The Laplacian Random Walk

Abstract

We show how to choose a stepping probability for an irreversible random walk such that only strictly self-avoiding configurations are generated. This probability is linked to the solution of the Laplace equation with appropriate boundary conditions and corresponds to the nonbranching version of the dielectric breakdown model. We have performed an exact enumeration for this so-called Laplacian random walk (LRW) on a square lattice (N ⩽ 18). We find a smooth variation of the exponent ν as a function of the parameter η. For η = 0 this walk corresponds to the IGSAW. A self-consistent scaling theory is formulated for η = 0 that predicts an upper critical dimensionality dc = 3.