A popular random walk model is that of a random walk on a regular lattice, where at each step the walk jumps to another site according to some probability distribution. In simple random walk, the walk can only jump to neighbouring sites of the lattice. In simple symmetric random walk on a locally finite lattice, the probabilities of walk jumping to any one of its neighbours are the same. The most well-studied example is of random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice) .
One-dimensional random walk
Imagine a one-dimensional length of something, a 'line'. Now imagine this line has numbers on it, spaced apart equally. A particularly elementary and concrete random walk is the random walk on the integer number line, Z , which starts at So=0 and at each step moves by ±1 with equal probability. To define this walk formally, take independent random variables Z1, Z2..., where each variable is either 1 or −1, with a 50% probability for either value, and set