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).