Labyrinthine patterns in a self-avoiding growing string

We propose a model of a self-avoiding growing string with excluded volume effects. In this model, the string develops to form a complex shape like a labyrinth. We find simple dynamical scaling relations for several characteristics quantities, including the radius of gyration and radius of curvature. Detailed numerical calculations show that there are two length scales and one time scale characterizing the shape transformations of the growing string.