A kinetic view of nonlocal self-avoiding processes

Abstract

Self-avoiding walks (SAWs) are central to modeling excluded-volume effects in polymers and related systems. SAWs are typically defined with local, nearest-neighbor steps, and their statistical behavior is well studied. In contrast, less is known about SAWs involving nonlocal motion. We study how introducing nonlocality — in the form of fixed-length jumps inspired by the knight’s move in chess — affects kinetic self-avoiding walks (GSAWs), where paths grow irreversibly. The resulting model, called the Self-Avoiding Random Knight (SARK), replaces local propagation with constrained long-range steps. Using large-scale simulations, we examine how this dynamic influences key properties of the walk. We find that nonlocality increases the walker’s lifetime and spatial extent. The end-to-end distance shows a crossover in scaling behavior, approaching that of the GSAW at long times. Clustering analysis reveals a dominant connected component in small lattices, which vanishes in larger ones. These results offer insight into how nonlocal constraints shape the geometry and growth of SAWs, with possible applications in ecological foraging and transport in constrained environments.