390 12 Random walks and the Metropolis algorithm
wi(tn) =∑
jWi j(tn)wj( 0 ),and defining
W(il−jl,nε) = (Wn(ε))i j
we obtain
wi(nε) =∑
j
(Wn(ε))i jwj( 0 ),or in matrix form
wˆ(nε) =Wˆn(ε)wˆ( 0 ). (12.4)
The matrixWˆ can be written in terms of two matrices
Wˆ=^1
2(ˆ
L+Rˆ
)
,
whereLˆandRˆrepresent the transition probabilities for a jump to the left or the right, respec-
tively. For a 4 × 4 case we could write these matrices as
Rˆ=
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
,
and
Lˆ=
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
.
However, in principle these are infinite dimensional matrices since the number of time steps
are very large or infinite. For the infinite case we can write these matricesRi j=δi,(j+ 1 )and
Li j=δ(i+ 1 ),j, implying that
LˆRˆ=RˆLˆ=I, (12.5)
which applies in the case of infinite matrices and
Lˆ=Rˆ−^1 (12.6)To see thatLˆRˆ=RˆLˆ= 1 , perform e.g., the matrix multiplication
LˆRˆ=∑
kLˆikRˆk j=∑
kδ(i+ 1 ),kδk,(j+ 1 )=δi+ 1 ,j+ 1 =δi,j,and only the diagonal matrix elements are different from zero.
For the first time step we have thus
Wˆ=^1
2(ˆ
L+Rˆ
)
,
and using the properties in Eqs. (12.5) and (12.6) we have after two time steps
Wˆ^2 ( 2 ε) =^1
4(ˆ
L^2 +Rˆ^2 + 2 RˆLˆ
)
,
and similarly after three time steps
Wˆ^3 ( 3 ε) =^1
8