Two-dimensional persistent random walk with linearly increasing step size

Abstract

The mean position and mean squared displacement for the two-dimensional persistent random walk with linearly increasing step size are calculated. Exact solutions are obtained for both quantities. For some values of turning angle probability, sample calculations are performed and the quantities are plotted as functions of the total number of steps taken. In all instances studied, the mean position spirals towards a fixed point, showing convergence of mean position, while the mean squared displacement increases indefinitely, with a cubic dependence on the number of steps in the limit of infinite steps.