The Smart Kinetic Self-Avoiding Walk and Schramm-Loewner Evolution

TL;DR

This paper investigates the scaling limits of the smart kinetic self-avoiding walk (SKSAW) on various lattices, providing evidence that it converges to Schramm-Loewner Evolution with kappa=6 (SLE_6), especially through simulations on the square lattice.

Contribution

It extends the understanding of SKSAW by conjecturing its universal convergence to SLE_6 across all regular lattices and supports this with simulation results on the square lattice.

Findings

Simulation results on the square lattice agree with SLE_6 predictions.

Conjecture that all variants of SKSAW converge to SLE_6.

Established equivalence of SKSAW and critical percolation interfaces on the hexagonal lattice.

Abstract

The smart kinetic self-avoiding walk (SKSAW) is a random walk which never intersects itself and grows forever when run in the full-plane. At each time step the walk chooses the next step uniformly from among the allowable nearest neighbors of the current endpoint of the walk. In the full-plane a nearest neighbor is allowable if it has not been visited before and there is a path from the nearest neighbor to infinity through sites that have not been visited before. It is well known that on the hexagonal lattice the SKSAW in a bounded domain between two boundary points is equivalent to an interface in critical percolation, and hence its scaling limit is the chordal Schramm-Loewner evolution with kappa=6 (SLE_6). Like SLE there are variants of the SKSAW depending on the domain and the initial and terminal points. On the hexagonal lattice these variants have been shown to converge to the corresponding version of SLE_6. We conjecture that the scaling limit of all these variants on any regular lattice is the corresponding version of SLE_6. We test this conjecture for the square lattice by simulating the SKSAW in the full-plane and find excellent agreement with the predictions of full-plane SLE_6.