \omega_1, \ldots, \omega_N\) on a lattice such that each point is a nearest neighbour of its predecessor. Let \(c_N\) be the number of \(N\)-step SAWs starting at the origin and ending anywhere, and \(c_N(x)\) – the number of those ones which end at \(x\). Then \(c_N\) and \(c_N(x)\) are believed to have the asymptotic behaviour \(c_N \sim \text{const} \times \mu^N N^{\gamma-1}\), \(c_N(x) \sim \text{const} \times \mu^N N^{\alpha-2}\) as \(N \to \infty\), where \(\mu\) is called the connective constant of the lattice, \(\gamma\) and \(\alpha\) are the critical exponents. Several measures of the size of an \(N\)-step SAW are considered, namely, the squared end-to-end distance \[ R_e^2 = \omega_N^2, \] the squared radius of gyration, \[ R_g^2 = \frac{1}{2(N+1)^2} \sum_{i,j=0}^N ( \omega_i-\omega_j )^2 , \] and the mean-square distance of a monomer from the endpoints \[ R_m^2 = \frac{1}{2(N+1)} \sum_{i=0}^N [\omega_i^2 + ( \omega_i-\omega_N)^2 ]. \] The corresponding mean values are believed to have the leading asymptotic behaviour \[ \langle R_e^2 \rangle_N, \langle R_g^2 \rangle_N, \langle R_m^2 \rangle_N \sim \text{const} \times N^{2 \nu} \] as \(N \to \infty\), where \(\nu\) is another (universal) critical exponent. Hyperscaling predicts that \(d \nu = 2 - \alpha\) holds for \(d\)-dimensional lattice. Exact enumeration of all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice are performed, and the above mean quantities are measured. Self-avoiding walks on square lattice are generated also by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to \(\approx0.01\).
Established theoretical values of the exponents for SAW in two dimensions are \(\nu=3/4\), \(\alpha=1/2\), and \(\gamma=43/32\). The paper addresses the problem of determining the leading non-analytic correction-to-scaling exponent \(\Delta_1\) for the two-dimensional self-avoiding walk and for the closely related problem of self-avoiding polygons. There are at least two theoretical predictions: \(\Delta_1 = 3/2\) based on Coulomb-gas arguments, and \(\Delta_1=11/16\) based on conformal invariance methods.
Based on the obtained data (coming both from exact enumerations and Monte Carlo), numerical evidences are provided that the first non-analytic correction term for two-dimensional self-avoiding walks is \(\Delta_1=3/2\). Monte Carlo analysis points also to a possible correction with \(\Delta_1 \approx 0.9\). The latter value, however, seems unlikely (after all, it does not agree with any theoretical prediction) and can be explained as an effective exponent arising from competition between two correction terms of opposite sign. The conclusion from exact enumeration data that \(\Delta_1=3/2\) looks convincing, although the method of analysis does not rule out a possibility that there is a correction term with, e.g., \(\Delta_1=1/2\) and an amplitude, which is by several orders of magnitude smaller than the amplitudes of other terms. The analysis assumes that the latter situation does not take place, since it is a priori unlikely.