Square lattice self-avoiding walks and biased differential approximants

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≥slant 2 the analysis yields a very accurate estimate for the critical exponent γ =1.343 7500(3) thus confirming the conjectured exact value γ =43/32 to eight 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 ν =0.750 0002(4) thus confirming the exact value ν =3/4 to seven significant digits.

Dedicated to Tony Guttmann on the occasion of his 70th birthday.