A striking feature of random walks is that the spatial dimension , little d, plays a crucial role. We normally think of random walks as moving about in one dimension, two dimensions, or three dimensions, but it behooves us to extend ourselves and think about a random walk in arbitrary spatial dimension d, and as we'll see as a function of d there is a transition in a fundamental property of a random walk. So let's think of a typical Pearson random walk and now let me draw in a continuum formulation, so it's some continuous trajectory that that starts somewhere and moves about in a random walk fashion, and let it go until time t. And as we've learned, the RMS displacement scales at the square root of t, so there's a characteristic distance that grows at the square root of time. So, we can define an exploration sphere as the range over which a typical random walk can be expected to extend up to time t. Lets now calculate the density of points visited inside this exploration sphere. So the density of visited sites So let me call that quantity Ρ. So what is this density of visited sites? It's the number of visited sites, divided by the volume of the exploration sphere. So in a time t, our random walker will visit t sites, and then we have to divide by the volume of the exploration sphere which is its radius, which is square root of t to the power, dimension of space, times some... ...constant numerical factors that're irrelevant for this consideration. the crucial point is that this quantity has a interesting dependence on time. It is t to the power of 1-d/2 So this density of visited sites has three fundamental behaviors that depend on the spacial dimension. So if I look in summary at how P depends on time, there are three answers. For d > 2 this density will go to 0, because for d > 2 this is a negative exponent so in the infinite time limit, this density goes to 0. So I should maybe write → here for approaching in limit of t going to infinity. So this happens for d > 2. For d < 2 on the other hand, then this is a positive exponent and so this density of visited sites is approaching inifinity. In d=2 this exponent is equal to 0 and so it would suggest that the density approach is a constant for d = 2. This change in behavior between infinite density and zero density is also what's known as a transition between transience and recurrence. So for d < 2, this regime is what's called "recurrent behavior". And what is meant by recurrent behavior is that because the density of points is infinite, it means a random walker as it's moving about will visit every single site. So in particular, if you start at some site, you are guaranteed that you must return eventually to that site. So the recurrent regime is also the regime where the return of the random walker is certain. Conversely, for d > 2, this is what's called the "transient regime". and in the transient regime, if you launch a random walker from a given point, because the density of points is going to 0, it's not certain that it's going to return to its starting point. So here, return is uncertain. The transition at d = 2 is actually lying in the recurrent regime, because it turns out that this constant behavior is the result of this very crude, is a flaw of this very crude line of reasoning, and it turns out that the correct behavior of the density is that it goes like log t in two dimensions. So the fundamental result of this discussion here is that for d ≤ 2, a random walk is recurrent and return is certain. For d > 2, a random walk is transient, and its return is uncertain. One important point to emphasize is that even though the return is certain for d ≤ 2, as we will learn at the end of this tutorial, the mean time to return to the origin is actually infinite. This dichotomy between certain return and infinite return time is what makes random walks so intriguing.