Suppose there is a horizontal straight line, starting from a certain position, there is a 50% probability of going left 1 m and a 50% probability of going right 1 m. What is the probability that you will eventually return to the starting point if you wander indefinitely like this? The answer is 100%. In the process of one-dimensional random walk, as long as the time is long enough, you can always return to the starting point.
Now consider a drunk walking randomly in the street. Assuming that the streets of the whole city are distributed in a grid, every time an alcoholic walks to a crossroads, he will wait for the probability to choose a road to continue.
So what are the chances that he can finally return to the starting point? The answer is still 100%. At first, the drunk may go further and further, but in the end he can always find his way home.
However, drunken birds are not so lucky. If a bird chooses a direction equally from up, down, left, right, front and back every time it flies, it will probably never return to its starting point. In fact, the probability of walking randomly in a three-dimensional grid and finally returning to the starting point is only about 34%.
This theorem was proved by the famous mathematician Paulia in 192 1. With the increase of dimensions, the probability of returning to the starting point will be lower and lower. In the four-dimensional grid, the probability of returning to the starting point is 19.3%I, while in the eight-dimensional space, the probability is only 7.3%.