skip to main content

DOE PAGESDOE PAGES

Title: On the growth constant for square-lattice self-avoiding walks

The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular lattices have been sought. For the square lattice, a possible conjecture was advanced by one of us (AJG) more than 20 years ago, based on the six significant digit estimate available at the time. This estimate has improved by a further six digits over the intervening decades, and the conjectured value continued to agree with the increasingly precise estimates. Here, we discuss the three most successful methods for estimating the growth constant, including the most recently developed topological transfer-matrix method, due to another of us (JLJ). We show this to be the most computationally efficient of the three methods, and by parallelising the algorithm we have estimated the growth constant significantly more precisely, incidentally ruling out the conjecture, which fails in the 12th digit. Our new estimate of the growth constant is μ(square) = 2.63815853032790 (3).
Authors:
ORCiD logo [1] ;  [2] ;  [3]
  1. Normal Superior School (ENS)-PSL Research Univ., Paris (France). Lab. for Theoretical Physics (LPTENS); Sorbonne Univ., CNRS, Paris (France); Atomic Energy Commission (CEA), Saclay (France). Inst. of Theoretical Physics
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  3. Univ. of Melbourne (Australia). ARC Center of Excellence for Mathematics and Statistics of Complex Systems and Dept. of Mathematics and Statistics
Publication Date:
Report Number(s):
LLNL-JRNL-747715
Journal ID: ISSN 1751-8113; 932555
Grant/Contract Number:
AC52-07NA27344; NuQFT; DP120100939
Type:
Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 49; Journal Issue: 49; Journal ID: ISSN 1751-8113
Publisher:
IOP Publishing
Research Org:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org:
USDOE National Nuclear Security Administration (NNSA); Univ. Inst. of France (IUF); European Research Council (ERC); Australian Research Council (ARC)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
OSTI Identifier:
1459139
Alternate Identifier(s):
OSTI ID: 1332598