1. Introduction

A geometric random walk starts at some point in ℝn and at each step, moves to a “neighboring” point chosen according to some distribution that depends only on the current point, e.g., a uniform random point within a fixed distance δ.The sequence of points visited is a random walk. The distribution of the current point, in particular, its convergence to a steady state (or stationary) distribution, turns out to be a very interesting phenomenon. By choosing the one-step distribution appropriately, one can ensure that the steady state distribution is, for example, the uniform distribution over a convex body, or indeed any reasonable distribution in ℝ n.

Geometric random walks are Markov chains, and the study of the existence and uniqueness of and the convergence to a steady state distribution is a classical field of mathematics. In the geometric setting, the dependence on the dimension (called n in this survey) is of particular interest. Polya proved that with probability 1, a random walk on an n-dimensional grid returns to its starting point infinitely often for n ≤ 2, but only a finite number of times for n ≥ 3. Random walks also provide a general approach to sampling a geometric distribution. To sample a given distribution, we set up a random walk whose steady state is the desired distribution. A random (or nearly random) sample is obtained by taking sufficiently many steps of the walk. Basic problems such as optimization and volume computation can be reduced to sampling.