Variations on the basic theme include self-avoiding random walks, important in the theory of polymer folding, and correlated random walks, important in describing the motion of atoms through condensed matter.
Random walks are important in a number of biological applications as well.
From a mathematical standpoint, there is a strong connection between the correlated random walk in one dimension and the behavior of systems of interacting spins, stochastic neural networks, and stochastic cellular automata. Recently, my collaborators and I have obtained a number of new exact results for the correlated random walk in one dimension. A fair amount of work now needs to be done on extending the results to two and three dimensions and applying the results to atomic motion in real materials. A second area of research interest involves electrochemical systems. To interpret the results of electrical measurements on electrochemical systems and to design electrochemical cells and sensors for a wide range of applications, we must understand how the coupling of chemical reactions and transport processes occurring in such systems is reflected in their overall electrical characteristics.