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

Understand

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.
• 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).