At the upcoming meeting of the Australian Mathematical Society, Prof. Jonathan Borwein will give a plenary talk on the mathematics of uniform random walks. This is in addition to the public lecture
The Life of Pi.
Abstract:
Following Pearson in 1905, we study the expected distance of a two-dimensional walk in the plane with n unit steps in random directions — what Pearson called a random walk or a “ramble”. While the statistics and large n behaviour are well understood, the precise behaviour of the first few steps is quite remarkable and less tractable. Series evaluations and recursions are obtained making it possible to explicitly determine this distance for small number of steps. Hypergeometric and elliptic hyper-closed-form expressions are given for the densities and all the moments of a 2, 3 or 4-step walk. Heavy use is made of analytic continuation of the integral and also of modern special functions and computer algebra systems.