Exact shapes of random walks in two dimensions

Since the random walk problem was first presented by Pearson in 1905, the shape of a walk which is either completely random or self-avoiding has attracted the attention of generations of researchers working in such diverse fields as chemistry, physics, biology and statistics. Among many advances in the field made in the past decade is the formulation of the three-dimensional shape distribution function of a random walk as a triple Fourier integral plus its numerical evaluation and graphical illustration. However, exact calculations of the averaged individual principal components of the shape tensor for a walk of a certain architectural type including an open walk have remained a challenge. Here we provide an exact analytical approach to the shapes of arbitrary random walks in two dimensions. Especially, we find that an end-looped random walk surprisingly has an even larger shape asymmetry than an open walk