When Probability Meets Complex Variables: Random Walks and Path Integrals
In the previous article we already saw that the probability distribution of a random walk is normal, and in probability theory one learns that the normal distribution is (almost) the most important distribution of all. Both random walk models and normal distributions have extremely wide applications, so we might ask: did random walks give rise to the normal distribution, or did the normal distribution give rise to random walks? In other words, which is more fundamental? Based on what I've read so far, I believe the random walk is the more fundamental of the two—random walks correspond precisely to the ubiquitous randomness and uncertainty found everywhere (such as the error in every measurement), and their distribution happens to be normal. That's why the normal distribution finds such widespread application: random uncertainty is everywhere.
Below we consider another way of describing random walks, one that is, in principle, more general and more profound—it goes by the grand name of "path integral." more
A Brief Introduction to Path Integrals
When talking about path integrals, we cannot avoid talking about Feynman. The originator of the path integral was none other than Feynman, the genius physicist I admire so much. At the time, the path integral emerged as a third way of formulating quantum mechanics, equivalent to the other two. Compared with Heisenberg's matrix mechanics and Schrödinger's wave mechanics, Feynman's path integral is mathematically the most cumbersome, but its underlying idea is the easiest to understand, and it has such generality that it has become the mainstream formulation of modern quantum field theory. This further illustrates that mathematical complexity is not where the real difficulty of a problem lies—if the idea is simple and the logic is clear, then no matter how tedious the calculations are, they remain, in essence, simple. In the words of physicists: without mathematical details, we can still get a lot done; but without physical insight, we can do nothing at all.
The idea behind the path integral really is very simple. What we normally consider is the probability density—that is, the probability of something happening within some small region. The path integral instead considers the probability of something happening along a particular path (in quantum mechanics, this is the probability amplitude), and then, using some suitable integration measure, sums up the probabilities over all possible paths to obtain the total probability.
That's really all there is to the idea of the path integral, but making it mathematically rigorous requires a good deal of additional work, and Scientific Spaces will have a dedicated series on this topic in the future. In the meantime, Feynman's Quantum Mechanics and Path Integrals is an excellent reference book (arguably the best one there is).
The Path Integral of a Random Walk
From the results above, we know that in a random walk, starting from $x_0$, the probability density of arriving at $x_n$ after time $t$ is
$$\frac{1}{\sqrt{2\pi \alpha t}}\exp\left(-\frac{(x_n-x_0)^2}{2\alpha t}\right)$$
Let's divide the time into $n$ equal parts, each of length $\Delta t=\frac{t}{n}$. At time $i\Delta t$, the particle's position is $x_i$. The probability density for the particle to go from $x_i$ to $x_{i+1}$ is
$$\frac{1}{\sqrt{2\pi \alpha \Delta t}}\exp\left(-\frac{(x_{i+1}-x_i)^2}{2\alpha \Delta t}\right)$$
So the probability density for the particle to pass through $x_1,x_2,\dots,x_{n-1},x_n$ in sequence is
$$\left(\frac{1}{\sqrt{2\pi \alpha \Delta t}}\right)^n\exp\left(-\frac{(x_1-x_0)^2+(x_2-x_1)^2+\dots+(x_n-x_{n-1})^2}{2\alpha \Delta t}\right)$$
Dropping the leading factor and then taking the limit $\Delta t\to 0$, we find that the probability of the particle traveling along the path $x=x(t)$ is proportional to
$$\exp\left(-\int\frac{\dot{x}^2}{2\alpha}dt\right)$$
This gives us the probability associated with a particular path. Using the method of path integrals from quantum mechanics, we can proceed from here and, working backwards, recover the probability density
$$\frac{1}{\sqrt{2\pi \alpha t}}\exp\left(-\frac{(x_n-x_0)^2}{2\alpha t}\right)=\int \exp\left(-\int\frac{\dot{x}^2}{2\alpha}dt\right)\mathcal{D}x(t)$$
This is the essential idea behind the path-integral treatment of random walks. As for the specific mathematical details, we will discuss those another time.
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.