symmetric random walk

A symmetric random walk is a mathematical process where a point moves in a space, taking steps of equal length. At each step, the direction of movement is random, with an equal probability of moving in any of the possible directions. For example, in one dimension, this is like a coin toss determining whether the point moves left or right.12

In higher dimensions, the particle moves to one of its nearest neighbors with equal probability. While in one or two dimensions, the particle is guaranteed to return to its starting point infinitely often, this is not the case in three or more dimensions. Even in lower dimensions, the expected time to return to the origin is infinite. This concept is related to Brownian motion, which models the irregular movement of particles.23