Random walks in one dimension : new results and applications

Abstract

The random walk is one of the paradigms of statistical physics, but it is also a very practical concept that has found applications in a wide variety of fields. Since the gambler's ruin problem can be formulated as a random walk problem (how many time steps does it take before the gambler exhausts his capital), one can argue that the concept of a random walk goes back as far as the seventeenth century, when the vivid interest in gambling led to the formulation of probability theory. However, we have to wait until the beginning of the twentieth century for the first explicit formulation of a random walk problem. ... Random walk theory continues to be, even today, a field of very active and exciting research. Our purpose here will be to add a few new technical results, in particular for the case of one-dimensional random walks, and to add some further applications to the long list of existing ones. ...