Square lattice self-avoiding walks and biased differential approximants

Abstract

The model of self-avoiding lattice walks and the asymptotic analysis of power-series have been two of the major research themes of Tony Guttmann. In this paper we bring the two together and perform a new analysis of the generating functions for the number of square lattice self-avoiding walks and some of their metric properties such as the mean-square end-to-end distance. The critical point $x_c$ for self-avoiding walks is known to a high degree of accuracy and we utilise this knowledge to undertake a new numerical analysis of the series using biased differential approximants. The new method is major advance in asymptotic power-series analysis in that it allows us to bias differential approximants to have a singularity of order $q$ at $x_c$. When biasing at $x_c$ with $q\geq 2$ the analysis yields a very accurate estimate for the critical exponent $\gamma=1.3437500(3)$ thus confirming the conjectured exact value $\gamma=43/32$ to 8 significant digits and removing a long-standing minor discrepancy between exact and numerical results. The analysis of the mean-square end-to-end distance yields $\nu=0.7500002(4)$ thus confirming the exact value $\nu=3/4$ to 7 significant digits.