1.1. Simple random walk 11
1.1.5. Several dimensions. We now consider a random walker on
the d-dimensional integer grid
Zd
= {(x1,...,xd) : xj integers} .
At each time step, the random walker chooses one of its 2d nearest
neighbors, each with probability 1/2d, and moves to that site. Again,
we let
Sn = x + X1 + · · · + Xn
denote the position of the particle. Here x, X1,...,Xn,Sn represent
points in
Zd,
i.e., they are d-dimensional vectors with integer compo-
nents. The increments X1,X2,... are unit vectors with one compo-
nent of absolute value 1. Note that Xj
· Xj = 1 and if j = k, then
Xj · Xk equals 1 with probability 1/(2d); equals −1 with probability
1/(2d); and otherwise equals zero. In particular, E[Xj · Xj] = 1 and
E[Xj · Xk] = 0 if j = k. Suppose S0 = 0. Then E[Sn] = 0, and a
calculation as in the one-dimensional case gives
E[|Sn|2]
= E[Sn · Sn] = E
⎡⎛
⎣⎝
n
j=1
Xj⎠

·


n
j=1
Xj⎠⎦
⎞⎤
= n.
o
Figure 2. The integer lattice Z2
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