A short walk can be beautiful

The story I tell is of research undertaken, with students and colleagues, in the last six or so years on short random walks. As the research progressed, my criteria for `beauty' changed. Things seemingly remarkable became `simple' and other seemingly simple things became more remarkable as our analytic and computational tools were refined, and understanding improved. I intentionally display some rather advanced mathematics as it is my contention -- as with classical music -- that one can learn to appreciate and enjoy complex formulas without needing to understand them deeply.