Twodimensional interacting selfavoiding walks: new estimates for critical temperatures and exponents
Abstract
We investigate, by series methods, the behaviour of interacting selfavoiding walks (ISAWs) on the honeycomb lattice and on the square lattice. This is the first such investigation of ISAWs on the honeycomb lattice. We have generated data for ISAWs up to 75 steps on this lattice, and 55 steps on the square lattice. For the hexagonal lattice we find the θpoint to be at u_{c} = 2.767 ± 0.002. The honeycomb lattice is unique among the regular twodimensional lattices in that the exact growth constant is known for noninteracting walks, and is $\sqrt{2+\sqrt{2}}$ (DuminilCopin H and Smirnov S 2014 Ann. Math. 175 165365), while for halfplane walks interacting with a surface, the critical fugacity, again for the honeycomb lattice, is $1+\sqrt{2}$ (Beaton N R et al 2014 Commun. Math. Phys. 326 72754). We could not help but notice that $\sqrt{2+4\sqrt{2}}=2.767\dots .$ We discuss the difficulties of trying to prove, or disprove, this possibility. For square lattice ISAWs we find u_{c} = 1.9474 ± 0.001, which is consistent with the best Monte Carlo analysis. We also study bridges and terminallyattached walks (TAWs) on the square lattice at the θpoint. We estimate the exponents to be γ_{b} = 0.00 ± 0.03, and γ_{1} = 0.55 ± 0.03 respectively. The latter result is consistent with the prediction (Duplantier B and Saleur H 1987 Phys. Rev. Lett. 59 53942; Seno F and Stella A L 1988 Europhys. Lett. 7 60510; Stella A L et al 1993 J. Stat. Phys. 73 2146) ${\gamma }_{1}\left(\theta \right)=\nu =\frac{4}{7}$ , albeit for a modified version of the problem, while the former estimate is predicted in [Duplantier B and Guttmann A J 2019 Statistical mechanics of confined polymer networks (in preparation)] to be zero. *All series data generated for this paper are available with the preprint at arXiv:1911.05852.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 April 2020
 DOI:
 10.1088/17518121/ab7ad1
 arXiv:
 arXiv:1911.05852
 Bibcode:
 2020JPhA...53p5002B
 Keywords:

 selfavoiding walks;
 polymers;
 thetapoint;
 critical exponents;
 series analysis;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 82B41;
 82B26;
 82B27
 EPrint:
 21 pages, 9 figures (v3 with series data included as ancillary files)