Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices

@article{Guttmann2022SelfavoidingWA,
  title={Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices},
  author={Anthony J. Guttmann and Iwan Jensen},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2022},
  volume={55}
}
  • A. GuttmannI. Jensen
  • Published 13 August 2022
  • Mathematics
  • Journal of Physics A: Mathematical and Theoretical
We have analysed the recently extended series for the number of self-avoiding walks (SAWs) CL(1) that cross an L × L square between diagonally opposed corners. The number of such walks is known to grow as λSL2. We have made more precise the estimate of λS, based on additional series coefficients provided by several authors, and refined analysis techniques. We estimate that λS=1.7445498±0.0000012. We have also studied the subdominant behaviour, and conjecture that CL(1)∼λSL2+bL+c⋅Lg, where b=−0… 

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References

SHOWING 1-10 OF 24 REFERENCES

Self-avoiding walks crossing a square

We study a restricted class of self-avoiding walks (SAWs) which start at the origin (0, 0), end at (L, L), and are entirely contained in the square [0 ,L ] × [0 ,L ] on the square lattice Z 2 . The

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

Self-avoiding walks which cross a square

The authors consider self-avoiding walks on the square lattice which are confined to lie in or on the boundary of a square with vertices at (0, 0), (0, L), (L, 0) and (L, L). They ask for the number

CRITICAL BEHAVIOUR OF SELF-AVOIDING WALKS THAT CROSS A SQUARE

Consider the set of all self-avoiding walks in the square lattice which start at (0, 0), end at (L, L), and are entirely contained in the square (0, L)*(0, L). Associate a fugacity x with each step

A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice

A parallel algorithm is developed that allows us to enumerate the number of self-avoiding polygons on the square lattice to perimeter length 110 confirming to a high degree of accuracy various predictions for universal amplitude combinations.

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

Large Deviations and Ratio Limit Theorems for Pattern-Avoiding Permutations

It is shown that some neighbourhood of the upper left corner of [0,1]2 is exponentially unlikely to contain a point of the graph if and only if τ starts with its largest element.

Algebraic techniques for enumerating self-avoiding walks on the square lattice

The authors describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of N steps is of order 3N/4 times a

Honeycomb lattice polygons and walks as a test of series analysis techniques

We have calculated long series expansions for self-avoiding walks and polygons on the honeycomb lattice, including series for metric properties such as mean-squared radius of gyration as well as

Generating functions for enumerating self-avoiding rings on the square lattice

It is shown that generating function techniques provide an efficient means of enumerating the number of self-avoiding rings (polygons) on the square lattice. The techniques can be applied to a number