Self-avoiding walks and polygons on quasiperiodic tilings

TL;DR

This paper extends enumeration of self-avoiding walks and polygons on quasiperiodic tilings, showing they likely share universality class with regular lattices, and discusses methods to improve asymptotic estimates.

Contribution

It provides new enumeration data for self-avoiding walks and polygons on Penrose and Ammann-Beenker tilings, and introduces averaging techniques to enhance convergence of critical parameters.

Findings

Enumeration results support universality class similarity with square lattice

Averaging methods improve convergence to asymptotic behavior

Polygon data insufficient for definitive exponent conclusions

Abstract

We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen start vertex. We compare different ways of counting and demonstrate that suitable averaging improves converge to the asymptotic regime. This leads to improved estimates for critical points and exponents, which support the conjecture that self-avoiding walks on quasiperiodic tilings belong to the same universality class as self-avoiding walks on the square lattice. For polygons, the obtained e numeration data does not allow to draw decisive conclusions about the exponent.