The d > 3

Informally, a random walk is a path that is created by some stochastic process. At each time unit, a walker flips
Random walk – the stochastic process formed by successive summation of independent, identically distributed random variables – is one of the most basic and well-studied topics in probability theory. Execute a random walk from the origin on the integer lattice, but bias the four compass-direction probabilities from $\frac{1}{4}$ each to prefer to step in a spiraling direction. At each time step, a random walker makes a random move of length one in one of the lattice directions. As a simple example, consider a person standing on the integer line who ips a coin and moves one unit to the right if it lands on heads, and one unit to the left if it lands on tails. Proof: I will prove this theorem for the d = 1,2 recur-rent cases and the d = 3 transient case. 1 Simple Random Walk We consider one of the basic models for random walk, simple random walk on the integer lattice Zd.

That is, for d = 1,2 it is “certain” to return to the ori-gin, but for d ≥ 3 it is not. The path that is created by the random movements of the walker is a random walk. 1.1 One dimension We start by studying simple random walk on the integers. A simple random walk on the d-dimensional lattice Zd is recurrent for d = 1 and d = 2, but is transient for d ≥ 3.
For random walks on the integer lattice Zd, the main reference is the classic book by Spitzer [16].