36 1. Random Walk and Discrete Heat Equation
If we choose y {1,...,N 1} and find constants c1,...,cN−1 such
that
N−1
j=1
cj sinh(βj) sin
jπk
N
= δ(y k),
then
H(N,y)(x) = HA,(N,y)(N, x)
N−1
j=1
cj hj(x);
but we have already seen that the correct choice is
cj =
2
(N 1) sinh(βj)
sin
jπy
N
.
Therefore,
(1.25) H(N,y)(x1,x2)
=
2
N 1
N−1
j=1
1
sinh(βj)
sin
jπy
N
sinh
βj x1
N
sin
jπx2
N
.
The formula (1.25) is somewhat complicated, but there are some
nice things that can be proved using this formula. Let AN denote the
square and let
ˆ
A
N
= (x1,x2) A :
N
4
xj
3N
4
.
Note that
ˆ
A
N
is a cube of (about) half the side length of AN in the
middle of AN . Let y {1,...,N −1} and consider H(N,y). In Exercise
1.13 you are asked to show the following: there exist c, c1 such
that the following is true for every N and every y and every x, ˜ x
ˆ
A
N
:

(1.26)
c−1
N
−1
sin(πy/N) HN,y(x) c N
−1
sin(πy/N).
In particular,
(1.27) HN,y(x)
c2
HN,y(˜). x

(1.28) |HN,y(x) HN,y(˜)| x
c1
|x ˜| x
N
Hn,y(x) c1 c
|x ˜| x
N
N
−1
sin(πy/N).
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