Self-avoiding walks on deterministic and random fractals: Numerical results

Abstract

This chapter introduces two major classes of numerical methods used to study self-avoiding walks (SAWs) on deterministic and random fractals, namely, Monte Carlo (MC) methods and the exact enumeration (EE) technique. The advantages and disadvantages of both approaches are highlighted in connection to the different substrates and the statistical quantities of interest. Self-avoiding walks (SAWs) constitute the simplest, yet non-trivial model for studying the static behavior of a linear polymer embedded in a good solvent. Many MC methods have been developed for studying SAWs on regular lattices. These diverse algorithms can be classified according to static, quasistatic, and dynamic MC methods. The exact enumeration (EE) technique allows enumeration and evaluation of all SAW configurations on a given substrate. The chapter also discusses SAWs on three different substrates and compares them with the results obtained for SAWs on regular lattices. The first two substrates, Sierpinski triangular and square lattices, are representative members of the class of deterministic fractals, whereas percolation is the standard model for random fractals. Deterministic fractals consist of two distinct subgroups: the finitely (Sierpinski triangular) and infinitely (Sierpinski square) ramified fractals. The behavior of SAWs is drastically affected by the underlying substrate but certain similarities emerge: SAWs on Sierpinski triangular and square lattices display a kind of intermediate behavior between SAWs on the corresponding regular lattices and on percolation. The spatial characteristics of SAWs on Sierpinski square lattices are closer to those observed for regular lattices, while SAWs on Sierpinski triangular lattices are more similar to their counterparts on percolation. Scaling forms known for SAWs on regular lattices seem to remain valid also on deterministic and random fractals, while the corresponding relations between the scaling exponents need to be modified in some cases.