1.6. Exercises 47

the union of T1 and T2 and adding one more connection: we say that

o and ˜ o are adjacent.

• Convince yourself that T is a connected tree, i.e., between

any two points of T there is a unique path in the tree that

does not go through any point more than once.

• Let Sn denote simple random walk on the tree, i.e., the

process that at each step chooses one of the three nearest

neighbors at random, each with probability 1/3, with the

choice being independent of all the previous moves. Show

that Sn is transient, i.e., with probability one Sn visits the

origin only finitely often. (Hint: Exercise 1.18 could be

helpful.)

• Show that with probability one the random walk does one

of the two following things: either the random walk visits

T1 only finitely often or it visits T2 only finitely often. Let

f(x) be the probability that the walk visits T1 finitely often.

Show that f is a nonconstant bounded harmonic function.

(A function f on T is harmonic if for every x ∈ T , f(x)

equals the average value of f on the nearest neighbors of x.)

• Consider the space of bounded harmonic functions on T .

Show that this is an infinite dimensional vector space.

Exercise 1.27. Show that if A ⊂ A1 are two subsets of

Zd,

then

λA ≤ λA1 . Show that if A1 is connected and A = A1, then λA λA1 .

Give an example with A1 disconnected and A a strict subset of A1

for which λA = λA1 .

Exercise 1.28. Consider β(j, N),j = 1,...,N − 1 where β(j, N) is

the unique positive number satisfying

cosh

β(j, N)

N

+ cos

jπ

N

= 2.