One-dimensional pinning-depinning transitions

Abstract

I present in this chapter two results on one-dimensional random walks interacting with a layer. This layer, for the random walk, is just the path identical to zero. The interaction presented in the two sections are slightly different, but the effects are quite similar. The first section discusses what in physics literature is called a wetting transition. Here the layer is acting as a hard wall in the sense that the random walk has to stay on one side, but there is also an attractive interaction between the random walk and the wall. There is a considerable literature around such wetting-transitions, an I present here only the very most simple case where such a transition occurs. Of considerable interest are cases where the random walk is replaced by a random surface, and then, of course, similar questions for more complicated random interfaces, like interfaces in Ising type models, but I leave this out.